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-{"documenter":{"julia_version":"1.10.4","generation_timestamp":"2024-06-09T18:41:22","documenter_version":"1.4.1"}}
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dev/networks/index.html

@@ -12,4 +12,4 @@
 R = RecurrenceMatrix(X, 0.25; metric = Chebyshev())
 network = SimpleGraph(R)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">{201, 2945} undirected simple Int64 graph</code></pre><p>There are various plotting tools that can be used to visualize such graphs. For instance, the following plot made with the package <a href="https://github.com/MakieOrg/GraphMakie.jl">GraphMakie.jl</a>.</p><pre><code class="language-julia hljs">using GraphMakie, CairoMakie
 graphplot(network)</code></pre><img src="c1865a9f.png" alt="Example block output"/><h2 id="Recurrence-Network-measures"><a class="docs-heading-anchor" href="#Recurrence-Network-measures">Recurrence Network measures</a><a id="Recurrence-Network-measures-1"></a><a class="docs-heading-anchor-permalink" href="#Recurrence-Network-measures" title="Permalink"></a></h2><p>LightGraphs has a large set of functions to extract local measures (associated to particular vertices or edges) and global coefficients associated to the whole network. For <code>SimpleGraph</code>s created from recurrence matrices, as the variable <code>network</code> in the previous example, the vertices are labelled with numeric indices following the same ordering as the rows or columns of the given matrix. So for instance <code>degree(network, i)</code> would give the <em>degree</em> of the <code>i</code>-th point of the timeseries (number of connections with other points), whereas <code>degree(network)</code> would give a vector of such measures ordered as the original timeseries.</p><p>As in RQA, we provide a function that computes a selection of commonly used global RNA measures, directly from the recurrence matrix:</p><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="RecurrenceAnalysis.rna" href="#RecurrenceAnalysis.rna"><code>RecurrenceAnalysis.rna</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">rna(R::AbstractRecurrenceMatrix)
-rna(args...; kwargs...)</code></pre><p>Calculate a set of Recurrence Network parameters. The input <code>R</code> can be a symmetric recurrence matrix that is interpreted as the adjacency matrix of an undirected complex network, such that linked vertices are neighboring points in the phase space.</p><p>Alternatively, the inputs can be a graph object or any valid inputs to the <code>SimpleGraph</code> constructor of the <a href="https://github.com/JuliaGraphs/Graphs.jl">Graphs</a> package.</p><p><strong>Return</strong></p><p>The returned value is a dictionary that contains the following entries, with the corresponding global network properties[1, 2]:</p><ul><li><code>:density</code>: edge density, approximately equivalent to the global recurrence rate in the phase space.</li><li><code>:transitivity</code>: network transitivity, which describes the</li></ul><p>global clustering of points following Barrat&#39;s and Weigt&#39;s formulation [3].</p><ul><li><code>:averagepath</code>: mean value of the shortest path lengths taken over</li></ul><p>all pairs of connected vertices, related to the average separation between points in the phase.</p><ul><li><code>:diameter</code>: maximum value of the shortest path lengths between</li></ul><p>pairs of connected vertices, related to the phase space diameter.</p><p><strong>References</strong></p><p>[1]: R.V. Donner <em>et al.</em> &quot;Recurrence networks — a novel paradigm for nonlinear time series analysis&quot;, <em>New Journal of Physics</em> 12, 033025 (2010) <a href="https://doi.org/10.1088/1367-2630/12/3/033025">DOI:10.1088/1367-2630/12/3/033025</a></p><p>[2]: R.V. Donner <em>et al.</em>, The geometry of chaotic dynamics — a complex network perspective, <em>Eur. Phys. J.</em> B 84, 653–672 (2011) <a href="https://doi.org/10.1140/epjb/e2011-10899-1">DOI:10.1140/epjb/e2011-10899-1</a></p><p>[3]: A. Barrat &amp; M. Weight, &quot;On the properties of small-world network models&quot;, <em>The European Physical Journal B</em> 13, 547–560 (2000) <a href="https://doi.org/10.1007/s100510050067">DOI:10.1007/s100510050067</a></p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/JuliaDynamics/RecurrenceAnalysis.jl/blob/64229ed2b15e99cebbe079e7da115ef7af17df5c/src/rna/rna.jl#L1-L40">source</a></section></article><div class="admonition is-info"><header class="admonition-header">Transitivity and global clustering coefficient</header><div class="admonition-body"><p>The concept of clustering coefficient at local level (for individual nodes of the network) is clearly defined as the fraction of connecting nodes that are also connected between them, forming &quot;triangles&quot;. But at global level it is a source of confusion: the term of &quot;global clustering coefficient&quot; was originally used by Watts and Strogatz<sup class="footnote-reference"><a id="citeref-1" href="#footnote-1">[1]</a></sup>, referred to the average of local clustering coefficient across all the graph&#39;s nodes. But Barrat and Weigt proposed an alternative definition<sup class="footnote-reference"><a id="citeref-2" href="#footnote-2">[2]</a></sup> that characterizes the effective global dimensionality of the system, giving equal weight to all triangles in the network<sup class="footnote-reference"><a id="citeref-3" href="#footnote-3">[3]</a></sup>.</p><p>This second definition is often named with the distinctive term of &quot;transitivity&quot;, as in the output of <code>rna</code>, whereas the corresponding function of the <code>LightGraphs</code> package is <code>global_clustering_coefficient</code>. The &quot;global clustering coefficient&quot; as by Watts and Strogatz could be obtained as <code>mean(local_clustering_coefficient(network))</code> – with <code>network</code> being a graph object as in the previous example. (The function <code>mean</code> is in the Julia standard library, and can be brought into scope with the command <code>using Statistics</code>.)</p></div></div><section class="footnotes is-size-7"><ul><li class="footnote" id="footnote-1"><a class="tag is-link" href="#citeref-1">1</a>D.J. Watts &amp; S.H. Strogatz, &quot;Collective dynamics of &#39;small-world&#39; networks&quot;, <em>Nature 393</em>(6684), 440–442 (1998) <a href="https://doi.org/10.1038%2F30918">DOI:10.1038%2F30918</a></li><li class="footnote" id="footnote-2"><a class="tag is-link" href="#citeref-2">2</a>A. Barrat &amp; M. Weight, &quot;On the properties of small-world network models&quot;, <em>The European Physical Journal B</em> 13, 547–560 (2000)  <a href="https://doi.org/10.1007/s100510050067">DOI:10.1007/s100510050067</a></li><li class="footnote" id="footnote-3"><a class="tag is-link" href="#citeref-3">3</a>R.V. Donner <em>et al.</em> &quot;Recurrence networks — a novel paradigm for nonlinear time series analysis&quot;, <em>New Journal of Physics</em> 12, 033025 (2010) <a href="https://doi.org/10.1088/1367-2630/12/3/033025">DOI:10.1088/1367-2630/12/3/033025</a></li></ul></section></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../quantification/">« Recurrence Quantification Analysis</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.4.1 on <span class="colophon-date" title="Sunday 9 June 2024 18:41">Sunday 9 June 2024</span>. Using Julia version 1.10.4.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
+rna(args...; kwargs...)</code></pre><p>Calculate a set of Recurrence Network parameters. The input <code>R</code> can be a symmetric recurrence matrix that is interpreted as the adjacency matrix of an undirected complex network, such that linked vertices are neighboring points in the phase space.</p><p>Alternatively, the inputs can be a graph object or any valid inputs to the <code>SimpleGraph</code> constructor of the <a href="https://github.com/JuliaGraphs/Graphs.jl">Graphs</a> package.</p><p><strong>Return</strong></p><p>The returned value is a dictionary that contains the following entries, with the corresponding global network properties[1, 2]:</p><ul><li><code>:density</code>: edge density, approximately equivalent to the global recurrence rate in the phase space.</li><li><code>:transitivity</code>: network transitivity, which describes the</li></ul><p>global clustering of points following Barrat&#39;s and Weigt&#39;s formulation [3].</p><ul><li><code>:averagepath</code>: mean value of the shortest path lengths taken over</li></ul><p>all pairs of connected vertices, related to the average separation between points in the phase.</p><ul><li><code>:diameter</code>: maximum value of the shortest path lengths between</li></ul><p>pairs of connected vertices, related to the phase space diameter.</p><p><strong>References</strong></p><p>[1]: R.V. Donner <em>et al.</em> &quot;Recurrence networks — a novel paradigm for nonlinear time series analysis&quot;, <em>New Journal of Physics</em> 12, 033025 (2010) <a href="https://doi.org/10.1088/1367-2630/12/3/033025">DOI:10.1088/1367-2630/12/3/033025</a></p><p>[2]: R.V. Donner <em>et al.</em>, The geometry of chaotic dynamics — a complex network perspective, <em>Eur. Phys. J.</em> B 84, 653–672 (2011) <a href="https://doi.org/10.1140/epjb/e2011-10899-1">DOI:10.1140/epjb/e2011-10899-1</a></p><p>[3]: A. Barrat &amp; M. Weight, &quot;On the properties of small-world network models&quot;, <em>The European Physical Journal B</em> 13, 547–560 (2000) <a href="https://doi.org/10.1007/s100510050067">DOI:10.1007/s100510050067</a></p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/JuliaDynamics/RecurrenceAnalysis.jl/blob/b87b5e2057cbbc723ac7bbe14f06e31f812ace8f/src/rna/rna.jl#L1-L40">source</a></section></article><div class="admonition is-info"><header class="admonition-header">Transitivity and global clustering coefficient</header><div class="admonition-body"><p>The concept of clustering coefficient at local level (for individual nodes of the network) is clearly defined as the fraction of connecting nodes that are also connected between them, forming &quot;triangles&quot;. But at global level it is a source of confusion: the term of &quot;global clustering coefficient&quot; was originally used by Watts and Strogatz<sup class="footnote-reference"><a id="citeref-1" href="#footnote-1">[1]</a></sup>, referred to the average of local clustering coefficient across all the graph&#39;s nodes. But Barrat and Weigt proposed an alternative definition<sup class="footnote-reference"><a id="citeref-2" href="#footnote-2">[2]</a></sup> that characterizes the effective global dimensionality of the system, giving equal weight to all triangles in the network<sup class="footnote-reference"><a id="citeref-3" href="#footnote-3">[3]</a></sup>.</p><p>This second definition is often named with the distinctive term of &quot;transitivity&quot;, as in the output of <code>rna</code>, whereas the corresponding function of the <code>LightGraphs</code> package is <code>global_clustering_coefficient</code>. The &quot;global clustering coefficient&quot; as by Watts and Strogatz could be obtained as <code>mean(local_clustering_coefficient(network))</code> – with <code>network</code> being a graph object as in the previous example. (The function <code>mean</code> is in the Julia standard library, and can be brought into scope with the command <code>using Statistics</code>.)</p></div></div><section class="footnotes is-size-7"><ul><li class="footnote" id="footnote-1"><a class="tag is-link" href="#citeref-1">1</a>D.J. Watts &amp; S.H. Strogatz, &quot;Collective dynamics of &#39;small-world&#39; networks&quot;, <em>Nature 393</em>(6684), 440–442 (1998) <a href="https://doi.org/10.1038%2F30918">DOI:10.1038%2F30918</a></li><li class="footnote" id="footnote-2"><a class="tag is-link" href="#citeref-2">2</a>A. Barrat &amp; M. Weight, &quot;On the properties of small-world network models&quot;, <em>The European Physical Journal B</em> 13, 547–560 (2000)  <a href="https://doi.org/10.1007/s100510050067">DOI:10.1007/s100510050067</a></li><li class="footnote" id="footnote-3"><a class="tag is-link" href="#citeref-3">3</a>R.V. Donner <em>et al.</em> &quot;Recurrence networks — a novel paradigm for nonlinear time series analysis&quot;, <em>New Journal of Physics</em> 12, 033025 (2010) <a href="https://doi.org/10.1088/1367-2630/12/3/033025">DOI:10.1088/1367-2630/12/3/033025</a></li></ul></section></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../quantification/">« Recurrence Quantification Analysis</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.4.1 on <span class="colophon-date" title="Monday 10 June 2024 10:31">Monday 10 June 2024</span>. Using Julia version 1.10.4.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>