From 26cf14b679b1e4dc9cf281088c0f0306ec399ff5 Mon Sep 17 00:00:00 2001 From: "Documenter.jl" Date: Mon, 16 Oct 2023 13:01:21 +0000 Subject: [PATCH] build based on da56a6d --- stable | 2 +- v0.8 | 2 +- v0.8.2/.documenter-siteinfo.json | 1 + v0.8.2/BasicConstructors/index.html | 54 ++ v0.8.2/Library/index.html | 2 + v0.8.2/Multithreading/index.html | 2 + v0.8.2/SDEVertex/5474df40.svg | 48 + v0.8.2/SDEVertex/dbaa6d16.svg | 48 + v0.8.2/SDEVertex/index.html | 40 + v0.8.2/accessing_edge_variables/index.html | 15 + v0.8.2/assets/documenter.js | 889 ++++++++++++++++++ v0.8.2/assets/themes/documenter-dark.css | 7 + v0.8.2/assets/themes/documenter-light.css | 9 + v0.8.2/assets/themeswap.js | 84 ++ v0.8.2/assets/warner.js | 52 + .../directed_and_weighted_graphs/f688b770.svg | 132 +++ .../directed_and_weighted_graphs/index.html | 58 ++ .../2029525f.svg | 82 ++ .../9c5f74e2.svg | 62 ++ .../index.html | 42 + v0.8.2/heterogeneous_system/50fddf9e.svg | 62 ++ v0.8.2/heterogeneous_system/9b6a91ac.svg | 62 ++ v0.8.2/heterogeneous_system/index.html | 58 ++ v0.8.2/index.html | 3 + v0.8.2/kuramoto_delay/3508ce7e.svg | 58 ++ v0.8.2/kuramoto_delay/index.html | 47 + v0.8.2/parameters/index.html | 4 + v0.8.2/search_index.js | 3 + v0.8.2/siteinfo.js | 1 + versions.js | 2 +- 30 files changed, 1928 insertions(+), 3 deletions(-) create mode 100644 v0.8.2/.documenter-siteinfo.json create mode 100644 v0.8.2/BasicConstructors/index.html create mode 100644 v0.8.2/Library/index.html create mode 100644 v0.8.2/Multithreading/index.html create mode 100644 v0.8.2/SDEVertex/5474df40.svg create mode 100644 v0.8.2/SDEVertex/dbaa6d16.svg create mode 100644 v0.8.2/SDEVertex/index.html create mode 100644 v0.8.2/accessing_edge_variables/index.html create mode 100644 v0.8.2/assets/documenter.js create mode 100644 v0.8.2/assets/themes/documenter-dark.css create mode 100644 v0.8.2/assets/themes/documenter-light.css create mode 100644 v0.8.2/assets/themeswap.js create mode 100644 v0.8.2/assets/warner.js create mode 100644 v0.8.2/directed_and_weighted_graphs/f688b770.svg create mode 100644 v0.8.2/directed_and_weighted_graphs/index.html create mode 100644 v0.8.2/getting_started_with_network_dynamics/2029525f.svg create mode 100644 v0.8.2/getting_started_with_network_dynamics/9c5f74e2.svg create mode 100644 v0.8.2/getting_started_with_network_dynamics/index.html create mode 100644 v0.8.2/heterogeneous_system/50fddf9e.svg create mode 100644 v0.8.2/heterogeneous_system/9b6a91ac.svg create mode 100644 v0.8.2/heterogeneous_system/index.html create mode 100644 v0.8.2/index.html create mode 100644 v0.8.2/kuramoto_delay/3508ce7e.svg create mode 100644 v0.8.2/kuramoto_delay/index.html create mode 100644 v0.8.2/parameters/index.html create mode 100644 v0.8.2/search_index.js create mode 100644 v0.8.2/siteinfo.js diff --git a/stable b/stable index 349ca0e1..fe5b6a9d 120000 --- a/stable +++ b/stable @@ -1 +1 @@ -v0.8.1 \ No newline at end of file +v0.8.2 \ No newline at end of file diff --git a/v0.8 b/v0.8 index 349ca0e1..fe5b6a9d 120000 --- a/v0.8 +++ b/v0.8 @@ -1 +1 @@ -v0.8.1 \ No newline at end of file +v0.8.2 \ No newline at end of file diff --git a/v0.8.2/.documenter-siteinfo.json b/v0.8.2/.documenter-siteinfo.json new file mode 100644 index 00000000..eabe0aa2 --- /dev/null +++ b/v0.8.2/.documenter-siteinfo.json @@ -0,0 +1 @@ +{"documenter":{"julia_version":"1.6.7","generation_timestamp":"2023-10-16T13:01:12","documenter_version":"1.1.1"}} \ No newline at end of file diff --git a/v0.8.2/BasicConstructors/index.html b/v0.8.2/BasicConstructors/index.html new file mode 100644 index 00000000..79c7df0c --- /dev/null +++ b/v0.8.2/BasicConstructors/index.html @@ -0,0 +1,54 @@ + +Functions · NetworkDynamics
GitHub

Functions

The key network_dynamics assembles the dynamics of the whole network from functions for the single vertices and edges of the graph g. These functions have to be wrapped in types VertexFunction and EdgeFunction which store additional information on dimension and coupling type. Both have several subtypes that account for the different classes of equations that may represent the local dynamics. At the moment, algebraic (static) equations, ordinary differential equations (ODEs) and delay differential equations (DDEs) are supported:

# VertexFunctions
+StaticVertex(vertexfunction!, dimension, symbol)
+ODEVertex(vertexfunction!, dimension, mass_matrix, symbol)
+DDEVertex(vertexfunction!, dimension, mass_matrix, symbol)
+
+# EdgeFunctions
+StaticEdge(edgefunction!, dimension, coupling, symbol)
+ODEEdge(edgefunction!, dimension, mass_matrix, coupling, symbol)
+StaticDelayEdge(edgefunction!, dimension, mass_matrix, coupling, symbol)

VertexFunctions

Given a set of (algebraic or differential) equations describing a node or an edge the first step is to turn them into a mutating function vertexfunction!. Depending on the class of the function vertexfunction!, the constructors StaticVertex, ODEVertex or DDEVertex are called in order to turn vertexfunction! into a VertexFunction object compatible with network_dynamics.

Since in general the state of a vertex depends on the vertex value itself as well as on the in- and outgoing edges, the function vertexfunction! has to respect one of the following calling syntaxes.

# For static nodes
+function vertexfunction!(v, edges, p, t) end
+# For dynamic nodes
+function vertexfunction!(dv, v, edges, p, t) end
+# For delay nodes
+function vertexfunction!(dv, v, edges, h_v, p, t) end

Here dv, v, p, and t are the usual ODE arguments, while edges is an array containing the edges for which the vertex is the destination (corresponding to incoming edges for directed graphs). In the delay case, the array h_v denotes the vertex history. The typical case of diffusive coupling on a directed graph could be described as

function vertex!(dv, v, edges, p, t)
+    dv .= 0.0
+    for e in edges
+        dv .+= e
+    end
+    nothing
+end

StaticVertex

If a vertex is described by an algebraic equation vertexfunction!(v, edges, p, t), i.e. dv = 0 the VertexFunction is constructed as

StaticVertex(vertexfunction!, dim, sym)

Here, dim is the number of independent variables in the vertex equations and sym is an array of symbols for these variables. For example, if a node models a constant input $I = p$, then dim = 1 and sym = [:I]. For more details on the use of symbols, check out the Getting started tutorial and the Julia documentation. The use of symbols makes it easier to select variables of interest from the solution object.

ODEVertex

If a vertex has local dynamics vertexfunction!(dv, v, edges, p, t) described by an ODE, the VertexFunction is contructed as

ODEVertex(vertexfunction!, dim, mass_matrix, sym)

As above, dim is the number of independent variables in the vertex equations and sym corresponds to the symbols of these variables.

mass_matrix is an optional argument that defaults to the identity matrix I. If a mass matrix M is given, the local system M * dv = vertexfunction! will be solved. network_dynamics assembles all local mass matrices into one global mass matrix that can be passed to a differential equation solver like Rodas4.

One may also call ODEVertex with keyword arguments, omitting optional arguments:

ODEVertex(; f=vertexfunction!, dim=dim)

The function then defaults to using the identity as mass matrix and [:v for i in 1:dimension] as symbols.

DDEVertex

If a vertex has local dynamics described by a delay differential equation (DDE) the local dynamics need to have the signature vertexfunction!(dv, v, edges, h_v, p, t), where h_v is an automatically generated wrapper of the global history function, providing history values forv. Then the VertexFunction is constructed as

DDEVertex(vertexfunction!, dim, mass_matrix, sym)

As before, dim is the number of independent variables in the vertex function and sym describes the symbols of the variables. The mass matrix is optional and defaults to the identity matrix I.

As above, the DDEVertex function can also be called without the optional arguments:

DDEVertex(; f=vertexfunction!, dim=dim)

EdgeFunctions

Similar to the case of vertices, an edge is described by mutating function edgefunction!. At the moment the constructors StaticEdge, ODEEdge, and StaticDelayEdge are available. edgefunction! has to respect one of the following syntaxes:

# For static edges
+function edgefunction!(e, v_s, v_d, p, t) end
+# For dynamics edges
+function edgefunction!(de, e, v_s, v_d, p, t) end
+# For static delay edges
+function edgefunction!(e, v_s, v_d, h_v_s, h_v_d, p, t) end

Just like above, de, e, p and t are the usual ODE arguments, while v_s and v_d are the source and destination vertices respectively and h_v_s and h_v_d the corresponding history arrays.

StaticEdge

Static here means, that the edge value described by edgefunction! only depends on the values of the vertices the edge connects to and that no derivative of the edge's internal state is involved. One very simple and natural example is a diffusive edge:

edgefunction! = (e, v_s, v_d, p, t) -> e .= v_s .- v_d

In this case the EdgeFunction is constructed by

StaticEdge(edgefunction!, dim, coupling, sym)

The keywords are the same as for the vertices, except from the additional keyword coupling, which describes if the EdgeFunction is intended for a (:directed) or for an undirected graph ({:undirected, :symmetric, :antisymmetric, :fiducial}). :directed is intended for directed graphs. :undirected is the default option and is only compatible with SimpleGraph. In this case the edgefunction! should specify the coupling from a source vertex to a destination vertex. :symmetric and :antisymmetric trigger performance optimizations, if edgefunction! has that symmetry property. :fiducial lets the user specify both the coupling from source to destination, as well as the coupling from destination to source and is intended for advanced users.

ODEEdge

For problems where edgefunction! describes the differential of an edge value, we use the ODEEdge function. An example for such a system is given by:

edgefunction! = (de, e, v_s, v_d, p, t) -> de .= 1000 * (v_s .- v_d .- e)

The EdgeFunction object is constructed as

ODEEdge(edgefunction!, dim, coupling, mass_matrix, sym)

The keywords are the same as above for the StaticEdge. For ODEEdge the same simplified construction rules apply when keyword arguments are used, except that the coupling type has to be specified explicitly to avoid ambiguities. The available coupling tyes are only :directed, :undirected, :fiducial. For :undirected coupling the internal dimension of the EdgeFunction is doubled in order to resolve both directions in an undirected network. Remember to specify 2 initial conditiosn for every edge in that case.

ODEEdge(; f=edgefunction!, dim=n, coupling=:undirected)

In this case the function defaults to using the identity matrix as mass matrix and [:e for in 1:dimension] as symbols.

StaticDelayEdge

This constructor is used when edge variables depend on past values of the vertex variables. In this case the edgefunction! has to accept two additional arguments h_v_s and h_v_d that hold the (automatically generated) localised history functions of v_s and v_d. Static means that the edge depends only on the dynamics of the vertices the edge is connected to and not on an internal derivative of the edge variables itself.

As an example for such system, we show a diffusive coupling with delay:

edgefunction! = (e, v_s, v_d, h_v_s, h_v_d, p, t) -> e .= 0.1 * (h_v_s(t - p) .- v_d)

The EdgeFunction object is constructed as

StaticDelayEdge(edgefunction!, dim, coupling, mass_matrix, sym)

Like for a StaticEdge, we can also leave out the optional keywords coupling, sym and mass_matrix. Available coupling types are :unspecified, :directed, :undirected, :fiducial.

StaticDelayEdge(; f=edgefunction!, dim=n)

Constructor

The central constructor is the function network_dynamics that takes in two arrays of EdgeFunctions and VertexFunctions describing the local dynamics on the edges and nodes of a graph g, given as a Graphs.jl object. It returns a composite function compatible with the DifferentialEquations.jl calling syntax.

nd = network_dynamics(vertices!::Array{VertexFunction},
+                      edges!::Array{EdgeFunction}, g)
+nd(dx, x, p, t)

If all vertices, respectively edges share the same VertexFunction or EdgeFunction, than network_dynamics can be conveniently called with these functions as arguments.

nd = network_dynamics(vertexfunction!::VertexFunction,
+                      edgefunction!::EdgeFunction, g)
+

The optional keyword argument parallel is false by default and can be set to true if a multi-threaded ODEFunction should be used. This may significantly improve performance on multi-core machines, for more details see section Multi-Threading.

nd = network_dynamics(vertexfunction!::VertexFunction,
+                      edgefunction!::EdgeFunction, g; parallel=true)
+

Example

Let's look at an example. First, we define our graph as well as the differential systems connected to its vertices and edges:

using NetworkDynamics, Graphs
+
+g = erdos_renyi(10, 25) # random graph with 10 vertices and 25 edges
+
+function vertexfunction!(dv, v, edges, p, t)
+    dv .= 0
+    for e in edges
+        dv .+= e
+    end
+end
+
+function edgefunction!(de, e, v_s, v_d, p, t)
+    de .= 1000 .* (v_s .- v_d .- e)
+    nothing
+end
+
+vertex = ODEVertex(; f=vertexfunction!, dim=1)
+vertexarr = [vertex for v in vertices(g)]
+
+edge = ODEEdge(; f=edgefunction!, dim=1, coupling=:undirected)
+edgearr = [edge for e in edges(g)]
+
+nd = network_dynamics(vertexarr, edgearr, g)

Now we have an ODEFunction nd that can be solved with the tools provided by DifferentialEquations.jl.

For more details check out the Tutorials section.

diff --git a/v0.8.2/Library/index.html b/v0.8.2/Library/index.html new file mode 100644 index 00000000..2421015d --- /dev/null +++ b/v0.8.2/Library/index.html @@ -0,0 +1,2 @@ + +Library · NetworkDynamics
GitHub

Library

Public Interface

The following functions are designed for public use

NetworkDynamics.network_dynamics — Function
network_dynamics(vertices!, edges!, g; parallel = false)

Assembles the the dynamical equations of the network problem into an ODEFunction compatible with the DifferentialEquations.jl solvers. Takes as arguments an array of VertexFunctions vertices!, an array of EdgeFunctions edges! and a Graphs.jl object g. The optional argument parallel is a boolean value that denotes if the central loop should be executed in parallel with the number of threads set by the environment variable JULIA_NUM_THREADS.

source
NetworkDynamics.StaticVertex — Function
StaticVertex(; f, dim, sym)

Wrapper for ODEVertex with 0 mass matrix, i.e. static behaviour / algebraic constraint in mass matrix form.

f describes the local behaviour at a static node and has to respect the following calling syntax

f(v, edges, p, t) -> nothing

Here v, p and t are the usual arguments, while edges is an arrays containing the edges for which the described vertex is the destination (in-edges for directed graphs).

dim is the number of independent variables in the vertex equations and sym is an array of symbols for these variables.

For more details see the documentation.

source
NetworkDynamics.ODEVertex — Type
ODEVertex(; f, dim, mass_matrix, sym)

Wrapper that ensures compatibility of a mutating function f with the key constructor network_dynamics.

f describes the local behaviour at a dynamic node and has to respect the following calling syntax

f(dv, v, edges, p, t) -> nothing

Here dv, v, p and t are the usual ODE arguments, while edges is an Array containing the edges for which the vertex is the destination (in-edges for directed graphs).

dim is the number of independent variables in the vertex equations and sym is an array of symbols for these variables. mass_matrix is an optional argument that defaults to the identity matrix I. If a mass matrix M is given the system M * dv = f will be solved.

For more details see the documentation.

source
NetworkDynamics.DDEVertex — Type
DDEVertex(; f, dim, mass_matrix, sym)

Wrapper that ensures compatibility of a mutating function f with the key constructor network_dynamics.

f describes the local behaviour at a dynamic node and has to respect the following calling syntax

f(dv, v, edges, h, p, t) -> nothing

Here dv, v, p and t are the usual ODE arguments, while edges is an Arry of incoming edges. h is the history array for v.

  • dim is the number of independent variables in the edge equations and
  • sym is an array of symbols for these variables.
  • mass_matrix is an optional argument that defaults to the identity matrix I. If a mass matrix M is given the system M * de = f will be solved.

For more details see the documentation.

source
NetworkDynamics.StaticEdge — Type
StaticEdge(; f, dim, coupling, sym)

Wrapper that ensures compatibility of a mutating function f with the key constructor network_dynamics.

f describes the local behaviour at a static edge and has to respect the following calling syntax

f(e, v_s, v_t, p, t) -> nothing

Here e, p and t are the usual arguments, while v_s and v_d are arrays containing the vertices which are the source and destination of the described edge.

  • dim is the number of independent variables in the edge equations and
  • sym is an array of symbols for these variables.
  • coupling is a Symbol describing if the EdgeFunction is intended for a directed graph (:directed) or for an undirected graph ({:undirected, :symmetric, :antisymmetric, :fiducial}). :directed is intended for directed graphs. :undirected is the default option and is only compatible with SimpleGraph. In this case f should specify the coupling from a source vertex to a destination vertex. :symmetric and :antisymmetric trigger performance optimizations, if f has that symmetry property. :fiducial lets the user specify both the coupling from src to dst, as well as the coupling from dst to src and is intended for advanced users, i.e. the edge gets passed a vector of 2dim edge states, where the first dim elements will be presented to the dst and the second dim elements will be presented to src,

For more details see the documentation.

source
NetworkDynamics.ODEEdge — Type
ODEEdge(; f, dim, mass_matrix, sym)

Wrapper that ensures compatibility of a mutating function f with the key constructor network_dynamics.

f describes the local behaviour at a dynamic edge and has to respect the following calling syntax

f(de, e, v_s, v_t, p, t) -> nothing

Here de, e, p and t are the usual arguments, while v_s and v_d are arrays containing the vertices which are the source and destination of the described edge.

dim is the number of independent variables in the edge equations and sym is an array of symbols for these variables. For more details see the documentation. mass_matrix is an optional argument that defaults to the identity matrix I. If a mass matrix M is given the system M * de = f will be solved.

For more details see the documentation.

source
NetworkDynamics.StaticDelayEdge — Type
StaticDelayEdge(; f, dim, coupling, sym)

Like a static edge but with extra arguments for the history of the source and destination vertices. This is NOT a DDEEdge.

source
NetworkDynamics.sum_coupling! — Function
sum_coupling!(e_sum, dst_edges)

A small utility function for writing diffusion dynamics. It provides the sum of all incoming edges.

source
diff --git a/v0.8.2/Multithreading/index.html b/v0.8.2/Multithreading/index.html new file mode 100644 index 00000000..6da6b08d --- /dev/null +++ b/v0.8.2/Multithreading/index.html @@ -0,0 +1,2 @@ + +Multi-Threading · NetworkDynamics
GitHub

Multi-Threading

Since version 0.3.0 multi-threading via the Threads.@threads macro is possible. This allows julia to integrate different nodes and edges in different threads, and can lead to significant performance gains on parallel architectures. To enable multi-threading call network_dynamics with the keyword argument parallel=true.

network_dynamics(vertices!, edges!, graph; parallel=true)

In order for this to take effect, multiple threads have to be available. This is achieved by setting the environment variable JULIA_NUM_THREADS before starting Julia. To start Julia from a bash shell and with 4 threads use:

$ env JULIA_NUM_THREADS=4 julia

If you are using Juno for the Atom text editor JULIA_NUM_THREADS is set to the number of physical cores of your processor by default. This is also the number of threads we recommend to use.

Note

The thread handling causes an overhead in the order of 20 μs per call to the ODE function which might impair performance on small networks (<100 nodes) or on single core machines. In theses cases network_dynamics can be called without any additional arguments, since parallel defaults to false.

For more information on setting environment varibales see the Julia documentation.

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GitHub

SDE Tutorial

An IJulia notebook corresponding to this tutorial will be available on GitHub soon.

Topics covered in this tutorial include:

  • the swing equation
  • fixpoint search of ODE systems
  • constructing an SDE problem with NetworkDynamics.jl

Example: Fluctuations in Power Grids

This tutorial explains the use of stochastic differential equations (SDE's) in NetworkDynamics.jl. As an example system we will simulate fluctuations in a simple power grid model.

The phase and frequency dynamics in power grids can be modeled by the swing equation. In the complex systems community this is also known as the Kuramoto model with inertia.

\[\begin{aligned} +\dot \theta_i &= \omega_i \\ +M_i \dot \omega_i &= P_i(t) - D_i \omega_i - \sum_{i=1}^N K_{ij} \sin(\theta_i - \theta_j) +\end{aligned}\]

Here, $M_i$ and $D_i$ are inertia and damping parameters, $K_{ij}$ is the power capacitiy of the line and $P_i(t)$ is the power infeed at the node. We assume that this power can be separated into a constant power setpoint $P^\circ_i$ and a stochastic fluctuation.

\[\begin{aligned} +P_i(t) = P^\circ_i + \sigma_i \cdot \xi_i(t) +\end{aligned}\]

In this tutorial we assume the fluctuations $\xi_i(t)$ to be white Gaussian noise. Separating the deterministic and stochastic part, we can write this problem as a stochastic differential equation.

\[\begin{aligned} +d\omega_i = \left[P_i - D_i \omega_i - \sum_{i=1}^N K_{ij} \sin(\theta_i - \theta_j)\right] dt + \sigma_i dW_i +\end{aligned}\]

Here, $dW_i = \xi_i dt$ is the infinitesimal increment of the Wiener process. Problems of this type can be numerically solved in Julia as described in the SDE Tutorial of DifferentialEquations.

Implementing the Swing Equation

First we will implement the node and edge functions for the deterministic case without the fluctuations. We assume the inertia and damping parameters to be $M_i = 1.0$ and $D_i = 0.1$.

function swing_equation!(dv, v, edges, P, t)
+    dv[1] = v[2]
+    dv[2] = P - 0.1 * v[2]
+    for e in edges
+        dv[2] += e[1]
+    end
+end
+
+function powerflow!(e, v_s, v_d, K, t)
+    e[1] = K * sin(v_s[1] - v_d[1])
+end

Contructing the Deterministic Dynamics

For the graph structure we will use a simple 4 node ring network.

using Graphs
+g = watts_strogatz(4, 2, 0.0)

Then we can construct the ODEFunction of the deterministic system by using network_dynamics().

using NetworkDynamics
+
+swing_vertex = ODEVertex(; f=swing_equation!, dim=2, sym=[:θ, :ω])
+powerflow_edge = StaticEdge(; f=powerflow!, dim=1)
+
+nd = network_dynamics(swing_vertex, powerflow_edge, g)
[ Info: Reconstructing EdgeFunction with :undefined coupling type..

Now we need to define the dynamic parameters of vertices and edges. For simplicity we assume homogeneous capacities on the lines. For the nodes we assume half of them to be net producers ($P = 1.0$) and half of them to be net consumers ($P = -1.0$) of power.

K = 6.0
+P = [1.0, -1.0, 1.0, -1.0]
+p = (P, K)

We want to simulate fluctuations around an equilibrium state of our model system. Therefore, we need to find a fixpoint of the determinitic system which can be done by using the utility function find_fixpoint(). As an initial guess we take all variables equal to zero.

u0 = find_fixpoint(nd, p, zeros(8))
+
+using StochasticDiffEq, OrdinaryDiffEq
+ode_prob = ODEProblem(nd, u0, (0.0, 500.0), p)
+ode_sol = solve(ode_prob, Tsit5())
+
+using Plots, LaTeXStrings
+plot(ode_sol; vars=syms_containing(nd, "ω"), ylims=(-1.0, 1.0), ylabel=L"\omega", legend=false, fmt=:png)
Example block output

We see that this is in fact a fixpoint solution. We will later use this as an initial condition for the numerical integration of the SDE system.

Adding a Stochastic Layer

For adding the stochastic part of the dynamics we have to define a second graph layer. In our example, the fluctuations at different nodes are independent of each other. Therefore, we define a second graph with the same number of vertices but without any edges.

h = SimpleGraph(4, 0)

The dynamics at the nodes has to have the same dimension as in the deterministic case. In our example we only have fluctuations in the second variable.

function fluctuation!(dx, x, edges, p, t)
+    dx[1] = 0.0
+    dx[2] = 0.05
+end

Now we can construct the dynamics of the second layer by using network_dynamics(). Since the graph structure of the stochastic layer has no edges we can take the edge function of the deterministic case as a placeholder.

fluctuation_vertex = ODEVertex(; f=fluctuation!, dim=2)
+nd_noise = network_dynamics(fluctuation_vertex, powerflow_edge, h)
[ Info: Reconstructing EdgeFunction with :undefined coupling type..

Simulating the SDE

Finally, we can create an SDEProblem and solve it with DifferentialEquations.

sde_prob = SDEProblem(nd, nd_noise, u0, (0.0, 500.0), p)
+sde_sol = solve(sde_prob, SOSRA())
+plot(sde_sol; vars=syms_containing(nd, "ω"), ylims=(-1.0, 1.0), ylabel=L"\omega", legend=false, fmt=:png)
Example block output

More details on SDE problems, e.g. how to include correlations or how to define an EnsembleProblem, can be found in the documentation of DifferentialEquations.

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GitHub

Accessing internal edge values

In the following, we will present two different ways how the edge values of an ODEFunction with StaticEdges can be retrieved. A more extensive demonstration is available in the examples folder.

Accessing edge values via GetGD

NetworkDynamics.jl includes the types GetGD and GetGS, which can be used to access to the underlying GraphData and GraphStruct objects by multiple dispatching of the generated ODEFunction. Let nd be the ODEFunction that has been formed by calling network_dynamics. To get access to the edge values at any time $t$, nd can be called as follows:

gd_nd = nd(x, p, t, GetGD) # exposes underlying graph data struct
+e_values = gd_nd.gdb.e_array
+nothing # hide

By adding GetGD as an argument to the call of nd, a GraphData instance at the given time, parameters and states will be returned. gdb is the internal GraphDataBuffer struct. The argument x has to be the precomputed state of the system at time t, e.g. from a solution object.

Accessing edge values using SavingCallback

Instead of recomputing the values of the edge variables they can be saved in parallel to an integration with DiffEqCallbacks.

using DiffEqCallbacks: SavingCallback, SavedValues
+
+saved_values = SavedValues(Float64, Vector{Float64})
+function saving_func(u, t, integrator)
+    edgevals = Float64[]
+    for i in 1:integrator.f.f.graph_structure.num_e
+        push!(edgevals, integrator.f.f.graph_data.gdb.e[i]...)
+    end
+    edgevals
+end
+cb = SavingCallback(saving_func, saved_values)
+sol = solve(prob, Tsit5(); callback=cb)

The variables saved_values will contain the stored edge values. At the moment this method requires detailed knowledge of the internal data structures. Our plan is to simplify this in a future release.

diff --git a/v0.8.2/assets/documenter.js b/v0.8.2/assets/documenter.js new file mode 100644 index 00000000..f5311607 --- /dev/null +++ b/v0.8.2/assets/documenter.js @@ -0,0 +1,889 @@ +// Generated by Documenter.jl +requirejs.config({ + paths: { + 'highlight-julia': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/languages/julia.min', + 'headroom': 'https://cdnjs.cloudflare.com/ajax/libs/headroom/0.12.0/headroom.min', + 'jqueryui': 'https://cdnjs.cloudflare.com/ajax/libs/jqueryui/1.13.2/jquery-ui.min', + 'minisearch': 'https://cdn.jsdelivr.net/npm/minisearch@6.1.0/dist/umd/index.min', + 'katex-auto-render': 'https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/contrib/auto-render.min', + 'jquery': 'https://cdnjs.cloudflare.com/ajax/libs/jquery/3.7.0/jquery.min', + 'headroom-jquery': 'https://cdnjs.cloudflare.com/ajax/libs/headroom/0.12.0/jQuery.headroom.min', + 'katex': 'https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/katex.min', + 'highlight': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/highlight.min', + 'highlight-julia-repl': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/languages/julia-repl.min', + }, + shim: { + "highlight-julia": { + "deps": [ + "highlight" + ] + }, + "katex-auto-render": { + "deps": [ + "katex" + ] + }, + "headroom-jquery": { + "deps": [ + "jquery", + "headroom" + ] + }, + "highlight-julia-repl": { + "deps": [ + "highlight" + ] + } +} +}); +//////////////////////////////////////////////////////////////////////////////// +require(['jquery', 'katex', 'katex-auto-render'], function($, katex, renderMathInElement) { +$(document).ready(function() { + renderMathInElement( + document.body, + { + "delimiters": [ + { + "left": "$", + "right": "$", + "display": false + }, + { + "left": "$$", + "right": "$$", + "display": true + }, + { + "left": "\\[", + "right": "\\]", + "display": true + } + ] +} + + ); +}) + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery', 'highlight', 'highlight-julia', 'highlight-julia-repl'], function($) { +$(document).ready(function() { + hljs.highlightAll(); +}) + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +let timer = 0; +var isExpanded = true; + +$(document).on("click", ".docstring header", function () { + let articleToggleTitle = "Expand docstring"; + + debounce(() => { + if ($(this).siblings("section").is(":visible")) { + $(this) + .find(".docstring-article-toggle-button") + .removeClass("fa-chevron-down") + .addClass("fa-chevron-right"); + } else { + $(this) + .find(".docstring-article-toggle-button") + .removeClass("fa-chevron-right") + .addClass("fa-chevron-down"); + + articleToggleTitle = "Collapse docstring"; + } + + $(this) + .find(".docstring-article-toggle-button") + .prop("title", articleToggleTitle); + $(this).siblings("section").slideToggle(); + }); +}); + +$(document).on("click", ".docs-article-toggle-button", function () { + let articleToggleTitle = "Expand docstring"; + let navArticleToggleTitle = "Expand all docstrings"; + + debounce(() => { + if (isExpanded) { + $(this).removeClass("fa-chevron-up").addClass("fa-chevron-down"); + $(".docstring-article-toggle-button") + .removeClass("fa-chevron-down") + .addClass("fa-chevron-right"); + + isExpanded = false; + + $(".docstring section").slideUp(); + } else { + $(this).removeClass("fa-chevron-down").addClass("fa-chevron-up"); + $(".docstring-article-toggle-button") + .removeClass("fa-chevron-right") + .addClass("fa-chevron-down"); + + isExpanded = true; + articleToggleTitle = "Collapse docstring"; + navArticleToggleTitle = "Collapse all docstrings"; + + $(".docstring section").slideDown(); + } + + $(this).prop("title", navArticleToggleTitle); + $(".docstring-article-toggle-button").prop("title", articleToggleTitle); + }); +}); + +function debounce(callback, timeout = 300) { + if (Date.now() - timer > timeout) { + callback(); + } + + clearTimeout(timer); + + timer = Date.now(); +} + +}) +//////////////////////////////////////////////////////////////////////////////// +require([], function() { +function addCopyButtonCallbacks() { + for (const el of document.getElementsByTagName("pre")) { + const button = document.createElement("button"); + button.classList.add("copy-button", "fa-solid", "fa-copy"); + button.setAttribute("aria-label", "Copy this code block"); + button.setAttribute("title", "Copy"); + + el.appendChild(button); + + const success = function () { + button.classList.add("success", "fa-check"); + button.classList.remove("fa-copy"); + }; + + const failure = function () { + button.classList.add("error", "fa-xmark"); + button.classList.remove("fa-copy"); + }; + + button.addEventListener("click", function () { + copyToClipboard(el.innerText).then(success, failure); + + setTimeout(function () { + button.classList.add("fa-copy"); + button.classList.remove("success", "fa-check", "fa-xmark"); + }, 5000); + }); + } +} + +function copyToClipboard(text) { + // clipboard API is only available in secure contexts + if (window.navigator && window.navigator.clipboard) { + return window.navigator.clipboard.writeText(text); + } else { + return new Promise(function (resolve, reject) { + try { + const el = document.createElement("textarea"); + el.textContent = text; + el.style.position = "fixed"; + el.style.opacity = 0; + document.body.appendChild(el); + el.select(); + document.execCommand("copy"); + + resolve(); + } catch (err) { + reject(err); + } finally { + document.body.removeChild(el); + } + }); + } +} + +if (document.readyState === "loading") { + document.addEventListener("DOMContentLoaded", addCopyButtonCallbacks); +} else { + addCopyButtonCallbacks(); +} + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery', 'headroom', 'headroom-jquery'], function($, Headroom) { + +// Manages the top navigation bar (hides it when the user starts scrolling down on the +// mobile). +window.Headroom = Headroom; // work around buggy module loading? +$(document).ready(function () { + $("#documenter .docs-navbar").headroom({ + tolerance: { up: 10, down: 10 }, + }); +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery', 'minisearch'], function($, minisearch) { + +// In general, most search related things will have "search" as a prefix. +// To get an in-depth about the thought process you can refer: https://hetarth02.hashnode.dev/series/gsoc + +let results = []; +let timer = undefined; + +let data = documenterSearchIndex["docs"].map((x, key) => { + x["id"] = key; // minisearch requires a unique for each object + return x; +}); + +// list below is the lunr 2.1.3 list minus the intersect with names(Base) +// (all, any, get, in, is, only, which) and (do, else, for, let, where, while, with) +// ideally we'd just filter the original list but it's not available as a variable +const stopWords = new Set([ + "a", + "able", + "about", + "across", + "after", + "almost", + "also", + "am", + "among", + "an", + "and", + "are", + "as", + "at", + "be", + "because", + "been", + "but", + "by", + "can", + "cannot", + "could", + "dear", + "did", + "does", + "either", + "ever", + "every", + "from", + "got", + "had", + "has", + "have", + "he", + "her", + "hers", + "him", + "his", + "how", + "however", + "i", + "if", + "into", + "it", + "its", + "just", + "least", + "like", + "likely", + "may", + "me", + "might", + "most", + "must", + "my", + "neither", + "no", + "nor", + "not", + "of", + "off", + "often", + "on", + "or", + "other", + "our", + "own", + "rather", + "said", + "say", + "says", + "she", + "should", + "since", + "so", + "some", + "than", + "that", + "the", + "their", + "them", + "then", + "there", + "these", + "they", + "this", + "tis", + "to", + "too", + "twas", + "us", + "wants", + "was", + "we", + "were", + "what", + "when", + "who", + "whom", + "why", + "will", + "would", + "yet", + "you", + "your", +]); + +let index = new minisearch({ + fields: ["title", "text"], // fields to index for full-text search + storeFields: ["location", "title", "text", "category", "page"], // fields to return with search results + processTerm: (term) => { + let word = stopWords.has(term) ? null : term; + if (word) { + // custom trimmer that doesn't strip @ and !, which are used in julia macro and function names + word = word + .replace(/^[^a-zA-Z0-9@!]+/, "") + .replace(/[^a-zA-Z0-9@!]+$/, ""); + } + + return word ?? null; + }, + // add . as a separator, because otherwise "title": "Documenter.Anchors.add!", would not find anything if searching for "add!", only for the entire qualification + tokenize: (string) => string.split(/[\s\-\.]+/), + // options which will be applied during the search + searchOptions: { + boost: { title: 100 }, + fuzzy: 2, + processTerm: (term) => { + let word = stopWords.has(term) ? null : term; + if (word) { + word = word + .replace(/^[^a-zA-Z0-9@!]+/, "") + .replace(/[^a-zA-Z0-9@!]+$/, ""); + } + + return word ?? null; + }, + tokenize: (string) => string.split(/[\s\-\.]+/), + }, +}); + +index.addAll(data); + +let filters = [...new Set(data.map((x) => x.category))]; +var modal_filters = make_modal_body_filters(filters); +var filter_results = []; + +$(document).on("keyup", ".documenter-search-input", function (event) { + // Adding a debounce to prevent disruptions from super-speed typing! + debounce(() => update_search(filter_results), 300); +}); + +$(document).on("click", ".search-filter", function () { + if ($(this).hasClass("search-filter-selected")) { + $(this).removeClass("search-filter-selected"); + } else { + $(this).addClass("search-filter-selected"); + } + + // Adding a debounce to prevent disruptions from crazy clicking! + debounce(() => get_filters(), 300); +}); + +/** + * A debounce function, takes a function and an optional timeout in milliseconds + * + * @function callback + * @param {number} timeout + */ +function debounce(callback, timeout = 300) { + clearTimeout(timer); + timer = setTimeout(callback, timeout); +} + +/** + * Make/Update the search component + * + * @param {string[]} selected_filters + */ +function update_search(selected_filters = []) { + let initial_search_body = ` +
Type something to get started!
+ `; + + let querystring = $(".documenter-search-input").val(); + + if (querystring.trim()) { + results = index.search(querystring, { + filter: (result) => { + // Filtering results + if (selected_filters.length === 0) { + return result.score >= 1; + } else { + return ( + result.score >= 1 && selected_filters.includes(result.category) + ); + } + }, + }); + + let search_result_container = ``; + let search_divider = `
`; + + if (results.length) { + let links = []; + let count = 0; + let search_results = ""; + + results.forEach(function (result) { + if (result.location) { + // Checking for duplication of results for the same page + if (!links.includes(result.location)) { + search_results += make_search_result(result, querystring); + count++; + } + + links.push(result.location); + } + }); + + let result_count = `
${count} result(s)
`; + + search_result_container = ` +
+ ${modal_filters} + ${search_divider} + ${result_count} +
+ ${search_results} +
+
+ `; + } else { + search_result_container = ` +
+ ${modal_filters} + ${search_divider} +
0 result(s)
+
+
No result found!
+ `; + } + + if ($(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").removeClass("is-justify-content-center"); + } + + $(".search-modal-card-body").html(search_result_container); + } else { + filter_results = []; + modal_filters = make_modal_body_filters(filters, filter_results); + + if (!$(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").addClass("is-justify-content-center"); + } + + $(".search-modal-card-body").html(initial_search_body); + } +} + +/** + * Make the modal filter html + * + * @param {string[]} filters + * @param {string[]} selected_filters + * @returns string + */ +function make_modal_body_filters(filters, selected_filters = []) { + let str = ``; + + filters.forEach((val) => { + if (selected_filters.includes(val)) { + str += `${val}`; + } else { + str += `${val}`; + } + }); + + let filter_html = ` +
+ Filters: + ${str} +
+ `; + + return filter_html; +} + +/** + * Make the result component given a minisearch result data object and the value of the search input as queryString. + * To view the result object structure, refer: https://lucaong.github.io/minisearch/modules/_minisearch_.html#searchresult + * + * @param {object} result + * @param {string} querystring + * @returns string + */ +function make_search_result(result, querystring) { + let search_divider = `
`; + let display_link = + result.location.slice(Math.max(0), Math.min(50, result.location.length)) + + (result.location.length > 30 ? "..." : ""); // To cut-off the link because it messes with the overflow of the whole div + + if (result.page !== "") { + display_link += ` (${result.page})`; + } + + let textindex = new RegExp(`\\b${querystring}\\b`, "i").exec(result.text); + let text = + textindex !== null + ? result.text.slice( + Math.max(textindex.index - 100, 0), + Math.min( + textindex.index + querystring.length + 100, + result.text.length + ) + ) + : ""; // cut-off text before and after from the match + + let display_result = text.length + ? "..." + + text.replace( + new RegExp(`\\b${querystring}\\b`, "i"), // For first occurrence + '$&' + ) + + "..." + : ""; // highlights the match + + let in_code = false; + if (!["page", "section"].includes(result.category.toLowerCase())) { + in_code = true; + } + + // We encode the full url to escape some special characters which can lead to broken links + let result_div = ` + +
+
${result.title}
+
${result.category}
+
+

+ ${display_result} +

+
+ ${display_link} +
+ + ${search_divider} + `; + + return result_div; +} + +/** + * Get selected filters, remake the filter html and lastly update the search modal + */ +function get_filters() { + let ele = $(".search-filters .search-filter-selected").get(); + filter_results = ele.map((x) => $(x).text().toLowerCase()); + modal_filters = make_modal_body_filters(filters, filter_results); + update_search(filter_results); +} + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// Modal settings dialog +$(document).ready(function () { + var settings = $("#documenter-settings"); + $("#documenter-settings-button").click(function () { + settings.toggleClass("is-active"); + }); + // Close the dialog if X is clicked + $("#documenter-settings button.delete").click(function () { + settings.removeClass("is-active"); + }); + // Close dialog if ESC is pressed + $(document).keyup(function (e) { + if (e.keyCode == 27) settings.removeClass("is-active"); + }); +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +let search_modal_header = ` +
+
+

+ + + + +

+
+
+ +
+
+`; + +let initial_search_body = ` +
Type something to get started!
+`; + +let search_modal_footer = ` + +`; + +$(document.body).append( + ` + + ` +); + +document.querySelector(".docs-search-query").addEventListener("click", () => { + openModal(); +}); + +document.querySelector(".close-search-modal").addEventListener("click", () => { + closeModal(); +}); + +$(document).on("click", ".search-result-link", function () { + closeModal(); +}); + +document.addEventListener("keydown", (event) => { + if ((event.ctrlKey || event.metaKey) && event.key === "/") { + openModal(); + } else if (event.key === "Escape") { + closeModal(); + } + + return false; +}); + +// Functions to open and close a modal +function openModal() { + let searchModal = document.querySelector("#search-modal"); + + searchModal.classList.add("is-active"); + document.querySelector(".documenter-search-input").focus(); +} + +function closeModal() { + let searchModal = document.querySelector("#search-modal"); + let initial_search_body = ` +
Type something to get started!
+ `; + + searchModal.classList.remove("is-active"); + document.querySelector(".documenter-search-input").blur(); + + if (!$(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").addClass("is-justify-content-center"); + } + + $(".documenter-search-input").val(""); + $(".search-modal-card-body").html(initial_search_body); +} + +document + .querySelector("#search-modal .modal-background") + .addEventListener("click", () => { + closeModal(); + }); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// Manages the showing and hiding of the sidebar. +$(document).ready(function () { + var sidebar = $("#documenter > .docs-sidebar"); + var sidebar_button = $("#documenter-sidebar-button"); + sidebar_button.click(function (ev) { + ev.preventDefault(); + sidebar.toggleClass("visible"); + if (sidebar.hasClass("visible")) { + // Makes sure that the current menu item is visible in the sidebar. + $("#documenter .docs-menu a.is-active").focus(); + } + }); + $("#documenter > .docs-main").bind("click", function (ev) { + if ($(ev.target).is(sidebar_button)) { + return; + } + if (sidebar.hasClass("visible")) { + sidebar.removeClass("visible"); + } + }); +}); + +// Resizes the package name / sitename in the sidebar if it is too wide. +// Inspired by: https://github.com/davatron5000/FitText.js +$(document).ready(function () { + e = $("#documenter .docs-autofit"); + function resize() { + var L = parseInt(e.css("max-width"), 10); + var L0 = e.width(); + if (L0 > L) { + var h0 = parseInt(e.css("font-size"), 10); + e.css("font-size", (L * h0) / L0); + // TODO: make sure it survives resizes? + } + } + // call once and then register events + resize(); + $(window).resize(resize); + $(window).on("orientationchange", resize); +}); + +// Scroll the navigation bar to the currently selected menu item +$(document).ready(function () { + var sidebar = $("#documenter .docs-menu").get(0); + var active = $("#documenter .docs-menu .is-active").get(0); + if (typeof active !== "undefined") { + sidebar.scrollTop = active.offsetTop - sidebar.offsetTop - 15; + } +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// Theme picker setup +$(document).ready(function () { + // onchange callback + $("#documenter-themepicker").change(function themepick_callback(ev) { + var themename = $("#documenter-themepicker option:selected").attr("value"); + if (themename === "auto") { + // set_theme(window.matchMedia('(prefers-color-scheme: dark)').matches ? 'dark' : 'light'); + window.localStorage.removeItem("documenter-theme"); + } else { + // set_theme(themename); + window.localStorage.setItem("documenter-theme", themename); + } + // We re-use the global function from themeswap.js to actually do the swapping. + set_theme_from_local_storage(); + }); + + // Make sure that the themepicker displays the correct theme when the theme is retrieved + // from localStorage + if (typeof window.localStorage !== "undefined") { + var theme = window.localStorage.getItem("documenter-theme"); + if (theme !== null) { + $("#documenter-themepicker option").each(function (i, e) { + e.selected = e.value === theme; + }); + } + } +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// update the version selector with info from the siteinfo.js and ../versions.js files +$(document).ready(function () { + // If the version selector is disabled with DOCUMENTER_VERSION_SELECTOR_DISABLED in the + // siteinfo.js file, we just return immediately and not display the version selector. + if ( + typeof DOCUMENTER_VERSION_SELECTOR_DISABLED === "boolean" && + DOCUMENTER_VERSION_SELECTOR_DISABLED + ) { + return; + } + + var version_selector = $("#documenter .docs-version-selector"); + var version_selector_select = $("#documenter .docs-version-selector select"); + + version_selector_select.change(function (x) { + target_href = version_selector_select + .children("option:selected") + .get(0).value; + window.location.href = target_href; + }); + + // add the current version to the selector based on siteinfo.js, but only if the selector is empty + if ( + typeof DOCUMENTER_CURRENT_VERSION !== "undefined" && + $("#version-selector > option").length == 0 + ) { + var option = $( + "" + ); + version_selector_select.append(option); + } + + if (typeof DOC_VERSIONS !== "undefined") { + var existing_versions = version_selector_select.children("option"); + var existing_versions_texts = existing_versions.map(function (i, x) { + return x.text; + }); + DOC_VERSIONS.forEach(function (each) { + var version_url = documenterBaseURL + "/../" + each + "/"; + var existing_id = $.inArray(each, existing_versions_texts); + // if not already in the version selector, add it as a new option, + // otherwise update the old option with the URL and enable it + if (existing_id == -1) { + var option = $( + "" + ); + version_selector_select.append(option); + } else { + var option = existing_versions[existing_id]; + option.value = version_url; + option.disabled = false; + } + }); + } + + // only show the version selector if the selector has been populated + if (version_selector_select.children("option").length > 0) { + version_selector.toggleClass("visible"); + } +}); + +}) diff --git a/v0.8.2/assets/themes/documenter-dark.css b/v0.8.2/assets/themes/documenter-dark.css new file mode 100644 index 00000000..ec054ecc --- /dev/null +++ b/v0.8.2/assets/themes/documenter-dark.css @@ -0,0 +1,7 @@ +html.theme--documenter-dark .pagination-previous,html.theme--documenter-dark .pagination-next,html.theme--documenter-dark .pagination-link,html.theme--documenter-dark .pagination-ellipsis,html.theme--documenter-dark .file-cta,html.theme--documenter-dark .file-name,html.theme--documenter-dark .select select,html.theme--documenter-dark .textarea,html.theme--documenter-dark .input,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input,html.theme--documenter-dark .button{-moz-appearance:none;-webkit-appearance:none;align-items:center;border:1px solid transparent;border-radius:.4em;box-shadow:none;display:inline-flex;font-size:1rem;height:2.5em;justify-content:flex-start;line-height:1.5;padding-bottom:calc(0.5em - 1px);padding-left:calc(0.75em - 1px);padding-right:calc(0.75em - 1px);padding-top:calc(0.5em - 1px);position:relative;vertical-align:top}html.theme--documenter-dark .pagination-previous:focus,html.theme--documenter-dark 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#1abc9c}html.theme--documenter-dark a.box:active{box-shadow:inset 0 1px 2px rgba(10,10,10,0.2),0 0 0 1px #1abc9c}html.theme--documenter-dark .button{background-color:#282f2f;border-color:#4c5759;border-width:1px;color:#375a7f;cursor:pointer;justify-content:center;padding-bottom:calc(0.5em - 1px);padding-left:1em;padding-right:1em;padding-top:calc(0.5em - 1px);text-align:center;white-space:nowrap}html.theme--documenter-dark .button strong{color:inherit}html.theme--documenter-dark .button .icon,html.theme--documenter-dark .button .icon.is-small,html.theme--documenter-dark .button #documenter .docs-sidebar form.docs-search>input.icon,html.theme--documenter-dark #documenter .docs-sidebar .button form.docs-search>input.icon,html.theme--documenter-dark .button .icon.is-medium,html.theme--documenter-dark .button .icon.is-large{height:1.5em;width:1.5em}html.theme--documenter-dark .button .icon:first-child:not(:last-child){margin-left:calc(-0.5em - 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.button.is-black.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--documenter-dark .button.is-black.is-inverted[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-black.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#0a0a0a}html.theme--documenter-dark .button.is-black.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-black.is-outlined{background-color:transparent;border-color:#0a0a0a;color:#0a0a0a}html.theme--documenter-dark .button.is-black.is-outlined:hover,html.theme--documenter-dark .button.is-black.is-outlined.is-hovered,html.theme--documenter-dark .button.is-black.is-outlined:focus,html.theme--documenter-dark .button.is-black.is-outlined.is-focused{background-color:#0a0a0a;border-color:#0a0a0a;color:#fff}html.theme--documenter-dark .button.is-black.is-outlined.is-loading::after{border-color:transparent transparent #0a0a0a #0a0a0a !important}html.theme--documenter-dark .button.is-black.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-black.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-black.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-black.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-black.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-black.is-outlined{background-color:transparent;border-color:#0a0a0a;box-shadow:none;color:#0a0a0a}html.theme--documenter-dark .button.is-black.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--documenter-dark .button.is-black.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-black.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-focused{background-color:#fff;color:#0a0a0a}html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #0a0a0a #0a0a0a !important}html.theme--documenter-dark .button.is-black.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-black.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--documenter-dark .button.is-light{background-color:#ecf0f1;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--documenter-dark 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.button.is-light.is-outlined.is-loading::after{border-color:transparent transparent #ecf0f1 #ecf0f1 !important}html.theme--documenter-dark .button.is-light.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-light.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-light.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-light.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--documenter-dark .button.is-light.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-light.is-outlined{background-color:transparent;border-color:#ecf0f1;box-shadow:none;color:#ecf0f1}html.theme--documenter-dark .button.is-light.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-light.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-light.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#ecf0f1}html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #ecf0f1 #ecf0f1 !important}html.theme--documenter-dark .button.is-light.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-light.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-dark,html.theme--documenter-dark .content kbd.button{background-color:#282f2f;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-dark:hover,html.theme--documenter-dark .content kbd.button:hover,html.theme--documenter-dark .button.is-dark.is-hovered,html.theme--documenter-dark .content kbd.button.is-hovered{background-color:#232829;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-dark:focus,html.theme--documenter-dark .content kbd.button:focus,html.theme--documenter-dark .button.is-dark.is-focused,html.theme--documenter-dark .content kbd.button.is-focused{border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-dark:focus:not(:active),html.theme--documenter-dark .content kbd.button:focus:not(:active),html.theme--documenter-dark .button.is-dark.is-focused:not(:active),html.theme--documenter-dark .content kbd.button.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(40,47,47,0.25)}html.theme--documenter-dark .button.is-dark:active,html.theme--documenter-dark .content kbd.button:active,html.theme--documenter-dark .button.is-dark.is-active,html.theme--documenter-dark .content kbd.button.is-active{background-color:#1d2122;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-dark[disabled],html.theme--documenter-dark .content kbd.button[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-dark,fieldset[disabled] html.theme--documenter-dark .content kbd.button{background-color:#282f2f;border-color:#282f2f;box-shadow:none}html.theme--documenter-dark .button.is-dark.is-inverted,html.theme--documenter-dark .content kbd.button.is-inverted{background-color:#fff;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-inverted:hover,html.theme--documenter-dark .content kbd.button.is-inverted:hover,html.theme--documenter-dark .button.is-dark.is-inverted.is-hovered,html.theme--documenter-dark .content kbd.button.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--documenter-dark .button.is-dark.is-inverted[disabled],html.theme--documenter-dark .content kbd.button.is-inverted[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-dark.is-inverted,fieldset[disabled] html.theme--documenter-dark .content kbd.button.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-loading::after,html.theme--documenter-dark .content kbd.button.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-dark.is-outlined,html.theme--documenter-dark .content kbd.button.is-outlined{background-color:transparent;border-color:#282f2f;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-outlined:hover,html.theme--documenter-dark .content kbd.button.is-outlined:hover,html.theme--documenter-dark .button.is-dark.is-outlined.is-hovered,html.theme--documenter-dark .content kbd.button.is-outlined.is-hovered,html.theme--documenter-dark .button.is-dark.is-outlined:focus,html.theme--documenter-dark .content kbd.button.is-outlined:focus,html.theme--documenter-dark .button.is-dark.is-outlined.is-focused,html.theme--documenter-dark .content kbd.button.is-outlined.is-focused{background-color:#282f2f;border-color:#282f2f;color:#fff}html.theme--documenter-dark .button.is-dark.is-outlined.is-loading::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading::after{border-color:transparent transparent #282f2f #282f2f !important}html.theme--documenter-dark .button.is-dark.is-outlined.is-loading:hover::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-dark.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-dark.is-outlined.is-loading:focus::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-dark.is-outlined.is-loading.is-focused::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-dark.is-outlined[disabled],html.theme--documenter-dark .content kbd.button.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-dark.is-outlined,fieldset[disabled] html.theme--documenter-dark .content kbd.button.is-outlined{background-color:transparent;border-color:#282f2f;box-shadow:none;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined,html.theme--documenter-dark .content 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.button.is-warning.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-warning.is-outlined{background-color:transparent;border-color:#ad8100;box-shadow:none;color:#ad8100}html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-focused{background-color:#fff;color:#ad8100}html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark 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.button.is-danger.is-outlined.is-loading::after{border-color:transparent transparent #9e1b0d #9e1b0d !important}html.theme--documenter-dark .button.is-danger.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-danger.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-danger.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-danger.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-danger.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-danger.is-outlined{background-color:transparent;border-color:#9e1b0d;box-shadow:none;color:#9e1b0d}html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-focused{background-color:#fff;color:#9e1b0d}html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #9e1b0d #9e1b0d !important}html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--documenter-dark .button.is-danger.is-light{background-color:#fdeeec;color:#ec311d}html.theme--documenter-dark .button.is-danger.is-light:hover,html.theme--documenter-dark .button.is-danger.is-light.is-hovered{background-color:#fce3e0;border-color:transparent;color:#ec311d}html.theme--documenter-dark .button.is-danger.is-light:active,html.theme--documenter-dark .button.is-danger.is-light.is-active{background-color:#fcd8d5;border-color:transparent;color:#ec311d}html.theme--documenter-dark .button.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button{font-size:.75rem}html.theme--documenter-dark .button.is-small:not(.is-rounded),html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button:not(.is-rounded){border-radius:3px}html.theme--documenter-dark .button.is-normal{font-size:1rem}html.theme--documenter-dark .button.is-medium{font-size:1.25rem}html.theme--documenter-dark .button.is-large{font-size:1.5rem}html.theme--documenter-dark .button[disabled],fieldset[disabled] html.theme--documenter-dark .button{background-color:#8c9b9d;border-color:#5e6d6f;box-shadow:none;opacity:.5}html.theme--documenter-dark .button.is-fullwidth{display:flex;width:100%}html.theme--documenter-dark .button.is-loading{color:transparent !important;pointer-events:none}html.theme--documenter-dark .button.is-loading::after{position:absolute;left:calc(50% - (1em * 0.5));top:calc(50% - (1em * 0.5));position:absolute !important}html.theme--documenter-dark .button.is-static{background-color:#282f2f;border-color:#5e6d6f;color:#dbdee0;box-shadow:none;pointer-events:none}html.theme--documenter-dark .button.is-rounded,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button{border-radius:9999px;padding-left:calc(1em + 0.25em);padding-right:calc(1em + 0.25em)}html.theme--documenter-dark .buttons{align-items:center;display:flex;flex-wrap:wrap;justify-content:flex-start}html.theme--documenter-dark .buttons .button{margin-bottom:0.5rem}html.theme--documenter-dark .buttons .button:not(:last-child):not(.is-fullwidth){margin-right:.5rem}html.theme--documenter-dark .buttons:last-child{margin-bottom:-0.5rem}html.theme--documenter-dark .buttons:not(:last-child){margin-bottom:1rem}html.theme--documenter-dark .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large){font-size:.75rem}html.theme--documenter-dark .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large):not(.is-rounded){border-radius:3px}html.theme--documenter-dark .buttons.are-medium .button:not(.is-small):not(.is-normal):not(.is-large){font-size:1.25rem}html.theme--documenter-dark .buttons.are-large .button:not(.is-small):not(.is-normal):not(.is-medium){font-size:1.5rem}html.theme--documenter-dark .buttons.has-addons .button:not(:first-child){border-bottom-left-radius:0;border-top-left-radius:0}html.theme--documenter-dark .buttons.has-addons .button:not(:last-child){border-bottom-right-radius:0;border-top-right-radius:0;margin-right:-1px}html.theme--documenter-dark .buttons.has-addons .button:last-child{margin-right:0}html.theme--documenter-dark .buttons.has-addons .button:hover,html.theme--documenter-dark .buttons.has-addons .button.is-hovered{z-index:2}html.theme--documenter-dark .buttons.has-addons .button:focus,html.theme--documenter-dark .buttons.has-addons .button.is-focused,html.theme--documenter-dark .buttons.has-addons .button:active,html.theme--documenter-dark .buttons.has-addons .button.is-active,html.theme--documenter-dark .buttons.has-addons .button.is-selected{z-index:3}html.theme--documenter-dark .buttons.has-addons .button:focus:hover,html.theme--documenter-dark .buttons.has-addons .button.is-focused:hover,html.theme--documenter-dark .buttons.has-addons .button:active:hover,html.theme--documenter-dark .buttons.has-addons .button.is-active:hover,html.theme--documenter-dark .buttons.has-addons .button.is-selected:hover{z-index:4}html.theme--documenter-dark .buttons.has-addons .button.is-expanded{flex-grow:1;flex-shrink:1}html.theme--documenter-dark .buttons.is-centered{justify-content:center}html.theme--documenter-dark .buttons.is-centered:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}html.theme--documenter-dark .buttons.is-right{justify-content:flex-end}html.theme--documenter-dark .buttons.is-right:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}@media screen and (max-width: 768px){html.theme--documenter-dark .button.is-responsive.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.5625rem}html.theme--documenter-dark .button.is-responsive,html.theme--documenter-dark .button.is-responsive.is-normal{font-size:.65625rem}html.theme--documenter-dark .button.is-responsive.is-medium{font-size:.75rem}html.theme--documenter-dark .button.is-responsive.is-large{font-size:1rem}}@media screen and (min-width: 769px) and (max-width: 1055px){html.theme--documenter-dark .button.is-responsive.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.65625rem}html.theme--documenter-dark .button.is-responsive,html.theme--documenter-dark .button.is-responsive.is-normal{font-size:.75rem}html.theme--documenter-dark .button.is-responsive.is-medium{font-size:1rem}html.theme--documenter-dark .button.is-responsive.is-large{font-size:1.25rem}}html.theme--documenter-dark .container{flex-grow:1;margin:0 auto;position:relative;width:auto}html.theme--documenter-dark .container.is-fluid{max-width:none !important;padding-left:32px;padding-right:32px;width:100%}@media screen and (min-width: 1056px){html.theme--documenter-dark .container{max-width:992px}}@media screen and (max-width: 1215px){html.theme--documenter-dark .container.is-widescreen:not(.is-max-desktop){max-width:1152px}}@media screen and (max-width: 1407px){html.theme--documenter-dark .container.is-fullhd:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}@media screen and (min-width: 1216px){html.theme--documenter-dark .container:not(.is-max-desktop){max-width:1152px}}@media screen and (min-width: 1408px){html.theme--documenter-dark .container:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}html.theme--documenter-dark .content li+li{margin-top:0.25em}html.theme--documenter-dark .content p:not(:last-child),html.theme--documenter-dark .content dl:not(:last-child),html.theme--documenter-dark .content ol:not(:last-child),html.theme--documenter-dark .content ul:not(:last-child),html.theme--documenter-dark .content blockquote:not(:last-child),html.theme--documenter-dark .content pre:not(:last-child),html.theme--documenter-dark .content table:not(:last-child){margin-bottom:1em}html.theme--documenter-dark .content h1,html.theme--documenter-dark .content h2,html.theme--documenter-dark .content h3,html.theme--documenter-dark .content h4,html.theme--documenter-dark .content h5,html.theme--documenter-dark .content h6{color:#f2f2f2;font-weight:600;line-height:1.125}html.theme--documenter-dark .content h1{font-size:2em;margin-bottom:0.5em}html.theme--documenter-dark .content h1:not(:first-child){margin-top:1em}html.theme--documenter-dark .content h2{font-size:1.75em;margin-bottom:0.5714em}html.theme--documenter-dark .content h2:not(:first-child){margin-top:1.1428em}html.theme--documenter-dark .content h3{font-size:1.5em;margin-bottom:0.6666em}html.theme--documenter-dark .content h3:not(:first-child){margin-top:1.3333em}html.theme--documenter-dark .content h4{font-size:1.25em;margin-bottom:0.8em}html.theme--documenter-dark .content h5{font-size:1.125em;margin-bottom:0.8888em}html.theme--documenter-dark .content h6{font-size:1em;margin-bottom:1em}html.theme--documenter-dark .content blockquote{background-color:#282f2f;border-left:5px solid #5e6d6f;padding:1.25em 1.5em}html.theme--documenter-dark .content ol{list-style-position:outside;margin-left:2em;margin-top:1em}html.theme--documenter-dark .content ol:not([type]){list-style-type:decimal}html.theme--documenter-dark .content ol.is-lower-alpha:not([type]){list-style-type:lower-alpha}html.theme--documenter-dark .content ol.is-lower-roman:not([type]){list-style-type:lower-roman}html.theme--documenter-dark .content ol.is-upper-alpha:not([type]){list-style-type:upper-alpha}html.theme--documenter-dark .content ol.is-upper-roman:not([type]){list-style-type:upper-roman}html.theme--documenter-dark .content ul{list-style:disc outside;margin-left:2em;margin-top:1em}html.theme--documenter-dark .content ul ul{list-style-type:circle;margin-top:0.5em}html.theme--documenter-dark .content ul ul ul{list-style-type:square}html.theme--documenter-dark .content dd{margin-left:2em}html.theme--documenter-dark .content figure{margin-left:2em;margin-right:2em;text-align:center}html.theme--documenter-dark .content figure:not(:first-child){margin-top:2em}html.theme--documenter-dark .content figure:not(:last-child){margin-bottom:2em}html.theme--documenter-dark .content figure img{display:inline-block}html.theme--documenter-dark .content figure figcaption{font-style:italic}html.theme--documenter-dark .content pre{-webkit-overflow-scrolling:touch;overflow-x:auto;padding:0;white-space:pre;word-wrap:normal}html.theme--documenter-dark .content sup,html.theme--documenter-dark .content sub{font-size:75%}html.theme--documenter-dark .content table{width:100%}html.theme--documenter-dark .content table td,html.theme--documenter-dark .content table th{border:1px solid #5e6d6f;border-width:0 0 1px;padding:0.5em 0.75em;vertical-align:top}html.theme--documenter-dark .content table th{color:#f2f2f2}html.theme--documenter-dark .content table th:not([align]){text-align:inherit}html.theme--documenter-dark .content table 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html.theme--documenter-dark .navbar-dropdown,html.theme--documenter-dark .navbar-dropdown.is-boxed{border-radius:8px;border-top:none;box-shadow:0 8px 8px rgba(10,10,10,0.1), 0 0 0 1px rgba(10,10,10,0.1);display:block;opacity:0;pointer-events:none;top:calc(100% + (-4px));transform:translateY(-5px);transition-duration:86ms;transition-property:opacity, transform}html.theme--documenter-dark .navbar-dropdown.is-right{left:auto;right:0}html.theme--documenter-dark .navbar-divider{display:block}html.theme--documenter-dark .navbar>.container .navbar-brand,html.theme--documenter-dark .container>.navbar .navbar-brand{margin-left:-.75rem}html.theme--documenter-dark .navbar>.container .navbar-menu,html.theme--documenter-dark .container>.navbar .navbar-menu{margin-right:-.75rem}html.theme--documenter-dark .navbar.is-fixed-bottom-desktop,html.theme--documenter-dark .navbar.is-fixed-top-desktop{left:0;position:fixed;right:0;z-index:30}html.theme--documenter-dark 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a.navbar-item.is-active:not(:focus):not(:hover),html.theme--documenter-dark .navbar-link.is-active:not(:focus):not(:hover){background-color:rgba(0,0,0,0)}html.theme--documenter-dark .navbar-item.has-dropdown:focus .navbar-link,html.theme--documenter-dark .navbar-item.has-dropdown:hover .navbar-link,html.theme--documenter-dark .navbar-item.has-dropdown.is-active .navbar-link{background-color:rgba(0,0,0,0)}}html.theme--documenter-dark .hero.is-fullheight-with-navbar{min-height:calc(100vh - 4rem)}html.theme--documenter-dark .pagination{font-size:1rem;margin:-.25rem}html.theme--documenter-dark .pagination.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.pagination{font-size:.75rem}html.theme--documenter-dark .pagination.is-medium{font-size:1.25rem}html.theme--documenter-dark .pagination.is-large{font-size:1.5rem}html.theme--documenter-dark .pagination.is-rounded .pagination-previous,html.theme--documenter-dark #documenter .docs-sidebar 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.hljs-string{color:#38a}.hljs-emphasis{font-style:italic}.hljs-strong{font-weight:bold}.gap-4{gap:1rem} diff --git a/v0.8.2/assets/themeswap.js b/v0.8.2/assets/themeswap.js new file mode 100644 index 00000000..9f5eebe6 --- /dev/null +++ b/v0.8.2/assets/themeswap.js @@ -0,0 +1,84 @@ +// Small function to quickly swap out themes. Gets put into the tag.. +function set_theme_from_local_storage() { + // Initialize the theme to null, which means default + var theme = null; + // If the browser supports the localstorage and is not disabled then try to get the + // documenter theme + if (window.localStorage != null) { + // Get the user-picked theme from localStorage. May be `null`, which means the default + // theme. + theme = window.localStorage.getItem("documenter-theme"); + } + // Check if the users preference is for dark color scheme + var darkPreference = + window.matchMedia("(prefers-color-scheme: dark)").matches === true; + // Initialize a few variables for the loop: + // + // - active: will contain the index of the theme that should be active. Note that there + // is no guarantee that localStorage contains sane values. If `active` stays `null` + // we either could not find the theme or it is the default (primary) theme anyway. + // Either way, we then need to stick to the primary theme. + // + // - disabled: style sheets that should be disabled (i.e. all the theme style sheets + // that are not the currently active theme) + var active = null; + var disabled = []; + var primaryLightTheme = null; + var primaryDarkTheme = null; + for (var i = 0; i < document.styleSheets.length; i++) { + var ss = document.styleSheets[i]; + // The tag of each style sheet is expected to have a data-theme-name attribute + // which must contain the name of the theme. The names in localStorage much match this. + var themename = ss.ownerNode.getAttribute("data-theme-name"); + // attribute not set => non-theme stylesheet => ignore + if (themename === null) continue; + // To distinguish the default (primary) theme, it needs to have the data-theme-primary + // attribute set. + if (ss.ownerNode.getAttribute("data-theme-primary") !== null) { + primaryLightTheme = themename; + } + // Check if the theme is primary dark theme so that we could store its name in darkTheme + if (ss.ownerNode.getAttribute("data-theme-primary-dark") !== null) { + primaryDarkTheme = themename; + } + // If we find a matching theme (and it's not the default), we'll set active to non-null + if (themename === theme) active = i; + // Store the style sheets of inactive themes so that we could disable them + if (themename !== theme) disabled.push(ss); + } + var activeTheme = null; + if (active !== null) { + // If we did find an active theme, we'll (1) add the theme--$(theme) class to + document.getElementsByTagName("html")[0].className = "theme--" + theme; + activeTheme = theme; + } else { + // If we did _not_ find an active theme, then we need to fall back to the primary theme + // which can either be dark or light, depending on the user's OS preference. + var activeTheme = darkPreference ? primaryDarkTheme : primaryLightTheme; + // In case it somehow happens that the relevant primary theme was not found in the + // preceding loop, we abort without doing anything. + if (activeTheme === null) { + console.error("Unable to determine primary theme."); + return; + } + // When switching to the primary light theme, then we must not have a class name + // for the tag. That's only for non-primary or the primary dark theme. + if (darkPreference) { + document.getElementsByTagName("html")[0].className = + "theme--" + activeTheme; + } else { + document.getElementsByTagName("html")[0].className = ""; + } + } + for (var i = 0; i < document.styleSheets.length; i++) { + var ss = document.styleSheets[i]; + // The tag of each style sheet is expected to have a data-theme-name attribute + // which must contain the name of the theme. The names in localStorage much match this. + var themename = ss.ownerNode.getAttribute("data-theme-name"); + // attribute not set => non-theme stylesheet => ignore + if (themename === null) continue; + // we'll disable all the stylesheets, except for the active one + ss.disabled = !(themename == activeTheme); + } +} +set_theme_from_local_storage(); diff --git a/v0.8.2/assets/warner.js b/v0.8.2/assets/warner.js new file mode 100644 index 00000000..3f6f5d00 --- /dev/null +++ b/v0.8.2/assets/warner.js @@ -0,0 +1,52 @@ +function maybeAddWarning() { + // DOCUMENTER_NEWEST is defined in versions.js, DOCUMENTER_CURRENT_VERSION and DOCUMENTER_STABLE + // in siteinfo.js. + // If either of these are undefined something went horribly wrong, so we abort. + if ( + window.DOCUMENTER_NEWEST === undefined || + window.DOCUMENTER_CURRENT_VERSION === undefined || + window.DOCUMENTER_STABLE === undefined + ) { + return; + } + + // Current version is not a version number, so we can't tell if it's the newest version. Abort. + if (!/v(\d+\.)*\d+/.test(window.DOCUMENTER_CURRENT_VERSION)) { + return; + } + + // Current version is newest version, so no need to add a warning. + if (window.DOCUMENTER_NEWEST === window.DOCUMENTER_CURRENT_VERSION) { + return; + } + + // Add a noindex meta tag (unless one exists) so that search engines don't index this version of the docs. + if (document.body.querySelector('meta[name="robots"]') === null) { + const meta = document.createElement("meta"); + meta.name = "robots"; + meta.content = "noindex"; + + document.getElementsByTagName("head")[0].appendChild(meta); + } + + const div = document.createElement("div"); + div.classList.add("outdated-warning-overlay"); + const closer = document.createElement("button"); + closer.classList.add("outdated-warning-closer", "delete"); + closer.addEventListener("click", function () { + document.body.removeChild(div); + }); + const href = window.documenterBaseURL + "/../" + window.DOCUMENTER_STABLE; + div.innerHTML = + 'This documentation is not for the latest stable release, but for either the development version or an older release.
Click here to go to the documentation for the latest stable release.'; + div.appendChild(closer); + document.body.appendChild(div); +} + +if (document.readyState === "loading") { + document.addEventListener("DOMContentLoaded", maybeAddWarning); +} else { + maybeAddWarning(); +} diff --git a/v0.8.2/directed_and_weighted_graphs/f688b770.svg b/v0.8.2/directed_and_weighted_graphs/f688b770.svg new file mode 100644 index 00000000..7c2789cc --- /dev/null +++ b/v0.8.2/directed_and_weighted_graphs/f688b770.svg @@ -0,0 +1,132 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/v0.8.2/directed_and_weighted_graphs/index.html b/v0.8.2/directed_and_weighted_graphs/index.html new file mode 100644 index 00000000..d13906e3 --- /dev/null +++ b/v0.8.2/directed_and_weighted_graphs/index.html @@ -0,0 +1,58 @@ + +Directed and weighted graphs · NetworkDynamics
GitHub

Neurodynamic model of synchronization in the human brain

Topics covered in this tutorial include:

  • constructing a directed, weighted graph from data
  • some useful macros
  • parameter handling
  • stiff equations

The FitzHugh-Nagumo model

Dynamics of spiking neurons have been described in a simplified manner by the FitzHugh-Nagumo model.

\[\begin{aligned} +\varepsilon \dot u & = u - \frac{u^3}{3} - v \\ +\dot v & = u + a +\end{aligned}\]

Here $u$ is a fast, excitatory variable corresponding to the membrane potential and $v$ is a slower, inhibitory varibale. $\varepsilon$ is a parameter separating these time-scales, and $a$ is a control parameter.

In simplified models of the brain, such relaxation oscillators may be used to model individual neurons, clusters of neurons or even larger areas in the brain. The FitzHugh-Nagumo model has been widely used for studying synchronization in neuronal activity, which in turn has been connected to physiological phenomena such as epileptic seizures.

Coupling relaxation oscillators

While different coupling schemes for FitzHugh-Nagumo oscillators have been proposed, in this tutorial we focus on coupling of the excitatory variables via electrical gap junctions, as described by the following system of equations.

\[\begin{aligned} +\varepsilon \dot u_i & = u_i - \frac{u_i^3}{3} - v_i - \sigma \sum_{j=1}^N G_{ij}(u_i - u_j) \\ +\dot v_i & = u_i + a +\end{aligned}\]

This is a simple diffusive coupling mediated by the difference between activation potentials in pairs of neurons. A similar coupling term was introduced in the "getting started" tutorial.

The network topology - a brain atlas

In the following we will use a directed and weigthed network encoding the strength and directionality of coupling between 90 different areas of the brain [Nathalie Tzourio-Mazoyer et al., 2002, Neuroimage].

The network weight matrix is given as a text file containing 90 lines with 90 numbers representing the coupling strength and separated by commas ,. The data can be conveniently read into a matrix with the DelimitedFiles module.

using DelimitedFiles
+# adjust the load path for your filesystem!
+G = readdlm(joinpath(@__DIR__, "../../examples/Norm_G_DTI.txt"), ',', Float64, '\n')

The data structure for directed, weighted graphs is provided by the package SimpleWeightedGraphs.jl which is based on Graphs.jl.

using SimpleWeightedGraphs, Graphs
+
+# First we construct a weighted, directed graph
+g_weighted = SimpleWeightedDiGraph(G)
+
+# For later use we extract the edge.weight attributes
+# . is the broadcasting operator and gets the attribute :weight for every edge
+edge_weights = getfield.(collect(edges(g_weighted)), :weight)
+
+# we promote the g_weighted graph as a directed graph (weights of the edges are included in parameters)
+g_directed = SimpleDiGraph(g_weighted)

Setting up the ODEProblem

Defining VertexFunction and EdgeFunction is similar to the example before. The macro Base.@propagate_inbounds tells the compiler to inline the function and propagate the inbounds context. For more details see the julia documentation.

using NetworkDynamics
+
+Base.@propagate_inbounds function fhn_electrical_vertex!(dv, v, edges, p, t)
+    dv[1] = v[1] - v[1]^3 / 3 - v[2]
+    dv[2] = (v[1] - a) * ϵ
+    for e in edges
+        dv[1] += e[1]
+    end
+    nothing
+end
+
+Base.@propagate_inbounds function electrical_edge!(e, v_s, v_d, p, t)
+    e[1] = p * (v_s[1] - v_d[1]) # * σ
+    nothing
+end
+
+odeelevertex = ODEVertex(; f=fhn_electrical_vertex!, dim=2, sym=[:u, :v]);
+electricaledge = StaticEdge(; f=electrical_edge!, dim=1, coupling=:directed)
+
+fhn_network! = network_dynamics(odeelevertex, electricaledge, g_directed)

Since this system is a directed one with thus directed edges, the keyword argument coupling is used to set the coupling of the edges to :directed.

Note that the multiplication with the coupling strength $\sigma$ has been commented out. Since $\sigma$ is the same for every edge we can absorb this multiplication into the edge weight parameter p. Since our network has almost 8000 edges, this saves 8000 multiplications at every function call and leads to an 8-fold increase in performance.

Parameter handling

# Defining global parameters
+
+N = 90         # number of nodes
+const ϵ = 0.05 # global variables that are accessed several times should be declared `const`
+const a = 0.5
+const σ = 0.5
+
+# Tuple of parameters for nodes and edges
+
+p = (nothing, σ * edge_weights)
+
+# Initial conditions
+
+x0 = randn(2N) * 5

The behaviour of network_dynamics changes with the type of parameters p being passed. When p is an Array, the entire Array will be passed to each VertexFunction and EdgeFunction. When p is a tuple of two Arrays with lengths corresponding to the number of nodes and number of edges respectively, then network_dynamics passes only the edge or node parameters with the index of the edge or node. When there are no parameters for either edges or nodes the value nothing may be used.

Solving the system

Now we are ready to create an ODEProblem. Since for some choices of parameters the FitzHugh-Nagumo model is stiff (i.e. numerically unstable), we use a solver with automated stiffness detection. Such a solver switches to a more stable solver only when the solution enters a region of phase space where the problem is numerically unstable. In this case we use Tsit5 and switch to TRBDF2 when necessary. AutoTsit5 is the switching version of the Tsit5 algorithm.

using OrdinaryDiffEq
+
+tspan = (0.0, 200.0)
+prob  = ODEProblem(fhn_network!, x0, tspan, p)
+sol = solve(prob, AutoTsit5(TRBDF2()));

Plotting

The plot of the excitatory variables shows that they synchronize for this choice of parameters.

using Plots
+
+plot(sol; vars=idx_containing(fhn_network!, :u), legend=false, ylim=(-5, 5), fmt=:png)
Example block output
diff --git a/v0.8.2/getting_started_with_network_dynamics/2029525f.svg b/v0.8.2/getting_started_with_network_dynamics/2029525f.svg new file mode 100644 index 00000000..70eb2975 --- /dev/null +++ b/v0.8.2/getting_started_with_network_dynamics/2029525f.svg @@ -0,0 +1,82 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/v0.8.2/getting_started_with_network_dynamics/9c5f74e2.svg b/v0.8.2/getting_started_with_network_dynamics/9c5f74e2.svg new file mode 100644 index 00000000..c33b0e43 --- /dev/null +++ b/v0.8.2/getting_started_with_network_dynamics/9c5f74e2.svg @@ -0,0 +1,62 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/v0.8.2/getting_started_with_network_dynamics/index.html b/v0.8.2/getting_started_with_network_dynamics/index.html new file mode 100644 index 00000000..8c1fd9e6 --- /dev/null +++ b/v0.8.2/getting_started_with_network_dynamics/index.html @@ -0,0 +1,42 @@ + +Getting started · NetworkDynamics
GitHub

Network diffusion

This introductory example explains the use of the basic types and constructors in NetworkDynamics.jl by modeling a simple diffusion on an undirected network. A corresponding IJulia notebook is available on GitHub.

Theoretical background

Diffusion processes are relevant for phenomena as diverse as heat conduction, electrical currents, and random walks. Generally speaking they describe the tendency of systems to evolve towards a state of equally distributed heat, charge or concentration. In such system the local temperature (or concentration) changes according to its difference with its neighborhood, i.e. the temperature gradient.

Let $g$ be a graph with $N$ nodes and adjacency matrix $A$. Let $v = (v_1, \dots, v_n)$ be a vector of (abstract) temperatures or concentrations at each node $i = 1, \dots, N$. Then the rate of change of state $v_i$ is described by its difference with its neighbors and we obtain the following ordinary differential equation

\[\dot v_i = \sum_{j=1}^N A_{ji} (v_j - v_i).\]

The sum on the right hand side plays the role of a (discrete) gradient. If the temperature at node $i$ is higher than at its neighboring node $j$ it will decrease along that edge.

Modeling diffusion in NetworkDynamics.jl

From the above considerations we see that in this model the nodes do not have any internal dynamics - if a node was disconnected from the rest of the network its state would never change, since then $A_{ji} = 0 \; \forall j$ and hence $\dot v_i = 0$. This means that the evolution of a node depends only on the interaction with its neighbors. In NetworkDynamics.jl, interactions with neighbors are described by equations for the edges.

function diffusionedge!(e, v_s, v_d, p, t)
+    # usually e, v_s, v_d are arrays, hence we use the broadcasting operator .
+    e .= v_s - v_d
+    nothing
+end

The function diffusionedge! takes as inputs the current state of the edge e, its source vertex v_s, its destination vertex v_d, a vector of parameters p and the time t. In order to comply with the syntax of NetworkDynamics.jl we always have to define functions for static edges with exactly these arguments, even though we do not need p and t for the diffusion example.

diffusionedge! is called a mutating function, since it modifies (or mutates) one of its inputs, namely the edge state e. As a convention in Julia names of mutating functions end with an !. The use of mutating functions reduces allocations and thereby speeds up computations. After the function call the edge's value e equals the difference between its source and its destination vertex (i.e. the discrete gradient along that edge).

The contributions of the different edges are then summed up in each vertex.

function diffusionvertex!(dv, v, edges, p, t)
+    # usually v, edges are arrays, hence we use the broadcasting operator .
+    dv .= 0.0
+    for e in edges
+        dv .+= e
+    end
+    nothing
+end

Just like above the input arguments v, edges, p, t are mandatory for the syntax of vertex functions. The additional input dv corresponding to the derivative of the vertex' state is mandatory for vertices described by ordinary differential equations.

For undirected graphs, the edgefunction! specifies the coupling from a source- to a destination vertex. The contributions of the connected edges are added to the destination of each edge, as defined in the diffusionvertex! function in the for loop.

Constructing the network

With the preliminaries out of the way, it only takes a few steps to assemble the network dynamics.

using Graphs
+
+N = 20 # number of nodes
+k = 4  # average degree
+g = barabasi_albert(N, k) # a little more exciting than a bare random graph

The Barabási–Albert model generates a scale-free random graph.

using NetworkDynamics
+
+nd_diffusion_vertex = ODEVertex(; f=diffusionvertex!, dim=1)
+nd_diffusion_edge = StaticEdge(; f=diffusionedge!, dim=1)
+
+nd = network_dynamics(nd_diffusion_vertex, nd_diffusion_edge, g)
[ Info: Reconstructing EdgeFunction with :undefined coupling type..

ODEVertex and StaticEdge are functions wrappers that equip the functions we defined above with additional information like dim and return objects of type VertexFunction and EdgeFunction. Then the key constructor network_dynamics combines them with the topological information contained in the graph g and returns an ODEFunction compatible with the solvers of DifferentialEquations.jl. The keyword dim specifies the number of variables at each edge or node.

using OrdinaryDiffEq
+
+x0 = randn(N) # random initial conditions
+ode_prob = ODEProblem(nd, x0, (0.0, 4.0))
+sol = solve(ode_prob, Tsit5());

We are solving the diffusion problem on the time interval $[0, 4]$ with the Tsit5() algorithm, which is recommended by the authors of DifferentialEquations.jl for most non-stiff problems.

using Plots
+plot(sol; vars=syms_containing(nd, "v"), fmt=:png)
Example block output

The plotting is straightforward. The vars keyword allows us to pass a list of indices or symbols specifiying the variables we want to plot. Symbols can be thought of as names given to the interal variables of an ODEFunction, much like the variables $x$ or $\phi$ in mathematical notation. The default symbol for vertex variables is v, however we are free to specify other symbols by passing them to the ODEVertex constructor.

syms_containing is a helper function that returns all symbols of an ODEProblem containg a specific string (or symbol).

To illustrate a very simple multi-dimensional case, in the following we simulate two independent diffusions on an identical network. The first uses the symbol x and is started with initial conditions drawn from the standard normal distribution $N(0,1)$, the second uses the symbol ϕ with squared standard normal inital conditions.

The symbols have to be passed with the keyword sym to ODEVertex.

N = 10 # number of nodes
+k = 4  # average degree
+g = barabasi_albert(N, k) # a little more exciting than a bare random graph
+
+# We will have two independent diffusions on the network, hence dim = 2
+nd_diffusion_vertex_2 = ODEVertex(; f=diffusionvertex!, dim=2, sym=[:x, :ϕ])
+nd_diffusion_edge_2 = StaticEdge(; f=diffusionedge!, dim=2)
+nd_2 = network_dynamics(nd_diffusion_vertex_2, nd_diffusion_edge_2, g)
+
+x0_2 = vec(transpose([randn(N) .^ 2 randn(N)])) # x ~ N(0,1)^2; ϕ ~ N(0,1)
+ode_prob_2 = ODEProblem(nd_2, x0_2, (0.0, 3.0))
+sol_2 = solve(ode_prob_2, Tsit5());
+
+
+# Try plotting the variables ϕ_i yourself. [To write ϕ type \phi and press TAB]
+plot(sol_2; vars=syms_containing(nd_2, "x"), fmt=:png)
Example block output

Appendix: The network Laplacian $L$

The diffusion equation on a network can be rewritten as

\[\dot v_i = \sum_{j=1}^N A_{ji} v_j - d_i v_i = e_i^T A v - d_i v_i \]

where $d_i$ is the degree of node $i$ and $e_i^T$ is the $i$-th standard basis vector. Introducing the diagonal matrix $D$ that has the degree of node $i$ in its $i$-th row and the Laplacian matrix $L = D - A$ we arrive at

\[\dot v = e_i^T(A - D) v\]

and finally

\[\dot v = - L v\]

This is a linear system of ODEs and its solution is a matrix exponential. To study the asymptotic behaviour of the system it suffices to analyze the eigenspectrum of $L$. For this reason $L$ is an important construction in network science.

diff --git a/v0.8.2/heterogeneous_system/50fddf9e.svg b/v0.8.2/heterogeneous_system/50fddf9e.svg new file mode 100644 index 00000000..d5e3b213 --- /dev/null +++ b/v0.8.2/heterogeneous_system/50fddf9e.svg @@ -0,0 +1,62 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/v0.8.2/heterogeneous_system/9b6a91ac.svg b/v0.8.2/heterogeneous_system/9b6a91ac.svg new file mode 100644 index 00000000..de891518 --- /dev/null +++ b/v0.8.2/heterogeneous_system/9b6a91ac.svg @@ -0,0 +1,62 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/v0.8.2/heterogeneous_system/index.html b/v0.8.2/heterogeneous_system/index.html new file mode 100644 index 00000000..30dba550 --- /dev/null +++ b/v0.8.2/heterogeneous_system/index.html @@ -0,0 +1,58 @@ + +Heterogeneous systems · NetworkDynamics
GitHub

Modeling a heterogeneous system

One of the main purposes of NetworkDynamics.jl is to facilitate modeling coupled systems with heterogenities. This means that components can differ in their parameters as well as in their dynamics.

Heterogenous parameters

We start by setting up a simple system of Kuramoto oscillators.

using NetworkDynamics, OrdinaryDiffEq, Plots, Graphs
+
+
+N = 8
+g = watts_strogatz(N, 2, 0) # ring network
+
+function kuramoto_edge!(e, θ_s, θ_d, K, t)
+    e[1] = K * sin(θ_s[1] - θ_d[1])
+end
+
+function kuramoto_vertex!(dθ, θ, edges, ω, t)
+    dθ[1] = ω
+    sum_coupling!(dθ, edges)
+end
+
+vertex! = ODEVertex(; f=kuramoto_vertex!, dim=1, sym=[:θ])
+edge!   = StaticEdge(; f=kuramoto_edge!, dim=1)
+nd! = network_dynamics(vertex!, edge!, g);
[ Info: Reconstructing EdgeFunction with :undefined coupling type..

Introducing heterogeneous parameters is as easy as defining an array. Here the vertex parameters are heterogeneous, while the edges share the same coupling parameter K.

ω = collect(1:N) ./ N
+ω .-= sum(ω) / N
+K = 3.0
+p = (ω, K); # p[1] vertex parameters, p[2] edge parameters

Integrate and plot

x0 = collect(1:N) ./ N
+x0 .-= sum(x0) ./ N
+tspan = (0.0, 10.0)
+prob = ODEProblem(nd!, x0, tspan, p)
+sol = solve(prob, Tsit5())
+plot(sol; ylabel="θ", fmt=:png)
Example block output

Heterogeneous dynamics

Two paradigmatic modifications of the node model above are static nodes and nodes with inertia. A static node has no internal dynamics and instead fixes the variable at a constant value. A Kuramoto model with inertia consists of two internal variables leading to more complicated (and for many applications more realistic) local dynamics.

static! = StaticVertex(; f=(θ, edges, c, t) -> θ .= c, dim=1, sym=[:θ])
+
+
+function kuramoto_inertia!(dv, v, edges, P, t)
+    dv[1] = v[2]
+    dv[2] = P - 1.0 * v[2]
+    for e in edges
+        dv[2] += e[1]
+    end
+end
+
+inertia! = ODEVertex(; f=kuramoto_inertia!, dim=2, sym=[:θ, :ω]);

Since now we model a system with heterogeneous node dynamics we can no longer straightforwardly pass a single VertexFunction to network_dynamics but instead have to hand over an Array.

vertex_array    = Array{VertexFunction}([vertex! for i in 1:N])
+vertex_array[1] = static!
+vertex_array[5] = inertia! # index should correspond to the node's index in the graph
+nd_hetero! = network_dynamics(vertex_array, edge!, g);
[ Info: Reconstructing EdgeFunction with :undefined coupling type..

Now we have to take a bit more care with defining initial conditions and parameters. For the static! node the initial condition has to match its parameter. For simplicity we use the same parameters as above.

x0[1] = ω[1];

The node with inertia is two-dimensional, hence we need to specify two initial conditions. For the first dimension we keep the initial conditions from above and insert! another one into x0 at the correct index.

inertia_ic_2 = 5
+insert!(x0, 6, inertia_ic_2);

x0[1:4] holds ic for nodes 1 to 4, x0[5:6] holds the two initial conditions for node 5, x0[7:9] holds initial conditions for nodes 6 to 8.

prob_hetero = ODEProblem(nd_hetero!, x0, tspan, p)
+sol_hetero = solve(prob_hetero, Rodas4());

For clarity we plot only the variables referring to the oscillator's angle θ and color them according to their type.

membership = ones(Int64, N)
+membership[1] = 2
+membership[5] = 3
+nodecolor = [colorant"lightseagreen", colorant"orange", colorant"darkred"];
+nodefillc = reshape(nodecolor[membership], 1, N);
+
+vars = syms_containing(nd_hetero!, :θ);
+plot(sol_hetero; ylabel="θ", vars=vars, lc=nodefillc, fmt=:png)
Example block output

Components with algebraic constraints

If one of the network components contains an algebraic as well as dynamical component, then there is the option to supply a mass matrix for the given component. In general this will look as follows:

function edgeA!(de, e, v_s, v_d, p, t)
+    de[1] = f(e, v_s, v_d, p, t) # dynamic variable
+    e[2]  = g(e, v_s, v_d, p, t) # static variable
+end
+
+M = zeros(2, 2)
+M[1, 1] = 1
+
+nd_edgeA! = ODEEdge(; f=edgeA!, dim=2, coupling=:undirected, mass_matrix=M);

This handles the second equations as 0 = M[2,2] * de[2] = g(e, v_s, v_d, p, t) - e[2].

See the example kuramoto_plasticity.jl and the discussion on github for more details.

This page was generated using Literate.jl.

diff --git a/v0.8.2/index.html b/v0.8.2/index.html new file mode 100644 index 00000000..8b86439d --- /dev/null +++ b/v0.8.2/index.html @@ -0,0 +1,3 @@ + +General · NetworkDynamics
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NetworkDynamics

A package for working with dynamical systems on complex networks. NetworkDynamics.jl provides an interface between Graphs.jl and DifferentialEquations.jl. It allows to define several types of dynamic and static nodes and edges and to link them up in order to create complex network dynamics.

The behavior of a node or an edge can be described by algebraic equations, by differential algebraic equation (DAEs) in mass matrix form, by ordinary differential equations (ODE) or by delay differential equations(DDE). Stochastic ordinary differential equations (SDE) can e.g. be implemented as a two-layer network. For detail see the tutorials section.

Installation

Installation is straightforward with Julia's package manager.

(v1.5) pkg> add NetworkDynamics

PowerDynamics

PowerDynamics.jl is an open-source framework for dynamic power grid modeling and analysis build on top of NetworkDynamics.jl.

Overview

The central construction is the function network_dynamics that receives functions describing the local dynamics on the edges and nodes of a graph g as inputs, and returns a composite function compatible with the DifferentialEquations.jl calling syntax.

nd = network_dynamics(vertices!,  edges!, g)
+nd(dx, x, p, t)
diff --git a/v0.8.2/kuramoto_delay/3508ce7e.svg b/v0.8.2/kuramoto_delay/3508ce7e.svg new file mode 100644 index 00000000..29b252cc --- /dev/null +++ b/v0.8.2/kuramoto_delay/3508ce7e.svg @@ -0,0 +1,58 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/v0.8.2/kuramoto_delay/index.html b/v0.8.2/kuramoto_delay/index.html new file mode 100644 index 00000000..3389ee2e --- /dev/null +++ b/v0.8.2/kuramoto_delay/index.html @@ -0,0 +1,47 @@ + +Delay differential equations · NetworkDynamics
GitHub

Network with time delays - Kuramoto model with delayed coupling

using DelayDiffEq
+using Random
+using Graphs
+using NetworkDynamics
+using Distributions

A common modification of the Kuramoto model is to include time-lags in the coupling function. In neuroscience this may be used to account for transmission delays along synapses connecting different neurons.

In this tutorial we solve the system

\[\begin{aligned} +\dot \theta_i(t) = \omega_i + \sum_{j=1}^N \kappa_{ji} (\theta_j(t - \tau_{ji}) - \theta_i(t)) +\end{aligned}\]

where $\kappa_{ji}$ denotes the coupling strength of the edge from $j$ to $i$, $\tau_{ji}$ its time delay and $\theta_j(t - \tau_{ji})$ the past state of its source vertex

For a general introduction to solving delay differential equations in Julia see this tutorial.

To implement this in NetworkDynamics.jl a StaticDelayEdge has to be defined. Such a coupling function receives two additional arguments h_v_s and h_v_d. These are wrappers of the global history function that directly compute the history values of the local variables at the source and destination vertex respectively. The delay time should be passed like any other parameter. The idxs keyword argument of the history function can be used to access specific local variables.

function delay_coupling(e, θ_s, θ_d, h_θ_s, h_θ_d, p, t)
+    τ, κ = p
+    hist1 = h_θ_s(t - τ; idxs=1)
+    e[1] = κ * sin(hist1 - θ_d[1])
+    return nothing
+end
+
+edge = StaticDelayEdge(; f=delay_coupling, dim=1, coupling=:directed);

The vertex dynamics are simple ODE vertices.

function kuramoto_vertex(dθ, θ, edges, p, t)
+    ω = p
+    dθ[1] = ω
+    for e in edges
+        dθ[1] += e[1]
+    end
+    return nothing
+end
+
+vertex = ODEVertex(; f=kuramoto_vertex, dim=1);

For this example we use a complete graph. Bear in mind however that the data structures of Network Dynamics are best suited for sparse problems and might introduce some additional overhead for dense graphs.

N = 6
+g = SimpleDiGraph(complete_graph(N))
+nd = network_dynamics(vertex, edge, g)
+
+Random.seed!(1)
+
+ω = rand(nv(g)) # internal frequency
+τ = rand(ne(g)) # time-lag per edge
+κ = rand(ne(g)) # coupling strength per edge
+p = (ω, [τ'; κ'])
+
+θ₀ = rand(Uniform(0, 2π), N); # initial conditions

Define random initial history function

const past = rand(Uniform(0, 2π), N)
+h(p, t; idxs=nothing) = typeof(idxs) <: Number ? past[idxs] : past;

When constructing the DDEProblem lags may be specified. The solver will then track discontinuities arising from the evaluation of the history function and step to each of the discontinuities. Since each multiplication combination of the lags may be connected to a discontinuity, this may be slow if many different lags are specified.

prob = DDEProblem(nd, θ₀, h, (0.0, 1.0), p; constant_lags=τ)
+@time solve(prob, MethodOfSteps(BS3()); abstol=1e-10, reltol=1e-5); # ~50000 steps because of discontinuities
+@time solve(prob, MethodOfSteps(BS3()); abstol=1e-10, reltol=1e-5);
  4.629521 seconds (15.21 M allocations: 791.631 MiB, 6.30% gc time, 87.17% compilation time)
+  0.496183 seconds (8.79 M allocations: 415.893 MiB, 19.10% gc time)

We recommend to solve with lowered absolute and relative error tolerances, since the Kuramoto system is highly multistable and the simulations may else produce very different results.

The discontinuities arise from the initial history function and quickly get smoothed out (i.e. reduced in order) when the integration time is larger than the maximum lag. If the asymptotic behaviour is more interesting than the correct solution for a specific initial condition, it is possible to trade accuracy for computational speed by leaving the constant_lags undeclared.

fast_prob = DDEProblem(nd, θ₀, h, (0.0, 1.0), p)
+@time solve(fast_prob, MethodOfSteps(BS3()); abstol=1e-10, reltol=1e-5); # ~200 steps
+@time solve(fast_prob, MethodOfSteps(BS3()); abstol=1e-10, reltol=1e-5);
  2.762453 seconds (4.81 M allocations: 276.193 MiB, 3.56% gc time, 99.65% compilation time)
+  0.009195 seconds (312.64 k allocations: 14.739 MiB)

The MethodOfSteps algortihm extends an ODE solver to DDEs. For an overview of available solvers consult the manual of DifferentialEquations.jl. For example, for stiff systems, such as this one, it might be beneficial to use a stiff solver such as TRBDF2.

@time solve(fast_prob, MethodOfSteps(TRBDF2()); abstol=1e-10, reltol=1e-5);
+@time solve(fast_prob, MethodOfSteps(TRBDF2()); abstol=1e-10, reltol=1e-5);
 10.914104 seconds (18.90 M allocations: 1.290 GiB, 4.57% gc time, 99.87% compilation time)
+  0.010333 seconds (238.61 k allocations: 13.143 MiB)

Some further helpful comments for dealing within initial discontinuities in DDEs may be found in the manual of the Python software JiTCDDE

using Plots
+fast_prob = DDEProblem(nd, θ₀, h, (0.0, 10.0), p)
+sol = solve(fast_prob, MethodOfSteps(BS3()); abstol=1e-10, reltol=1e-5, saveat=0.01)
+plot(sol; fmt=:png)
Example block output

This page was generated using Literate.jl.

diff --git a/v0.8.2/parameters/index.html b/v0.8.2/parameters/index.html new file mode 100644 index 00000000..beb5c9fa --- /dev/null +++ b/v0.8.2/parameters/index.html @@ -0,0 +1,4 @@ + +Parameter handling · NetworkDynamics
GitHub

Parameter handling

Let nd! be an ODEFunction returned by network_dynamics, e.g.

nd! = network_dynamics(vertices!, edges!, graph)

then the behaviour of nd! has the signature (dx, x, t, p) and its behaviour changes with the type of parameters p being passed.

  • When p is an Array, a Dict, a struct... then the entire object is passed to each VertexFunction and EdgeFunction.
  • When p = (p_v, p_e) is a Tuple of two values, then the first value will be passed to all vertices and the second to all edges.
  • If p = (p_v_arr, p_e_arr) is a Tuple of two Arrays with lengths corresponding to the number of nodes and number of edges respectively, then the edges or nodes receive only the parameter with the corresponding index.
  • If all nodes and/or edges have no internal parameters the value nothing may be passed. Using nothing instead of dummy parameters is usually faster, since then less data are copied.

Another option for specifying heterogeneous parameters is to make each VertexFunction a callable struct with the parameters hardcoded as fields. This approach is used in PowerDynamics.jl. However it provides considerably less flexibility and interoperability with other packages.

For its greater speed and flexibility in modeling we recommend to use the tuple syntax.

Compatability with specific packages

Some other packages from the Julia ecosystem, e.g. DiffEqFlux, assume that the parameters p are a subtype of AbstractArray. Therefore they are not fully compatible with the tuple syntax.

However, wrapping the function in such a way that it accepts arrays of parameters that are later pasted into the tuple syntax sidesteps potential issues. Depending on the use case such a wrapper might look like this:

function nd_wrapper!(dx, x, p, t)
+  nd!(dx, x, (p, nothing), t)
+end

At the moment we recommend this way for combining NetworkDynamics and DiffEqFlux.

diff --git a/v0.8.2/search_index.js b/v0.8.2/search_index.js new file mode 100644 index 00000000..7dd922fb --- /dev/null +++ b/v0.8.2/search_index.js @@ -0,0 +1,3 @@ +var documenterSearchIndex = {"docs": +[{"location":"kuramoto_delay/#Network-with-time-delays-Kuramoto-model-with-delayed-coupling","page":"Delay differential equations","title":"Network with time delays - Kuramoto model with delayed coupling","text":"","category":"section"},{"location":"kuramoto_delay/","page":"Delay differential equations","title":"Delay differential equations","text":"using DelayDiffEq\nusing Random\nusing Graphs\nusing NetworkDynamics\nusing Distributions","category":"page"},{"location":"kuramoto_delay/","page":"Delay differential equations","title":"Delay differential equations","text":"A common modification of the Kuramoto model is to include time-lags in the coupling function. 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Bear in mind however that the data structures of Network Dynamics are best suited for sparse problems and might introduce some additional overhead for dense graphs.","category":"page"},{"location":"kuramoto_delay/","page":"Delay differential equations","title":"Delay differential equations","text":"N = 6\ng = SimpleDiGraph(complete_graph(N))\nnd = network_dynamics(vertex, edge, g)\n\nRandom.seed!(1)\n\nω = rand(nv(g)) # internal frequency\nτ = rand(ne(g)) # time-lag per edge\nκ = rand(ne(g)) # coupling strength per edge\np = (ω, [τ'; κ'])\n\nθ₀ = rand(Uniform(0, 2π), N); # initial conditions\nnothing #hide","category":"page"},{"location":"kuramoto_delay/","page":"Delay differential equations","title":"Delay differential equations","text":"Define random initial history function","category":"page"},{"location":"kuramoto_delay/","page":"Delay differential equations","title":"Delay differential equations","text":"const past = rand(Uniform(0, 2π), N)\nh(p, t; idxs=nothing) = typeof(idxs) <: Number ? past[idxs] : past;\nnothing #hide","category":"page"},{"location":"kuramoto_delay/","page":"Delay differential equations","title":"Delay differential equations","text":"When constructing the DDEProblem lags may be specified. The solver will then track discontinuities arising from the evaluation of the history function and step to each of the discontinuities. Since each multiplication combination of the lags may be connected to a discontinuity, this may be slow if many different lags are specified.","category":"page"},{"location":"kuramoto_delay/","page":"Delay differential equations","title":"Delay differential equations","text":"prob = DDEProblem(nd, θ₀, h, (0.0, 1.0), p; constant_lags=τ)\n@time solve(prob, MethodOfSteps(BS3()); abstol=1e-10, reltol=1e-5); # ~50000 steps because of discontinuities\n@time solve(prob, MethodOfSteps(BS3()); abstol=1e-10, reltol=1e-5);\nnothing #hide","category":"page"},{"location":"kuramoto_delay/","page":"Delay differential equations","title":"Delay differential equations","text":"We recommend to solve with lowered absolute and relative error tolerances, since the Kuramoto system is highly multistable and the simulations may else produce very different results.","category":"page"},{"location":"kuramoto_delay/","page":"Delay differential equations","title":"Delay differential equations","text":"The discontinuities arise from the initial history function and quickly get smoothed out (i.e. reduced in order) when the integration time is larger than the maximum lag. If the asymptotic behaviour is more interesting than the correct solution for a specific initial condition, it is possible to trade accuracy for computational speed by leaving the constant_lags undeclared.","category":"page"},{"location":"kuramoto_delay/","page":"Delay differential equations","title":"Delay differential equations","text":"fast_prob = DDEProblem(nd, θ₀, h, (0.0, 1.0), p)\n@time solve(fast_prob, MethodOfSteps(BS3()); abstol=1e-10, reltol=1e-5); # ~200 steps\n@time solve(fast_prob, MethodOfSteps(BS3()); abstol=1e-10, reltol=1e-5);\nnothing #hide","category":"page"},{"location":"kuramoto_delay/","page":"Delay differential equations","title":"Delay differential equations","text":"The MethodOfSteps algortihm extends an ODE solver to DDEs. For an overview of available solvers consult the manual of DifferentialEquations.jl. For example, for stiff systems, such as this one, it might be beneficial to use a stiff solver such as TRBDF2.","category":"page"},{"location":"kuramoto_delay/","page":"Delay differential equations","title":"Delay differential equations","text":"@time solve(fast_prob, MethodOfSteps(TRBDF2()); abstol=1e-10, reltol=1e-5);\n@time solve(fast_prob, MethodOfSteps(TRBDF2()); abstol=1e-10, reltol=1e-5);\nnothing #hide","category":"page"},{"location":"kuramoto_delay/","page":"Delay differential equations","title":"Delay differential equations","text":"Some further helpful comments for dealing within initial discontinuities in DDEs may be found in the manual of the Python software JiTCDDE","category":"page"},{"location":"kuramoto_delay/","page":"Delay differential equations","title":"Delay differential equations","text":"using Plots\nfast_prob = DDEProblem(nd, θ₀, h, (0.0, 10.0), p)\nsol = solve(fast_prob, MethodOfSteps(BS3()); abstol=1e-10, reltol=1e-5, saveat=0.01)\nplot(sol; fmt=:png)","category":"page"},{"location":"kuramoto_delay/","page":"Delay differential equations","title":"Delay differential equations","text":"","category":"page"},{"location":"kuramoto_delay/","page":"Delay differential equations","title":"Delay differential equations","text":"This page was generated using Literate.jl.","category":"page"},{"location":"parameters/#parameters","page":"Parameter handling","title":"Parameter handling","text":"","category":"section"},{"location":"parameters/","page":"Parameter handling","title":"Parameter handling","text":"Let nd! be an ODEFunction returned by network_dynamics, e.g.","category":"page"},{"location":"parameters/","page":"Parameter handling","title":"Parameter handling","text":"nd! = network_dynamics(vertices!, edges!, graph)","category":"page"},{"location":"parameters/","page":"Parameter handling","title":"Parameter handling","text":"then the behaviour of nd! has the signature (dx, x, t, p) and its behaviour changes with the type of parameters p being passed.","category":"page"},{"location":"parameters/","page":"Parameter handling","title":"Parameter handling","text":"When p is an Array, a Dict, a struct... then the entire object is passed to each VertexFunction and EdgeFunction.\nWhen p = (p_v, p_e) is a Tuple of two values, then the first value will be passed to all vertices and the second to all edges.\nIf p = (p_v_arr, p_e_arr) is a Tuple of two Arrays with lengths corresponding to the number of nodes and number of edges respectively, then the edges or nodes receive only the parameter with the corresponding index.\nIf all nodes and/or edges have no internal parameters the value nothing may be passed. Using nothing instead of dummy parameters is usually faster, since then less data are copied.","category":"page"},{"location":"parameters/","page":"Parameter handling","title":"Parameter handling","text":"Another option for specifying heterogeneous parameters is to make each VertexFunction a callable struct with the parameters hardcoded as fields. This approach is used in PowerDynamics.jl. 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Therefore they are not fully compatible with the tuple syntax.","category":"page"},{"location":"parameters/","page":"Parameter handling","title":"Parameter handling","text":"However, wrapping the function in such a way that it accepts arrays of parameters that are later pasted into the tuple syntax sidesteps potential issues. Depending on the use case such a wrapper might look like this:","category":"page"},{"location":"parameters/","page":"Parameter handling","title":"Parameter handling","text":"function nd_wrapper!(dx, x, p, t)\n nd!(dx, x, (p, nothing), t)\nend","category":"page"},{"location":"parameters/","page":"Parameter handling","title":"Parameter handling","text":"At the moment we recommend this way for combining NetworkDynamics and DiffEqFlux.","category":"page"},{"location":"accessing_edge_variables/#Accessing-internal-edge-values","page":"Accessing internal edge values","title":"Accessing internal edge values","text":"","category":"section"},{"location":"accessing_edge_variables/","page":"Accessing internal edge values","title":"Accessing internal edge values","text":"In the following, we will present two different ways how the edge values of an ODEFunction with StaticEdges can be retrieved. 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The argument x has to be the precomputed state of the system at time t, e.g. from a solution object.","category":"page"},{"location":"accessing_edge_variables/#Accessing-edge-values-using-SavingCallback","page":"Accessing internal edge values","title":"Accessing edge values using SavingCallback","text":"","category":"section"},{"location":"accessing_edge_variables/","page":"Accessing internal edge values","title":"Accessing internal edge values","text":"Instead of recomputing the values of the edge variables they can be saved in parallel to an integration with DiffEqCallbacks.","category":"page"},{"location":"accessing_edge_variables/","page":"Accessing internal edge values","title":"Accessing internal edge values","text":"using DiffEqCallbacks: SavingCallback, SavedValues\n\nsaved_values = SavedValues(Float64, Vector{Float64})\nfunction saving_func(u, t, integrator)\n edgevals = Float64[]\n for i in 1:integrator.f.f.graph_structure.num_e\n push!(edgevals, integrator.f.f.graph_data.gdb.e[i]...)\n end\n edgevals\nend\ncb = SavingCallback(saving_func, saved_values)\nsol = solve(prob, Tsit5(); callback=cb)","category":"page"},{"location":"accessing_edge_variables/","page":"Accessing internal edge values","title":"Accessing internal edge values","text":"The variables saved_values will contain the stored edge values. At the moment this method requires detailed knowledge of the internal data structures. Our plan is to simplify this in a future release.","category":"page"},{"location":"Library/#Library","page":"Library","title":"Library","text":"","category":"section"},{"location":"Library/#Public-Interface","page":"Library","title":"Public Interface","text":"","category":"section"},{"location":"Library/","page":"Library","title":"Library","text":"The following functions are designed for public use","category":"page"},{"location":"Library/","page":"Library","title":"Library","text":"","category":"page"},{"location":"Library/","page":"Library","title":"Library","text":"network_dynamics\nStaticVertex\nODEVertex\nDDEVertex\nStaticEdge\nODEEdge\nStaticDelayEdge\nsyms_containing\nidx_containing\nsum_coupling!","category":"page"},{"location":"Library/#NetworkDynamics.network_dynamics","page":"Library","title":"NetworkDynamics.network_dynamics","text":"network_dynamics(vertices!, edges!, g; parallel = false)\n\nAssembles the the dynamical equations of the network problem into an ODEFunction compatible with the DifferentialEquations.jl solvers. Takes as arguments an array of VertexFunctions vertices!, an array of EdgeFunctions edges! and a Graphs.jl object g. The optional argument parallel is a boolean value that denotes if the central loop should be executed in parallel with the number of threads set by the environment variable JULIA_NUM_THREADS.\n\n\n\n\n\n","category":"function"},{"location":"Library/#NetworkDynamics.StaticVertex","page":"Library","title":"NetworkDynamics.StaticVertex","text":"StaticVertex(; f, dim, sym)\n\nWrapper for ODEVertex with 0 mass matrix, i.e. static behaviour / algebraic constraint in mass matrix form.\n\nf describes the local behaviour at a static node and has to respect the following calling syntax\n\nf(v, edges, p, t) -> nothing\n\nHere v, p and t are the usual arguments, while edges is an arrays containing the edges for which the described vertex is the destination (in-edges for directed graphs).\n\ndim is the number of independent variables in the vertex equations and sym is an array of symbols for these variables.\n\nFor more details see the documentation.\n\n\n\n\n\n","category":"function"},{"location":"Library/#NetworkDynamics.ODEVertex","page":"Library","title":"NetworkDynamics.ODEVertex","text":"ODEVertex(; f, dim, mass_matrix, sym)\n\nWrapper that ensures compatibility of a mutating function f with the key constructor network_dynamics.\n\nf describes the local behaviour at a dynamic node and has to respect the following calling syntax\n\nf(dv, v, edges, p, t) -> nothing\n\nHere dv, v, p and t are the usual ODE arguments, while edges is an Array containing the edges for which the vertex is the destination (in-edges for directed graphs).\n\ndim is the number of independent variables in the vertex equations and sym is an array of symbols for these variables. mass_matrix is an optional argument that defaults to the identity matrix I. If a mass matrix M is given the system M * dv = f will be solved.\n\nFor more details see the documentation.\n\n\n\n\n\n","category":"type"},{"location":"Library/#NetworkDynamics.DDEVertex","page":"Library","title":"NetworkDynamics.DDEVertex","text":"DDEVertex(; f, dim, mass_matrix, sym)\n\nWrapper that ensures compatibility of a mutating function f with the key constructor network_dynamics.\n\nf describes the local behaviour at a dynamic node and has to respect the following calling syntax\n\nf(dv, v, edges, h, p, t) -> nothing\n\nHere dv, v, p and t are the usual ODE arguments, while edges is an Arry of incoming edges. h is the history array for v.\n\ndim is the number of independent variables in the edge equations and\nsym is an array of symbols for these variables.\nmass_matrix is an optional argument that defaults to the identity matrix I. If a mass matrix M is given the system M * de = f will be solved.\n\nFor more details see the documentation.\n\n\n\n\n\n","category":"type"},{"location":"Library/#NetworkDynamics.StaticEdge","page":"Library","title":"NetworkDynamics.StaticEdge","text":"StaticEdge(; f, dim, coupling, sym)\n\nWrapper that ensures compatibility of a mutating function f with the key constructor network_dynamics.\n\nf describes the local behaviour at a static edge and has to respect the following calling syntax\n\nf(e, v_s, v_t, p, t) -> nothing\n\nHere e, p and t are the usual arguments, while v_s and v_d are arrays containing the vertices which are the source and destination of the described edge.\n\ndim is the number of independent variables in the edge equations and\nsym is an array of symbols for these variables.\ncoupling is a Symbol describing if the EdgeFunction is intended for a directed graph (:directed) or for an undirected graph ({:undirected, :symmetric, :antisymmetric, :fiducial}). :directed is intended for directed graphs. :undirected is the default option and is only compatible with SimpleGraph. In this case f should specify the coupling from a source vertex to a destination vertex. :symmetric and :antisymmetric trigger performance optimizations, if f has that symmetry property. :fiducial lets the user specify both the coupling from src to dst, as well as the coupling from dst to src and is intended for advanced users, i.e. the edge gets passed a vector of 2dim edge states, where the first dim elements will be presented to the dst and the second dim elements will be presented to src,\n\nFor more details see the documentation.\n\n\n\n\n\n","category":"type"},{"location":"Library/#NetworkDynamics.ODEEdge","page":"Library","title":"NetworkDynamics.ODEEdge","text":"ODEEdge(; f, dim, mass_matrix, sym)\n\nWrapper that ensures compatibility of a mutating function f with the key constructor network_dynamics.\n\nf describes the local behaviour at a dynamic edge and has to respect the following calling syntax\n\nf(de, e, v_s, v_t, p, t) -> nothing\n\nHere de, e, p and t are the usual arguments, while v_s and v_d are arrays containing the vertices which are the source and destination of the described edge.\n\ndim is the number of independent variables in the edge equations and sym is an array of symbols for these variables. For more details see the documentation. mass_matrix is an optional argument that defaults to the identity matrix I. If a mass matrix M is given the system M * de = f will be solved.\n\nFor more details see the documentation.\n\n\n\n\n\n","category":"type"},{"location":"Library/#NetworkDynamics.StaticDelayEdge","page":"Library","title":"NetworkDynamics.StaticDelayEdge","text":"StaticDelayEdge(; f, dim, coupling, sym)\n\nLike a static edge but with extra arguments for the history of the source and destination vertices. This is NOT a DDEEdge.\n\n\n\n\n\n","category":"type"},{"location":"Library/#NetworkDynamics.syms_containing","page":"Library","title":"NetworkDynamics.syms_containing","text":"syms_containing(nd, expr)\n\nFind all symbols present in a network_dynamics that contain the string, regex or symbol expr.\n\n\n\n\n\n","category":"function"},{"location":"Library/#NetworkDynamics.idx_containing","page":"Library","title":"NetworkDynamics.idx_containing","text":"idx_containing(nd, expr)\n\nFind all indices of variables with symbols containing the string, regex or symbol expr\n\n\n\n\n\n","category":"function"},{"location":"Library/#NetworkDynamics.sum_coupling!","page":"Library","title":"NetworkDynamics.sum_coupling!","text":"sum_coupling!(e_sum, dst_edges)\n\nA small utility function for writing diffusion dynamics. It provides the sum of all incoming edges.\n\n\n\n\n\n","category":"function"},{"location":"SDEVertex/#SDE-Tutorial","page":"Stochastic differential equations","title":"SDE Tutorial","text":"","category":"section"},{"location":"SDEVertex/","page":"Stochastic differential equations","title":"Stochastic differential equations","text":"An IJulia notebook corresponding to this tutorial will be available on GitHub soon.","category":"page"},{"location":"SDEVertex/#Topics-covered-in-this-tutorial-include:","page":"Stochastic differential equations","title":"Topics covered in this tutorial include:","text":"","category":"section"},{"location":"SDEVertex/","page":"Stochastic differential equations","title":"Stochastic differential equations","text":"the swing equation\nfixpoint search of ODE systems\nconstructing an SDE problem with NetworkDynamics.jl","category":"page"},{"location":"SDEVertex/#Example:-Fluctuations-in-Power-Grids","page":"Stochastic differential equations","title":"Example: Fluctuations in Power Grids","text":"","category":"section"},{"location":"SDEVertex/","page":"Stochastic differential equations","title":"Stochastic differential equations","text":"This tutorial explains the use of stochastic differential equations (SDE's) in NetworkDynamics.jl. As an example system we will simulate fluctuations in a simple power grid model.","category":"page"},{"location":"SDEVertex/","page":"Stochastic differential equations","title":"Stochastic differential equations","text":"The phase and frequency dynamics in power grids can be modeled by the swing equation. In the complex systems community this is also known as the Kuramoto model with inertia.","category":"page"},{"location":"SDEVertex/","page":"Stochastic differential equations","title":"Stochastic differential equations","text":"beginaligned\ndot theta_i = omega_i \nM_i dot omega_i = P_i(t) - D_i omega_i - sum_i=1^N K_ij sin(theta_i - theta_j)\nendaligned","category":"page"},{"location":"SDEVertex/","page":"Stochastic differential equations","title":"Stochastic differential equations","text":"Here, M_i and D_i are inertia and damping parameters, K_ij is the power capacitiy of the line and P_i(t) is the power infeed at the node. We assume that this power can be separated into a constant power setpoint P^circ_i and a stochastic fluctuation.","category":"page"},{"location":"SDEVertex/","page":"Stochastic differential equations","title":"Stochastic differential equations","text":"beginaligned\nP_i(t) = P^circ_i + sigma_i cdot xi_i(t)\nendaligned","category":"page"},{"location":"SDEVertex/","page":"Stochastic differential equations","title":"Stochastic differential equations","text":"In this tutorial we assume the fluctuations xi_i(t) to be white Gaussian noise. Separating the deterministic and stochastic part, we can write this problem as a stochastic differential equation.","category":"page"},{"location":"SDEVertex/","page":"Stochastic differential equations","title":"Stochastic differential equations","text":"beginaligned\ndomega_i = leftP_i - D_i omega_i - sum_i=1^N K_ij sin(theta_i - theta_j)right dt + sigma_i dW_i\nendaligned","category":"page"},{"location":"SDEVertex/","page":"Stochastic differential equations","title":"Stochastic differential equations","text":"Here, dW_i = xi_i dt is the infinitesimal increment of the Wiener process. Problems of this type can be numerically solved in Julia as described in the SDE Tutorial of DifferentialEquations.","category":"page"},{"location":"SDEVertex/#Implementing-the-Swing-Equation","page":"Stochastic differential equations","title":"Implementing the Swing Equation","text":"","category":"section"},{"location":"SDEVertex/","page":"Stochastic differential equations","title":"Stochastic differential equations","text":"First we will implement the node and edge functions for the deterministic case without the fluctuations. We assume the inertia and damping parameters to be M_i = 10 and D_i = 01.","category":"page"},{"location":"SDEVertex/","page":"Stochastic differential equations","title":"Stochastic differential equations","text":"function swing_equation!(dv, v, edges, P, t)\n dv[1] = v[2]\n dv[2] = P - 0.1 * v[2]\n for e in edges\n dv[2] += e[1]\n end\nend\n\nfunction powerflow!(e, v_s, v_d, K, t)\n e[1] = K * sin(v_s[1] - v_d[1])\nend\nnothing # hide","category":"page"},{"location":"SDEVertex/#Contructing-the-Deterministic-Dynamics","page":"Stochastic differential equations","title":"Contructing the Deterministic Dynamics","text":"","category":"section"},{"location":"SDEVertex/","page":"Stochastic differential equations","title":"Stochastic differential equations","text":"For the graph structure we will use a simple 4 node ring network.","category":"page"},{"location":"SDEVertex/","page":"Stochastic differential equations","title":"Stochastic differential equations","text":"using Graphs\ng = watts_strogatz(4, 2, 0.0)\nnothing # hide","category":"page"},{"location":"SDEVertex/","page":"Stochastic differential equations","title":"Stochastic differential equations","text":"Then we can construct the ODEFunction of the deterministic system by using network_dynamics().","category":"page"},{"location":"SDEVertex/","page":"Stochastic differential equations","title":"Stochastic differential equations","text":"using NetworkDynamics\n\nswing_vertex = ODEVertex(; f=swing_equation!, dim=2, sym=[:θ, :ω])\npowerflow_edge = StaticEdge(; f=powerflow!, dim=1)\n\nnd = network_dynamics(swing_vertex, powerflow_edge, g)\nnothing # hide","category":"page"},{"location":"SDEVertex/#Fixpoint-Search","page":"Stochastic differential equations","title":"Fixpoint Search","text":"","category":"section"},{"location":"SDEVertex/","page":"Stochastic differential equations","title":"Stochastic differential equations","text":"Now we need to define the dynamic parameters of vertices and edges. For simplicity we assume homogeneous capacities on the lines. For the nodes we assume half of them to be net producers (P = 10) and half of them to be net consumers (P = -10) of power.","category":"page"},{"location":"SDEVertex/","page":"Stochastic differential equations","title":"Stochastic differential equations","text":"K = 6.0\nP = [1.0, -1.0, 1.0, -1.0]\np = (P, K)\nnothing # hide","category":"page"},{"location":"SDEVertex/","page":"Stochastic differential equations","title":"Stochastic differential equations","text":"We want to simulate fluctuations around an equilibrium state of our model system. Therefore, we need to find a fixpoint of the determinitic system which can be done by using the utility function find_fixpoint(). As an initial guess we take all variables equal to zero.","category":"page"},{"location":"SDEVertex/","page":"Stochastic differential equations","title":"Stochastic differential equations","text":"u0 = find_fixpoint(nd, p, zeros(8))\n\nusing StochasticDiffEq, OrdinaryDiffEq\node_prob = ODEProblem(nd, u0, (0.0, 500.0), p)\node_sol = solve(ode_prob, Tsit5())\n\nusing Plots, LaTeXStrings\nplot(ode_sol; vars=syms_containing(nd, \"ω\"), ylims=(-1.0, 1.0), ylabel=L\"\\omega\", legend=false, fmt=:png)","category":"page"},{"location":"SDEVertex/","page":"Stochastic differential equations","title":"Stochastic differential equations","text":"We see that this is in fact a fixpoint solution. We will later use this as an initial condition for the numerical integration of the SDE system.","category":"page"},{"location":"SDEVertex/#Adding-a-Stochastic-Layer","page":"Stochastic differential equations","title":"Adding a Stochastic Layer","text":"","category":"section"},{"location":"SDEVertex/","page":"Stochastic differential equations","title":"Stochastic differential equations","text":"For adding the stochastic part of the dynamics we have to define a second graph layer. In our example, the fluctuations at different nodes are independent of each other. Therefore, we define a second graph with the same number of vertices but without any edges.","category":"page"},{"location":"SDEVertex/","page":"Stochastic differential equations","title":"Stochastic differential equations","text":"h = SimpleGraph(4, 0)\nnothing # hide","category":"page"},{"location":"SDEVertex/","page":"Stochastic differential equations","title":"Stochastic differential equations","text":"The dynamics at the nodes has to have the same dimension as in the deterministic case. In our example we only have fluctuations in the second variable.","category":"page"},{"location":"SDEVertex/","page":"Stochastic differential equations","title":"Stochastic differential equations","text":"function fluctuation!(dx, x, edges, p, t)\n dx[1] = 0.0\n dx[2] = 0.05\nend\nnothing # hide","category":"page"},{"location":"SDEVertex/","page":"Stochastic differential equations","title":"Stochastic differential equations","text":"Now we can construct the dynamics of the second layer by using network_dynamics(). Since the graph structure of the stochastic layer has no edges we can take the edge function of the deterministic case as a placeholder.","category":"page"},{"location":"SDEVertex/","page":"Stochastic differential equations","title":"Stochastic differential equations","text":"fluctuation_vertex = ODEVertex(; f=fluctuation!, dim=2)\nnd_noise = network_dynamics(fluctuation_vertex, powerflow_edge, h)\nnothing # hide","category":"page"},{"location":"SDEVertex/#Simulating-the-SDE","page":"Stochastic differential equations","title":"Simulating the SDE","text":"","category":"section"},{"location":"SDEVertex/","page":"Stochastic differential equations","title":"Stochastic differential equations","text":"Finally, we can create an SDEProblem and solve it with DifferentialEquations.","category":"page"},{"location":"SDEVertex/","page":"Stochastic differential equations","title":"Stochastic differential equations","text":"sde_prob = SDEProblem(nd, nd_noise, u0, (0.0, 500.0), p)\nsde_sol = solve(sde_prob, SOSRA())\nplot(sde_sol; vars=syms_containing(nd, \"ω\"), ylims=(-1.0, 1.0), ylabel=L\"\\omega\", legend=false, fmt=:png)","category":"page"},{"location":"SDEVertex/","page":"Stochastic differential equations","title":"Stochastic differential equations","text":"More details on SDE problems, e.g. how to include correlations or how to define an EnsembleProblem, can be found in the documentation of DifferentialEquations.","category":"page"},{"location":"getting_started_with_network_dynamics/#getting_started","page":"Getting started","title":"Network diffusion","text":"","category":"section"},{"location":"getting_started_with_network_dynamics/","page":"Getting started","title":"Getting started","text":"This introductory example explains the use of the basic types and constructors in NetworkDynamics.jl by modeling a simple diffusion on an undirected network. A corresponding IJulia notebook is available on GitHub.","category":"page"},{"location":"getting_started_with_network_dynamics/#Theoretical-background","page":"Getting started","title":"Theoretical background","text":"","category":"section"},{"location":"getting_started_with_network_dynamics/","page":"Getting started","title":"Getting started","text":"Diffusion processes are relevant for phenomena as diverse as heat conduction, electrical currents, and random walks. Generally speaking they describe the tendency of systems to evolve towards a state of equally distributed heat, charge or concentration. In such system the local temperature (or concentration) changes according to its difference with its neighborhood, i.e. the temperature gradient.","category":"page"},{"location":"getting_started_with_network_dynamics/","page":"Getting started","title":"Getting started","text":"Let g be a graph with N nodes and adjacency matrix A. Let v = (v_1 dots v_n) be a vector of (abstract) temperatures or concentrations at each node i = 1 dots N. Then the rate of change of state v_i is described by its difference with its neighbors and we obtain the following ordinary differential equation","category":"page"},{"location":"getting_started_with_network_dynamics/","page":"Getting started","title":"Getting started","text":"dot v_i = sum_j=1^N A_ji (v_j - v_i)","category":"page"},{"location":"getting_started_with_network_dynamics/","page":"Getting started","title":"Getting started","text":"The sum on the right hand side plays the role of a (discrete) gradient. If the temperature at node i is higher than at its neighboring node j it will decrease along that edge.","category":"page"},{"location":"getting_started_with_network_dynamics/#Modeling-diffusion-in-NetworkDynamics.jl","page":"Getting started","title":"Modeling diffusion in NetworkDynamics.jl","text":"","category":"section"},{"location":"getting_started_with_network_dynamics/","page":"Getting started","title":"Getting started","text":"From the above considerations we see that in this model the nodes do not have any internal dynamics - if a node was disconnected from the rest of the network its state would never change, since then A_ji = 0 forall j and hence dot v_i = 0. This means that the evolution of a node depends only on the interaction with its neighbors. In NetworkDynamics.jl, interactions with neighbors are described by equations for the edges.","category":"page"},{"location":"getting_started_with_network_dynamics/","page":"Getting started","title":"Getting started","text":"function diffusionedge!(e, v_s, v_d, p, t)\n # usually e, v_s, v_d are arrays, hence we use the broadcasting operator .\n e .= v_s - v_d\n nothing\nend\nnothing # hide","category":"page"},{"location":"getting_started_with_network_dynamics/","page":"Getting started","title":"Getting started","text":"The function diffusionedge! takes as inputs the current state of the edge e, its source vertex v_s, its destination vertex v_d, a vector of parameters p and the time t. In order to comply with the syntax of NetworkDynamics.jl we always have to define functions for static edges with exactly these arguments, even though we do not need p and t for the diffusion example.","category":"page"},{"location":"getting_started_with_network_dynamics/","page":"Getting started","title":"Getting started","text":"diffusionedge! is called a mutating function, since it modifies (or mutates) one of its inputs, namely the edge state e. As a convention in Julia names of mutating functions end with an !. The use of mutating functions reduces allocations and thereby speeds up computations. After the function call the edge's value e equals the difference between its source and its destination vertex (i.e. the discrete gradient along that edge).","category":"page"},{"location":"getting_started_with_network_dynamics/","page":"Getting started","title":"Getting started","text":"The contributions of the different edges are then summed up in each vertex.","category":"page"},{"location":"getting_started_with_network_dynamics/","page":"Getting started","title":"Getting started","text":"function diffusionvertex!(dv, v, edges, p, t)\n # usually v, edges are arrays, hence we use the broadcasting operator .\n dv .= 0.0\n for e in edges\n dv .+= e\n end\n nothing\nend\nnothing # hide","category":"page"},{"location":"getting_started_with_network_dynamics/","page":"Getting started","title":"Getting started","text":"Just like above the input arguments v, edges, p, t are mandatory for the syntax of vertex functions. The additional input dv corresponding to the derivative of the vertex' state is mandatory for vertices described by ordinary differential equations.","category":"page"},{"location":"getting_started_with_network_dynamics/","page":"Getting started","title":"Getting started","text":"For undirected graphs, the edgefunction! specifies the coupling from a source- to a destination vertex. The contributions of the connected edges are added to the destination of each edge, as defined in the diffusionvertex! function in the for loop.","category":"page"},{"location":"getting_started_with_network_dynamics/#Constructing-the-network","page":"Getting started","title":"Constructing the network","text":"","category":"section"},{"location":"getting_started_with_network_dynamics/","page":"Getting started","title":"Getting started","text":"With the preliminaries out of the way, it only takes a few steps to assemble the network dynamics.","category":"page"},{"location":"getting_started_with_network_dynamics/","page":"Getting started","title":"Getting started","text":"using Graphs\n\nN = 20 # number of nodes\nk = 4 # average degree\ng = barabasi_albert(N, k) # a little more exciting than a bare random graph\n\nnothing # hide","category":"page"},{"location":"getting_started_with_network_dynamics/","page":"Getting started","title":"Getting started","text":"The Barabási–Albert model generates a scale-free random graph.","category":"page"},{"location":"getting_started_with_network_dynamics/","page":"Getting started","title":"Getting started","text":"using NetworkDynamics\n\nnd_diffusion_vertex = ODEVertex(; f=diffusionvertex!, dim=1)\nnd_diffusion_edge = StaticEdge(; f=diffusionedge!, dim=1)\n\nnd = network_dynamics(nd_diffusion_vertex, nd_diffusion_edge, g)\n\nnothing # hide","category":"page"},{"location":"getting_started_with_network_dynamics/","page":"Getting started","title":"Getting started","text":"ODEVertex and StaticEdge are functions wrappers that equip the functions we defined above with additional information like dim and return objects of type VertexFunction and EdgeFunction. Then the key constructor network_dynamics combines them with the topological information contained in the graph g and returns an ODEFunction compatible with the solvers of DifferentialEquations.jl. The keyword dim specifies the number of variables at each edge or node.","category":"page"},{"location":"getting_started_with_network_dynamics/","page":"Getting started","title":"Getting started","text":"using OrdinaryDiffEq\n\nx0 = randn(N) # random initial conditions\node_prob = ODEProblem(nd, x0, (0.0, 4.0))\nsol = solve(ode_prob, Tsit5());\nnothing # hide","category":"page"},{"location":"getting_started_with_network_dynamics/","page":"Getting started","title":"Getting started","text":"We are solving the diffusion problem on the time interval 0 4 with the Tsit5() algorithm, which is recommended by the authors of DifferentialEquations.jl for most non-stiff problems.","category":"page"},{"location":"getting_started_with_network_dynamics/","page":"Getting started","title":"Getting started","text":"using Plots\nplot(sol; vars=syms_containing(nd, \"v\"), fmt=:png)","category":"page"},{"location":"getting_started_with_network_dynamics/","page":"Getting started","title":"Getting started","text":"The plotting is straightforward. The vars keyword allows us to pass a list of indices or symbols specifiying the variables we want to plot. Symbols can be thought of as names given to the interal variables of an ODEFunction, much like the variables x or phi in mathematical notation. The default symbol for vertex variables is v, however we are free to specify other symbols by passing them to the ODEVertex constructor.","category":"page"},{"location":"getting_started_with_network_dynamics/","page":"Getting started","title":"Getting started","text":"syms_containing is a helper function that returns all symbols of an ODEProblem containg a specific string (or symbol).","category":"page"},{"location":"getting_started_with_network_dynamics/","page":"Getting started","title":"Getting started","text":"To illustrate a very simple multi-dimensional case, in the following we simulate two independent diffusions on an identical network. The first uses the symbol x and is started with initial conditions drawn from the standard normal distribution N(01), the second uses the symbol ϕ with squared standard normal inital conditions.","category":"page"},{"location":"getting_started_with_network_dynamics/","page":"Getting started","title":"Getting started","text":"The symbols have to be passed with the keyword sym to ODEVertex.","category":"page"},{"location":"getting_started_with_network_dynamics/","page":"Getting started","title":"Getting started","text":"N = 10 # number of nodes\nk = 4 # average degree\ng = barabasi_albert(N, k) # a little more exciting than a bare random graph\n\n# We will have two independent diffusions on the network, hence dim = 2\nnd_diffusion_vertex_2 = ODEVertex(; f=diffusionvertex!, dim=2, sym=[:x, :ϕ])\nnd_diffusion_edge_2 = StaticEdge(; f=diffusionedge!, dim=2)\nnd_2 = network_dynamics(nd_diffusion_vertex_2, nd_diffusion_edge_2, g)\n\nx0_2 = vec(transpose([randn(N) .^ 2 randn(N)])) # x ~ N(0,1)^2; ϕ ~ N(0,1)\node_prob_2 = ODEProblem(nd_2, x0_2, (0.0, 3.0))\nsol_2 = solve(ode_prob_2, Tsit5());\n\n\n# Try plotting the variables ϕ_i yourself. [To write ϕ type \\phi and press TAB]\nplot(sol_2; vars=syms_containing(nd_2, \"x\"), fmt=:png)","category":"page"},{"location":"getting_started_with_network_dynamics/#Appendix:-The-network-Laplacian-L","page":"Getting started","title":"Appendix: The network Laplacian L","text":"","category":"section"},{"location":"getting_started_with_network_dynamics/","page":"Getting started","title":"Getting started","text":"The diffusion equation on a network can be rewritten as","category":"page"},{"location":"getting_started_with_network_dynamics/","page":"Getting started","title":"Getting started","text":"dot v_i = sum_j=1^N A_ji v_j - d_i v_i = e_i^T A v - d_i v_i ","category":"page"},{"location":"getting_started_with_network_dynamics/","page":"Getting started","title":"Getting started","text":"where d_i is the degree of node i and e_i^T is the i-th standard basis vector. Introducing the diagonal matrix D that has the degree of node i in its i-th row and the Laplacian matrix L = D - A we arrive at","category":"page"},{"location":"getting_started_with_network_dynamics/","page":"Getting started","title":"Getting started","text":"dot v = e_i^T(A - D) v","category":"page"},{"location":"getting_started_with_network_dynamics/","page":"Getting started","title":"Getting started","text":"and finally","category":"page"},{"location":"getting_started_with_network_dynamics/","page":"Getting started","title":"Getting started","text":"dot v = - L v","category":"page"},{"location":"getting_started_with_network_dynamics/","page":"Getting started","title":"Getting started","text":"This is a linear system of ODEs and its solution is a matrix exponential. To study the asymptotic behaviour of the system it suffices to analyze the eigenspectrum of L. For this reason L is an important construction in network science.","category":"page"},{"location":"Multithreading/#Multi-Threading","page":"Multi-Threading","title":"Multi-Threading","text":"","category":"section"},{"location":"Multithreading/","page":"Multi-Threading","title":"Multi-Threading","text":"Since version 0.3.0 multi-threading via the Threads.@threads macro is possible. This allows julia to integrate different nodes and edges in different threads, and can lead to significant performance gains on parallel architectures. To enable multi-threading call network_dynamics with the keyword argument parallel=true.","category":"page"},{"location":"Multithreading/","page":"Multi-Threading","title":"Multi-Threading","text":"network_dynamics(vertices!, edges!, graph; parallel=true)","category":"page"},{"location":"Multithreading/","page":"Multi-Threading","title":"Multi-Threading","text":"In order for this to take effect, multiple threads have to be available. This is achieved by setting the environment variable JULIA_NUM_THREADS before starting Julia. To start Julia from a bash shell and with 4 threads use:","category":"page"},{"location":"Multithreading/","page":"Multi-Threading","title":"Multi-Threading","text":"$ env JULIA_NUM_THREADS=4 julia","category":"page"},{"location":"Multithreading/","page":"Multi-Threading","title":"Multi-Threading","text":"If you are using Juno for the Atom text editor JULIA_NUM_THREADS is set to the number of physical cores of your processor by default. This is also the number of threads we recommend to use.","category":"page"},{"location":"Multithreading/","page":"Multi-Threading","title":"Multi-Threading","text":"note: Note\nThe thread handling causes an overhead in the order of 20 μs per call to the ODE function which might impair performance on small networks (<100 nodes) or on single core machines. In theses cases network_dynamics can be called without any additional arguments, since parallel defaults to false.","category":"page"},{"location":"Multithreading/","page":"Multi-Threading","title":"Multi-Threading","text":"For more information on setting environment varibales see the Julia documentation.","category":"page"},{"location":"heterogeneous_system/#Modeling-a-heterogeneous-system","page":"Heterogeneous systems","title":"Modeling a heterogeneous system","text":"","category":"section"},{"location":"heterogeneous_system/","page":"Heterogeneous systems","title":"Heterogeneous systems","text":"One of the main purposes of NetworkDynamics.jl is to facilitate modeling coupled systems with heterogenities. This means that components can differ in their parameters as well as in their dynamics.","category":"page"},{"location":"heterogeneous_system/#Heterogenous-parameters","page":"Heterogeneous systems","title":"Heterogenous parameters","text":"","category":"section"},{"location":"heterogeneous_system/","page":"Heterogeneous systems","title":"Heterogeneous systems","text":"We start by setting up a simple system of Kuramoto oscillators.","category":"page"},{"location":"heterogeneous_system/","page":"Heterogeneous systems","title":"Heterogeneous systems","text":"using NetworkDynamics, OrdinaryDiffEq, Plots, Graphs\n\n\nN = 8\ng = watts_strogatz(N, 2, 0) # ring network\n\nfunction kuramoto_edge!(e, θ_s, θ_d, K, t)\n e[1] = K * sin(θ_s[1] - θ_d[1])\nend\n\nfunction kuramoto_vertex!(dθ, θ, edges, ω, t)\n dθ[1] = ω\n sum_coupling!(dθ, edges)\nend\n\nvertex! = ODEVertex(; f=kuramoto_vertex!, dim=1, sym=[:θ])\nedge! = StaticEdge(; f=kuramoto_edge!, dim=1)\nnd! = network_dynamics(vertex!, edge!, g);\nnothing #hide","category":"page"},{"location":"heterogeneous_system/","page":"Heterogeneous systems","title":"Heterogeneous systems","text":"Introducing heterogeneous parameters is as easy as defining an array. Here the vertex parameters are heterogeneous, while the edges share the same coupling parameter K.","category":"page"},{"location":"heterogeneous_system/","page":"Heterogeneous systems","title":"Heterogeneous systems","text":"ω = collect(1:N) ./ N\nω .-= sum(ω) / N\nK = 3.0\np = (ω, K); # p[1] vertex parameters, p[2] edge parameters\nnothing #hide","category":"page"},{"location":"heterogeneous_system/","page":"Heterogeneous systems","title":"Heterogeneous systems","text":"Integrate and plot","category":"page"},{"location":"heterogeneous_system/","page":"Heterogeneous systems","title":"Heterogeneous systems","text":"x0 = collect(1:N) ./ N\nx0 .-= sum(x0) ./ N\ntspan = (0.0, 10.0)\nprob = ODEProblem(nd!, x0, tspan, p)\nsol = solve(prob, Tsit5())\nplot(sol; ylabel=\"θ\", fmt=:png)","category":"page"},{"location":"heterogeneous_system/#Heterogeneous-dynamics","page":"Heterogeneous systems","title":"Heterogeneous dynamics","text":"","category":"section"},{"location":"heterogeneous_system/","page":"Heterogeneous systems","title":"Heterogeneous systems","text":"Two paradigmatic modifications of the node model above are static nodes and nodes with inertia. A static node has no internal dynamics and instead fixes the variable at a constant value. A Kuramoto model with inertia consists of two internal variables leading to more complicated (and for many applications more realistic) local dynamics.","category":"page"},{"location":"heterogeneous_system/","page":"Heterogeneous systems","title":"Heterogeneous systems","text":"static! = StaticVertex(; f=(θ, edges, c, t) -> θ .= c, dim=1, sym=[:θ])\n\n\nfunction kuramoto_inertia!(dv, v, edges, P, t)\n dv[1] = v[2]\n dv[2] = P - 1.0 * v[2]\n for e in edges\n dv[2] += e[1]\n end\nend\n\ninertia! = ODEVertex(; f=kuramoto_inertia!, dim=2, sym=[:θ, :ω]);\nnothing #hide","category":"page"},{"location":"heterogeneous_system/","page":"Heterogeneous systems","title":"Heterogeneous systems","text":"Since now we model a system with heterogeneous node dynamics we can no longer straightforwardly pass a single VertexFunction to network_dynamics but instead have to hand over an Array.","category":"page"},{"location":"heterogeneous_system/","page":"Heterogeneous systems","title":"Heterogeneous systems","text":"vertex_array = Array{VertexFunction}([vertex! for i in 1:N])\nvertex_array[1] = static!\nvertex_array[5] = inertia! # index should correspond to the node's index in the graph\nnd_hetero! = network_dynamics(vertex_array, edge!, g);\nnothing #hide","category":"page"},{"location":"heterogeneous_system/","page":"Heterogeneous systems","title":"Heterogeneous systems","text":"Now we have to take a bit more care with defining initial conditions and parameters. For the static! node the initial condition has to match its parameter. For simplicity we use the same parameters as above.","category":"page"},{"location":"heterogeneous_system/","page":"Heterogeneous systems","title":"Heterogeneous systems","text":"x0[1] = ω[1];\nnothing #hide","category":"page"},{"location":"heterogeneous_system/","page":"Heterogeneous systems","title":"Heterogeneous systems","text":"The node with inertia is two-dimensional, hence we need to specify two initial conditions. For the first dimension we keep the initial conditions from above and insert! another one into x0 at the correct index.","category":"page"},{"location":"heterogeneous_system/","page":"Heterogeneous systems","title":"Heterogeneous systems","text":"inertia_ic_2 = 5\ninsert!(x0, 6, inertia_ic_2);\nnothing #hide","category":"page"},{"location":"heterogeneous_system/","page":"Heterogeneous systems","title":"Heterogeneous systems","text":"x0[1:4] holds ic for nodes 1 to 4, x0[5:6] holds the two initial conditions for node 5, x0[7:9] holds initial conditions for nodes 6 to 8.","category":"page"},{"location":"heterogeneous_system/","page":"Heterogeneous systems","title":"Heterogeneous systems","text":"prob_hetero = ODEProblem(nd_hetero!, x0, tspan, p)\nsol_hetero = solve(prob_hetero, Rodas4());\nnothing #hide","category":"page"},{"location":"heterogeneous_system/","page":"Heterogeneous systems","title":"Heterogeneous systems","text":"For clarity we plot only the variables referring to the oscillator's angle θ and color them according to their type.","category":"page"},{"location":"heterogeneous_system/","page":"Heterogeneous systems","title":"Heterogeneous systems","text":"membership = ones(Int64, N)\nmembership[1] = 2\nmembership[5] = 3\nnodecolor = [colorant\"lightseagreen\", colorant\"orange\", colorant\"darkred\"];\nnodefillc = reshape(nodecolor[membership], 1, N);\n\nvars = syms_containing(nd_hetero!, :θ);\nplot(sol_hetero; ylabel=\"θ\", vars=vars, lc=nodefillc, fmt=:png)","category":"page"},{"location":"heterogeneous_system/#Components-with-algebraic-constraints","page":"Heterogeneous systems","title":"Components with algebraic constraints","text":"","category":"section"},{"location":"heterogeneous_system/","page":"Heterogeneous systems","title":"Heterogeneous systems","text":"If one of the network components contains an algebraic as well as dynamical component, then there is the option to supply a mass matrix for the given component. In general this will look as follows:","category":"page"},{"location":"heterogeneous_system/","page":"Heterogeneous systems","title":"Heterogeneous systems","text":"function edgeA!(de, e, v_s, v_d, p, t)\n de[1] = f(e, v_s, v_d, p, t) # dynamic variable\n e[2] = g(e, v_s, v_d, p, t) # static variable\nend\n\nM = zeros(2, 2)\nM[1, 1] = 1\n\nnd_edgeA! = ODEEdge(; f=edgeA!, dim=2, coupling=:undirected, mass_matrix=M);\nnothing #hide","category":"page"},{"location":"heterogeneous_system/","page":"Heterogeneous systems","title":"Heterogeneous systems","text":"This handles the second equations as 0 = M[2,2] * de[2] = g(e, v_s, v_d, p, t) - e[2].","category":"page"},{"location":"heterogeneous_system/","page":"Heterogeneous systems","title":"Heterogeneous systems","text":"See the example kuramoto_plasticity.jl and the discussion on github for more details.","category":"page"},{"location":"heterogeneous_system/","page":"Heterogeneous systems","title":"Heterogeneous systems","text":"","category":"page"},{"location":"heterogeneous_system/","page":"Heterogeneous systems","title":"Heterogeneous systems","text":"This page was generated using Literate.jl.","category":"page"},{"location":"directed_and_weighted_graphs/#Neurodynamic-model-of-synchronization-in-the-human-brain","page":"Directed and weighted graphs","title":"Neurodynamic model of synchronization in the human brain","text":"","category":"section"},{"location":"directed_and_weighted_graphs/#Topics-covered-in-this-tutorial-include:","page":"Directed and weighted graphs","title":"Topics covered in this tutorial include:","text":"","category":"section"},{"location":"directed_and_weighted_graphs/","page":"Directed and weighted graphs","title":"Directed and weighted graphs","text":"constructing a directed, weighted graph from data\nsome useful macros\nparameter handling\nstiff equations","category":"page"},{"location":"directed_and_weighted_graphs/#The-FitzHugh-Nagumo-model","page":"Directed and weighted graphs","title":"The FitzHugh-Nagumo model","text":"","category":"section"},{"location":"directed_and_weighted_graphs/","page":"Directed and weighted graphs","title":"Directed and weighted graphs","text":"Dynamics of spiking neurons have been described in a simplified manner by the FitzHugh-Nagumo model.","category":"page"},{"location":"directed_and_weighted_graphs/","page":"Directed and weighted graphs","title":"Directed and weighted graphs","text":"beginaligned\nvarepsilon dot u = u - fracu^33 - v \ndot v = u + a\nendaligned","category":"page"},{"location":"directed_and_weighted_graphs/","page":"Directed and weighted graphs","title":"Directed and weighted graphs","text":"Here u is a fast, excitatory variable corresponding to the membrane potential and v is a slower, inhibitory varibale. varepsilon is a parameter separating these time-scales, and a is a control parameter.","category":"page"},{"location":"directed_and_weighted_graphs/","page":"Directed and weighted graphs","title":"Directed and weighted graphs","text":"In simplified models of the brain, such relaxation oscillators may be used to model individual neurons, clusters of neurons or even larger areas in the brain. The FitzHugh-Nagumo model has been widely used for studying synchronization in neuronal activity, which in turn has been connected to physiological phenomena such as epileptic seizures.","category":"page"},{"location":"directed_and_weighted_graphs/#Coupling-relaxation-oscillators","page":"Directed and weighted graphs","title":"Coupling relaxation oscillators","text":"","category":"section"},{"location":"directed_and_weighted_graphs/","page":"Directed and weighted graphs","title":"Directed and weighted graphs","text":"While different coupling schemes for FitzHugh-Nagumo oscillators have been proposed, in this tutorial we focus on coupling of the excitatory variables via electrical gap junctions, as described by the following system of equations.","category":"page"},{"location":"directed_and_weighted_graphs/","page":"Directed and weighted graphs","title":"Directed and weighted graphs","text":"beginaligned\nvarepsilon dot u_i = u_i - fracu_i^33 - v_i - sigma sum_j=1^N G_ij(u_i - u_j) \ndot v_i = u_i + a\nendaligned","category":"page"},{"location":"directed_and_weighted_graphs/","page":"Directed and weighted graphs","title":"Directed and weighted graphs","text":"This is a simple diffusive coupling mediated by the difference between activation potentials in pairs of neurons. A similar coupling term was introduced in the \"getting started\" tutorial.","category":"page"},{"location":"directed_and_weighted_graphs/#The-network-topology-a-brain-atlas","page":"Directed and weighted graphs","title":"The network topology - a brain atlas","text":"","category":"section"},{"location":"directed_and_weighted_graphs/","page":"Directed and weighted graphs","title":"Directed and weighted graphs","text":"In the following we will use a directed and weigthed network encoding the strength and directionality of coupling between 90 different areas of the brain [Nathalie Tzourio-Mazoyer et al., 2002, Neuroimage].","category":"page"},{"location":"directed_and_weighted_graphs/","page":"Directed and weighted graphs","title":"Directed and weighted graphs","text":"The network weight matrix is given as a text file containing 90 lines with 90 numbers representing the coupling strength and separated by commas ,. The data can be conveniently read into a matrix with the DelimitedFiles module.","category":"page"},{"location":"directed_and_weighted_graphs/","page":"Directed and weighted graphs","title":"Directed and weighted graphs","text":"using DelimitedFiles\n# adjust the load path for your filesystem!\nG = readdlm(joinpath(@__DIR__, \"../../examples/Norm_G_DTI.txt\"), ',', Float64, '\\n')\nnothing # hide","category":"page"},{"location":"directed_and_weighted_graphs/","page":"Directed and weighted graphs","title":"Directed and weighted graphs","text":"The data structure for directed, weighted graphs is provided by the package SimpleWeightedGraphs.jl which is based on Graphs.jl.","category":"page"},{"location":"directed_and_weighted_graphs/","page":"Directed and weighted graphs","title":"Directed and weighted graphs","text":"using SimpleWeightedGraphs, Graphs\n\n# First we construct a weighted, directed graph\ng_weighted = SimpleWeightedDiGraph(G)\n\n# For later use we extract the edge.weight attributes\n# . is the broadcasting operator and gets the attribute :weight for every edge\nedge_weights = getfield.(collect(edges(g_weighted)), :weight)\n\n# we promote the g_weighted graph as a directed graph (weights of the edges are included in parameters)\ng_directed = SimpleDiGraph(g_weighted)\n\nnothing # hide","category":"page"},{"location":"directed_and_weighted_graphs/#Setting-up-the-ODEProblem","page":"Directed and weighted graphs","title":"Setting up the ODEProblem","text":"","category":"section"},{"location":"directed_and_weighted_graphs/","page":"Directed and weighted graphs","title":"Directed and weighted graphs","text":"Defining VertexFunction and EdgeFunction is similar to the example before. The macro Base.@propagate_inbounds tells the compiler to inline the function and propagate the inbounds context. For more details see the julia documentation.","category":"page"},{"location":"directed_and_weighted_graphs/","page":"Directed and weighted graphs","title":"Directed and weighted graphs","text":"using NetworkDynamics\n\nBase.@propagate_inbounds function fhn_electrical_vertex!(dv, v, edges, p, t)\n dv[1] = v[1] - v[1]^3 / 3 - v[2]\n dv[2] = (v[1] - a) * ϵ\n for e in edges\n dv[1] += e[1]\n end\n nothing\nend\n\nBase.@propagate_inbounds function electrical_edge!(e, v_s, v_d, p, t)\n e[1] = p * (v_s[1] - v_d[1]) # * σ\n nothing\nend\n\nodeelevertex = ODEVertex(; f=fhn_electrical_vertex!, dim=2, sym=[:u, :v]);\nelectricaledge = StaticEdge(; f=electrical_edge!, dim=1, coupling=:directed)\n\nfhn_network! = network_dynamics(odeelevertex, electricaledge, g_directed)\n\nnothing # hide","category":"page"},{"location":"directed_and_weighted_graphs/","page":"Directed and weighted graphs","title":"Directed and weighted graphs","text":"Since this system is a directed one with thus directed edges, the keyword argument coupling is used to set the coupling of the edges to :directed.","category":"page"},{"location":"directed_and_weighted_graphs/","page":"Directed and weighted graphs","title":"Directed and weighted graphs","text":"Note that the multiplication with the coupling strength sigma has been commented out. Since sigma is the same for every edge we can absorb this multiplication into the edge weight parameter p. Since our network has almost 8000 edges, this saves 8000 multiplications at every function call and leads to an 8-fold increase in performance.","category":"page"},{"location":"directed_and_weighted_graphs/#Parameter-handling","page":"Directed and weighted graphs","title":"Parameter handling","text":"","category":"section"},{"location":"directed_and_weighted_graphs/","page":"Directed and weighted graphs","title":"Directed and weighted graphs","text":"# Defining global parameters\n\nN = 90 # number of nodes\nconst ϵ = 0.05 # global variables that are accessed several times should be declared `const`\nconst a = 0.5\nconst σ = 0.5\n\n# Tuple of parameters for nodes and edges\n\np = (nothing, σ * edge_weights)\n\n# Initial conditions\n\nx0 = randn(2N) * 5\n\nnothing # hide","category":"page"},{"location":"directed_and_weighted_graphs/","page":"Directed and weighted graphs","title":"Directed and weighted graphs","text":"The behaviour of network_dynamics changes with the type of parameters p being passed. When p is an Array, the entire Array will be passed to each VertexFunction and EdgeFunction. When p is a tuple of two Arrays with lengths corresponding to the number of nodes and number of edges respectively, then network_dynamics passes only the edge or node parameters with the index of the edge or node. When there are no parameters for either edges or nodes the value nothing may be used.","category":"page"},{"location":"directed_and_weighted_graphs/#Solving-the-system","page":"Directed and weighted graphs","title":"Solving the system","text":"","category":"section"},{"location":"directed_and_weighted_graphs/","page":"Directed and weighted graphs","title":"Directed and weighted graphs","text":"Now we are ready to create an ODEProblem. Since for some choices of parameters the FitzHugh-Nagumo model is stiff (i.e. numerically unstable), we use a solver with automated stiffness detection. Such a solver switches to a more stable solver only when the solution enters a region of phase space where the problem is numerically unstable. In this case we use Tsit5 and switch to TRBDF2 when necessary. AutoTsit5 is the switching version of the Tsit5 algorithm.","category":"page"},{"location":"directed_and_weighted_graphs/","page":"Directed and weighted graphs","title":"Directed and weighted graphs","text":"using OrdinaryDiffEq\n\ntspan = (0.0, 200.0)\nprob = ODEProblem(fhn_network!, x0, tspan, p)\nsol = solve(prob, AutoTsit5(TRBDF2()));\nnothing # hide","category":"page"},{"location":"directed_and_weighted_graphs/#Plotting","page":"Directed and weighted graphs","title":"Plotting","text":"","category":"section"},{"location":"directed_and_weighted_graphs/","page":"Directed and weighted graphs","title":"Directed and weighted graphs","text":"The plot of the excitatory variables shows that they synchronize for this choice of parameters.","category":"page"},{"location":"directed_and_weighted_graphs/","page":"Directed and weighted graphs","title":"Directed and weighted graphs","text":"using Plots\n\nplot(sol; vars=idx_containing(fhn_network!, :u), legend=false, ylim=(-5, 5), fmt=:png)","category":"page"},{"location":"#NetworkDynamics","page":"General","title":"NetworkDynamics","text":"","category":"section"},{"location":"","page":"General","title":"General","text":"A package for working with dynamical systems on complex networks. NetworkDynamics.jl provides an interface between Graphs.jl and DifferentialEquations.jl. It allows to define several types of dynamic and static nodes and edges and to link them up in order to create complex network dynamics.","category":"page"},{"location":"","page":"General","title":"General","text":"The behavior of a node or an edge can be described by algebraic equations, by differential algebraic equation (DAEs) in mass matrix form, by ordinary differential equations (ODE) or by delay differential equations(DDE). Stochastic ordinary differential equations (SDE) can e.g. be implemented as a two-layer network. For detail see the tutorials section.","category":"page"},{"location":"#Installation","page":"General","title":"Installation","text":"","category":"section"},{"location":"","page":"General","title":"General","text":"Installation is straightforward with Julia's package manager.","category":"page"},{"location":"","page":"General","title":"General","text":"(v1.5) pkg> add NetworkDynamics","category":"page"},{"location":"#PowerDynamics","page":"General","title":"PowerDynamics","text":"","category":"section"},{"location":"","page":"General","title":"General","text":"PowerDynamics.jl is an open-source framework for dynamic power grid modeling and analysis build on top of NetworkDynamics.jl.","category":"page"},{"location":"#Overview","page":"General","title":"Overview","text":"","category":"section"},{"location":"","page":"General","title":"General","text":"The central construction is the function network_dynamics that receives functions describing the local dynamics on the edges and nodes of a graph g as inputs, and returns a composite function compatible with the DifferentialEquations.jl calling syntax.","category":"page"},{"location":"","page":"General","title":"General","text":"nd = network_dynamics(vertices!, edges!, g)\nnd(dx, x, p, t)","category":"page"},{"location":"BasicConstructors/#Functions","page":"Functions","title":"Functions","text":"","category":"section"},{"location":"BasicConstructors/","page":"Functions","title":"Functions","text":"The key network_dynamics assembles the dynamics of the whole network from functions for the single vertices and edges of the graph g. These functions have to be wrapped in types VertexFunction and EdgeFunction which store additional information on dimension and coupling type. Both have several subtypes that account for the different classes of equations that may represent the local dynamics. At the moment, algebraic (static) equations, ordinary differential equations (ODEs) and delay differential equations (DDEs) are supported:","category":"page"},{"location":"BasicConstructors/","page":"Functions","title":"Functions","text":"# VertexFunctions\nStaticVertex(vertexfunction!, dimension, symbol)\nODEVertex(vertexfunction!, dimension, mass_matrix, symbol)\nDDEVertex(vertexfunction!, dimension, mass_matrix, symbol)\n\n# EdgeFunctions\nStaticEdge(edgefunction!, dimension, coupling, symbol)\nODEEdge(edgefunction!, dimension, mass_matrix, coupling, symbol)\nStaticDelayEdge(edgefunction!, dimension, mass_matrix, coupling, symbol)","category":"page"},{"location":"BasicConstructors/#VertexFunctions","page":"Functions","title":"VertexFunctions","text":"","category":"section"},{"location":"BasicConstructors/","page":"Functions","title":"Functions","text":"Given a set of (algebraic or differential) equations describing a node or an edge the first step is to turn them into a mutating function vertexfunction!. Depending on the class of the function vertexfunction!, the constructors StaticVertex, ODEVertex or DDEVertex are called in order to turn vertexfunction! into a VertexFunction object compatible with network_dynamics.","category":"page"},{"location":"BasicConstructors/","page":"Functions","title":"Functions","text":"Since in general the state of a vertex depends on the vertex value itself as well as on the in- and outgoing edges, the function vertexfunction! has to respect one of the following calling syntaxes.","category":"page"},{"location":"BasicConstructors/","page":"Functions","title":"Functions","text":"# For static nodes\nfunction vertexfunction!(v, edges, p, t) end\n# For dynamic nodes\nfunction vertexfunction!(dv, v, edges, p, t) end\n# For delay nodes\nfunction vertexfunction!(dv, v, edges, h_v, p, t) end","category":"page"},{"location":"BasicConstructors/","page":"Functions","title":"Functions","text":"Here dv, v, p, and t are the usual ODE arguments, while edges is an array containing the edges for which the vertex is the destination (corresponding to incoming edges for directed graphs). In the delay case, the array h_v denotes the vertex history. The typical case of diffusive coupling on a directed graph could be described as","category":"page"},{"location":"BasicConstructors/","page":"Functions","title":"Functions","text":"function vertex!(dv, v, edges, p, t)\n dv .= 0.0\n for e in edges\n dv .+= e\n end\n nothing\nend","category":"page"},{"location":"BasicConstructors/#[StaticVertex](@ref)","page":"Functions","title":"StaticVertex","text":"","category":"section"},{"location":"BasicConstructors/","page":"Functions","title":"Functions","text":"If a vertex is described by an algebraic equation vertexfunction!(v, edges, p, t), i.e. dv = 0 the VertexFunction is constructed as","category":"page"},{"location":"BasicConstructors/","page":"Functions","title":"Functions","text":"StaticVertex(vertexfunction!, dim, sym)","category":"page"},{"location":"BasicConstructors/","page":"Functions","title":"Functions","text":"Here, dim is the number of independent variables in the vertex equations and sym is an array of symbols for these variables. For example, if a node models a constant input I = p, then dim = 1 and sym = [:I]. For more details on the use of symbols, check out the Getting started tutorial and the Julia documentation. The use of symbols makes it easier to select variables of interest from the solution object.","category":"page"},{"location":"BasicConstructors/#[ODEVertex](@ref)","page":"Functions","title":"ODEVertex","text":"","category":"section"},{"location":"BasicConstructors/","page":"Functions","title":"Functions","text":"If a vertex has local dynamics vertexfunction!(dv, v, edges, p, t) described by an ODE, the VertexFunction is contructed as","category":"page"},{"location":"BasicConstructors/","page":"Functions","title":"Functions","text":"ODEVertex(vertexfunction!, dim, mass_matrix, sym)","category":"page"},{"location":"BasicConstructors/","page":"Functions","title":"Functions","text":"As above, dim is the number of independent variables in the vertex equations and sym corresponds to the symbols of these variables.","category":"page"},{"location":"BasicConstructors/","page":"Functions","title":"Functions","text":"mass_matrix is an optional argument that defaults to the identity matrix I. If a mass matrix M is given, the local system M * dv = vertexfunction! will be solved. network_dynamics assembles all local mass matrices into one global mass matrix that can be passed to a differential equation solver like Rodas4.","category":"page"},{"location":"BasicConstructors/","page":"Functions","title":"Functions","text":"One may also call ODEVertex with keyword arguments, omitting optional arguments:","category":"page"},{"location":"BasicConstructors/","page":"Functions","title":"Functions","text":"ODEVertex(; f=vertexfunction!, dim=dim)","category":"page"},{"location":"BasicConstructors/","page":"Functions","title":"Functions","text":"The function then defaults to using the identity as mass matrix and [:v for i in 1:dimension] as symbols.","category":"page"},{"location":"BasicConstructors/#[DDEVertex](@ref)","page":"Functions","title":"DDEVertex","text":"","category":"section"},{"location":"BasicConstructors/","page":"Functions","title":"Functions","text":"If a vertex has local dynamics described by a delay differential equation (DDE) the local dynamics need to have the signature vertexfunction!(dv, v, edges, h_v, p, t), where h_v is an automatically generated wrapper of the global history function, providing history values forv. Then the VertexFunction is constructed as","category":"page"},{"location":"BasicConstructors/","page":"Functions","title":"Functions","text":"DDEVertex(vertexfunction!, dim, mass_matrix, sym)","category":"page"},{"location":"BasicConstructors/","page":"Functions","title":"Functions","text":"As before, dim is the number of independent variables in the vertex function and sym describes the symbols of the variables. The mass matrix is optional and defaults to the identity matrix I.","category":"page"},{"location":"BasicConstructors/","page":"Functions","title":"Functions","text":"As above, the DDEVertex function can also be called without the optional arguments:","category":"page"},{"location":"BasicConstructors/","page":"Functions","title":"Functions","text":"DDEVertex(; f=vertexfunction!, dim=dim)","category":"page"},{"location":"BasicConstructors/#EdgeFunctions","page":"Functions","title":"EdgeFunctions","text":"","category":"section"},{"location":"BasicConstructors/","page":"Functions","title":"Functions","text":"Similar to the case of vertices, an edge is described by mutating function edgefunction!. At the moment the constructors StaticEdge, ODEEdge, and StaticDelayEdge are available. edgefunction! has to respect one of the following syntaxes:","category":"page"},{"location":"BasicConstructors/","page":"Functions","title":"Functions","text":"# For static edges\nfunction edgefunction!(e, v_s, v_d, p, t) end\n# For dynamics edges\nfunction edgefunction!(de, e, v_s, v_d, p, t) end\n# For static delay edges\nfunction edgefunction!(e, v_s, v_d, h_v_s, h_v_d, p, t) end","category":"page"},{"location":"BasicConstructors/","page":"Functions","title":"Functions","text":"Just like above, de, e, p and t are the usual ODE arguments, while v_s and v_d are the source and destination vertices respectively and h_v_s and h_v_d the corresponding history arrays.","category":"page"},{"location":"BasicConstructors/#[StaticEdge](@ref)","page":"Functions","title":"StaticEdge","text":"","category":"section"},{"location":"BasicConstructors/","page":"Functions","title":"Functions","text":"Static here means, that the edge value described by edgefunction! only depends on the values of the vertices the edge connects to and that no derivative of the edge's internal state is involved. One very simple and natural example is a diffusive edge:","category":"page"},{"location":"BasicConstructors/","page":"Functions","title":"Functions","text":"edgefunction! = (e, v_s, v_d, p, t) -> e .= v_s .- v_d","category":"page"},{"location":"BasicConstructors/","page":"Functions","title":"Functions","text":"In this case the EdgeFunction is constructed by","category":"page"},{"location":"BasicConstructors/","page":"Functions","title":"Functions","text":"StaticEdge(edgefunction!, dim, coupling, sym)","category":"page"},{"location":"BasicConstructors/","page":"Functions","title":"Functions","text":"The keywords are the same as for the vertices, except from the additional keyword coupling, which describes if the EdgeFunction is intended for a (:directed) or for an undirected graph ({:undirected, :symmetric, :antisymmetric, :fiducial}). :directed is intended for directed graphs. :undirected is the default option and is only compatible with SimpleGraph. In this case the edgefunction! should specify the coupling from a source vertex to a destination vertex. :symmetric and :antisymmetric trigger performance optimizations, if edgefunction! has that symmetry property. :fiducial lets the user specify both the coupling from source to destination, as well as the coupling from destination to source and is intended for advanced users.","category":"page"},{"location":"BasicConstructors/#[ODEEdge](@ref)","page":"Functions","title":"ODEEdge","text":"","category":"section"},{"location":"BasicConstructors/","page":"Functions","title":"Functions","text":"For problems where edgefunction! describes the differential of an edge value, we use the ODEEdge function. An example for such a system is given by:","category":"page"},{"location":"BasicConstructors/","page":"Functions","title":"Functions","text":"edgefunction! = (de, e, v_s, v_d, p, t) -> de .= 1000 * (v_s .- v_d .- e)","category":"page"},{"location":"BasicConstructors/","page":"Functions","title":"Functions","text":"The EdgeFunction object is constructed as","category":"page"},{"location":"BasicConstructors/","page":"Functions","title":"Functions","text":"ODEEdge(edgefunction!, dim, coupling, mass_matrix, sym)","category":"page"},{"location":"BasicConstructors/","page":"Functions","title":"Functions","text":"The keywords are the same as above for the StaticEdge. For ODEEdge the same simplified construction rules apply when keyword arguments are used, except that the coupling type has to be specified explicitly to avoid ambiguities. The available coupling tyes are only :directed, :undirected, :fiducial. For :undirected coupling the internal dimension of the EdgeFunction is doubled in order to resolve both directions in an undirected network. Remember to specify 2 initial conditiosn for every edge in that case.","category":"page"},{"location":"BasicConstructors/","page":"Functions","title":"Functions","text":"ODEEdge(; f=edgefunction!, dim=n, coupling=:undirected)","category":"page"},{"location":"BasicConstructors/","page":"Functions","title":"Functions","text":"In this case the function defaults to using the identity matrix as mass matrix and [:e for in 1:dimension] as symbols.","category":"page"},{"location":"BasicConstructors/#[StaticDelayEdge](@ref)","page":"Functions","title":"StaticDelayEdge","text":"","category":"section"},{"location":"BasicConstructors/","page":"Functions","title":"Functions","text":"This constructor is used when edge variables depend on past values of the vertex variables. In this case the edgefunction! has to accept two additional arguments h_v_s and h_v_d that hold the (automatically generated) localised history functions of v_s and v_d. Static means that the edge depends only on the dynamics of the vertices the edge is connected to and not on an internal derivative of the edge variables itself.","category":"page"},{"location":"BasicConstructors/","page":"Functions","title":"Functions","text":"As an example for such system, we show a diffusive coupling with delay:","category":"page"},{"location":"BasicConstructors/","page":"Functions","title":"Functions","text":"edgefunction! = (e, v_s, v_d, h_v_s, h_v_d, p, t) -> e .= 0.1 * (h_v_s(t - p) .- v_d)","category":"page"},{"location":"BasicConstructors/","page":"Functions","title":"Functions","text":"The EdgeFunction object is constructed as","category":"page"},{"location":"BasicConstructors/","page":"Functions","title":"Functions","text":"StaticDelayEdge(edgefunction!, dim, coupling, mass_matrix, sym)","category":"page"},{"location":"BasicConstructors/","page":"Functions","title":"Functions","text":"Like for a StaticEdge, we can also leave out the optional keywords coupling, sym and mass_matrix. Available coupling types are :unspecified, :directed, :undirected, :fiducial.","category":"page"},{"location":"BasicConstructors/","page":"Functions","title":"Functions","text":"StaticDelayEdge(; f=edgefunction!, dim=n)","category":"page"},{"location":"BasicConstructors/#Constructor","page":"Functions","title":"Constructor","text":"","category":"section"},{"location":"BasicConstructors/","page":"Functions","title":"Functions","text":"The central constructor is the function network_dynamics that takes in two arrays of EdgeFunctions and VertexFunctions describing the local dynamics on the edges and nodes of a graph g, given as a Graphs.jl object. It returns a composite function compatible with the DifferentialEquations.jl calling syntax.","category":"page"},{"location":"BasicConstructors/","page":"Functions","title":"Functions","text":"nd = network_dynamics(vertices!::Array{VertexFunction},\n edges!::Array{EdgeFunction}, g)\nnd(dx, x, p, t)","category":"page"},{"location":"BasicConstructors/","page":"Functions","title":"Functions","text":"If all vertices, respectively edges share the same VertexFunction or EdgeFunction, than network_dynamics can be conveniently called with these functions as arguments.","category":"page"},{"location":"BasicConstructors/","page":"Functions","title":"Functions","text":"nd = network_dynamics(vertexfunction!::VertexFunction,\n edgefunction!::EdgeFunction, g)\n","category":"page"},{"location":"BasicConstructors/","page":"Functions","title":"Functions","text":"The optional keyword argument parallel is false by default and can be set to true if a multi-threaded ODEFunction should be used. This may significantly improve performance on multi-core machines, for more details see section Multi-Threading.","category":"page"},{"location":"BasicConstructors/","page":"Functions","title":"Functions","text":"nd = network_dynamics(vertexfunction!::VertexFunction,\n edgefunction!::EdgeFunction, g; parallel=true)\n","category":"page"},{"location":"BasicConstructors/#Example","page":"Functions","title":"Example","text":"","category":"section"},{"location":"BasicConstructors/","page":"Functions","title":"Functions","text":"Let's look at an example. First, we define our graph as well as the differential systems connected to its vertices and edges:","category":"page"},{"location":"BasicConstructors/","page":"Functions","title":"Functions","text":"using NetworkDynamics, Graphs\n\ng = erdos_renyi(10, 25) # random graph with 10 vertices and 25 edges\n\nfunction vertexfunction!(dv, v, edges, p, t)\n dv .= 0\n for e in edges\n dv .+= e\n end\nend\n\nfunction edgefunction!(de, e, v_s, v_d, p, t)\n de .= 1000 .* (v_s .- v_d .- e)\n nothing\nend\n\nvertex = ODEVertex(; f=vertexfunction!, dim=1)\nvertexarr = [vertex for v in vertices(g)]\n\nedge = ODEEdge(; f=edgefunction!, dim=1, coupling=:undirected)\nedgearr = [edge for e in edges(g)]\n\nnd = network_dynamics(vertexarr, edgearr, g)\n\nnothing # hide","category":"page"},{"location":"BasicConstructors/","page":"Functions","title":"Functions","text":"Now we have an ODEFunction nd that can be solved with the tools provided by DifferentialEquations.jl.","category":"page"},{"location":"BasicConstructors/","page":"Functions","title":"Functions","text":"For more details check out the Tutorials section.","category":"page"}] +} diff --git a/v0.8.2/siteinfo.js b/v0.8.2/siteinfo.js new file mode 100644 index 00000000..003e2724 --- /dev/null +++ b/v0.8.2/siteinfo.js @@ -0,0 +1 @@ +var DOCUMENTER_CURRENT_VERSION = "v0.8.2"; diff --git a/versions.js b/versions.js index 629c0546..fc3d220f 100644 --- a/versions.js +++ b/versions.js @@ -6,5 +6,5 @@ var DOC_VERSIONS = [ "v0.5", "dev", ]; -var DOCUMENTER_NEWEST = "v0.8.1"; +var DOCUMENTER_NEWEST = "v0.8.2"; var DOCUMENTER_STABLE = "stable";