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updated the glossary, so it should be good now
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brendapraggastis committed May 16, 2024
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Expand Up @@ -15,8 +15,10 @@ is a tuple of three sets, :math:`H = (V, E, \mathcal{I})`.
- :math:`E` a set of *edges* (aka hyperedges), distinguished by unique identifiers
- :math:`\mathcal{I}`, a set of *incidences*, which form a subset of :math:`E \times V`, distinguished by the pairing of unique identifiers of edges in :math:`E` and nodes in :math:`V`

The incidences :math:`\mathcal{I}` may be though of as a boolean function, :math:`\mathcal{I} : E \times V \rightarrow \{0, 1\}`, indicating whether or not a pair is included in the hypergraph.
The :term:`incidence matrix` defines the function in terms of the cross product of the two sets.

**Note: For all definitions below, assume :math:`H = (V, E, \mathcal{I})` is a
**Note: For all definitions below, assume** :math:`H = (V, E, \mathcal{I})` **is a
hypergraph.**


Expand All @@ -33,144 +35,150 @@ These hypergraph components are instantiate with three hypergraph objects for th
.. glossary::
:sorted:

.. // scan hypergraph.py
PropertyStore
Class in property_store.py. Each of the basic sets in a hypergraph, (Nodes, Edges, Incidences), have metadata stored in a
PropertyStore. By storing the data and metadata in a single place, updates and references have a single source of
truth.

IncidenceStore
Class in icidence_store.py. The minimal amount of data required to instantiate a hypergraph is a set of Incidences, :math:`\mathcal{I}`. The
Edges and Nodes can be inferred from the pairs :math:`(e,v)` in the Incidences.

hypergraph
There are many definitions of a *hypergraph*. In HNX a hypergraph
is a tuple of three sets, :math:`H = (V, E, \mathcal{I})`.

- :math:`V`, a set of *nodes* (aka hypernodes, vertices), distinguished by unique identifiers
- :math:`E` a set of *edges* (aka hyperedges), distinguished by unique identifiers
- :math:`\mathcal{I}`, a set of *incidences*, which form a subset of :math:`E \times V`, distinguished by the pairing of unique identifiers of edges in :math:`E` and nodes in :math:`V`

**Note: For all definitions below, assume :math:`H = (V, E, \mathcal{I})` is a
hypergraph.**

multihypergraph
HNX hypergraphs may be multihypergraphs. A multihypergraph is a hypergraph that allows distinct edges to contain the same set of *elements* and distinct nodes to belong to the same set of edges (aka *memberships*). When collapsing a hypergraph,
edges incident with the same set of nodes or nodes incident with the same set of edges are collapsed to single objects.

incidences in HNX
A set of ordered pairs of Edges and Nodes.
A subset of Edges :math:`\times` Nodes.
Each ordered pair uniquely identifies a single
incidence. Each incidence has metadata assigned to it. Incidences
in a hypergraph are assigned a weight either by default or specified by a user.

elements of an edge
The set of nodes incident to the edge in the Hypergraph.

memberships of a node
The set of edges incident to the node in the Hypergraph.

incidence matrix
A rectangular matrix constructed from a hypergraph, :math:`H = (V, E, \mathcal{I})`. The rows of the matrix are indexed and ordering of :math:`V`. The columns of the matrix are indexed by an ordering of :math:`E`. An entry in the matrix at
position :math:`(v,e)` for some :math:`v \in V` and :math:`e \in E :math:` is nonzero if and only if :math:`(e,v) \in I`.
A *weighted* incidence matrix uses the incidence weight associated with :math:`(e,v)` for the nonzero entry. An *unweighted* incidence
matrix has the integer :math:`1` in all nonzero entries.
If :math:`(e,v) \in \mathcal{I}` then :math:`e` *contains* :math:`v`, :math:`v` is an
`element` of :math:`e`, and :math:`v` has membership in :math:`e`.

edges in HNX, (aka hyperedges)
A set of objects distinguished by unique identifiers (uids). Each edge has
metadata associated with it. Edges are assigned a weight either by default or
specified by the user.

nodes in HNX, (aka hypernodes, vertices)
A set of objects distinguished by unique identifiers (uids). Each node has
metadata associated with it. Nodes are assigned a weight either by default or
specified by the user.

subhypergraph
A subhypergraph of a hypergraph, :math:`H = (V, E, \mathcal{I})`, is a hypergraph, :math:`H' = (V', E', \mathcal{I'})` such that :math:`(e',v') \in \mathcal{I'}` if and only if :math:`e' \in E' \subset E`, :math:`v' \in V' \subset V` and :math:`(e,v) \in \mathcal{I}`.

degree
Given a hypergraph (Nodes, Edges, Incidence), the degree of a node in Nodes is the number of edges in Edges to which the node is incident.
See also: :term:`s-degree`

dual
The dual of a hypergraph exchanges the roles of the edges and nodes in the hypergraph.
For a hypergraph :math:`H = (V, E, \mathcal{I})` the dual is
`H_D = (E, V, \mathcal{I}^T)` where the ordered pairs in :math:`\mathcal{I}^T)` are the transposes of the ordered pairs in :math:`\mathcal{I}`. The :term:`incidence matrix` of :math:`H_D` is the transpose of the incidence matrix of :math:`H`.

toplex
A toplex in a hypergraph, :math:`H = (V, E, \mathcal{I})`, is an edge :math:`e \in E` whose set of elements is not properly contained in any other edge in :math:`E`. That is, if :math:`f \in E` and the elements of :math:`e` are all elements of :math:`f` then the elements of :math:`f` are all elements of :math:`e`.

simple hypergraph
A hypergraph for which no edge is completely contained in another.

s-adjacency, s-edge-adjacency
For a hypergraph, :math:`H = (V, E, \mathcal{I})`, and positive integer s,
two nodes in :math:`V` are s-adjacent if there are at least s edges in :math:`E`, which contain both of them. Two edges are s-edge-adjacent if
they there are at least s nodes in :math:`V` belonging to both of them.
Another way of saying this is two edges are s-edge-adjacent if
they are s-adjacent in the dual of :math:`H`.

s-adjacency matrix, s-edge-adjacency matrix
For a positive integer s, a square matrix for a hypergraph, :math:`H = (V, E, \mathcal{I})`, indexed by :math:`V` such that an
entry :math:`(v_1,v_2)` is nonzero if only if :math:`v_1, v_2 \in V` are s-adjacent. An s-adjacency matrix can be weighted or unweighted, in which case all entries are 0's and 1's.

An s-edge-adjacency matrix is the s-adjacency matrix for the dual
of :math:`H`.

s-auxiliary matrix, s-edge-auxiliary matrix
For a hypergraph, :math:`H = (V, E, \mathcal{I})`, and positive integer s, the submatrix of the :term:`s-adjacency matrix <s-adjacency matrix>` or the :term:`s-edge-adjacency matrix <s-edge-adjacency matrix>` obtained by removing all 0-rows and 0-columns.

s-node-walk
For a hypergraph, :math:`H = (V, E, \mathcal{I})`, and positive integer s, a sequence of nodes in :math:`V` such that each successive pair of nodes are s-adjacent. The length of the
s-node-walk is the number of adjacent pairs in the sequence.

s-edge-walk
For a hypergraph, :math:`H = (V, E, \mathcal{I})`, and positive integer s, a sequence of edges in :math:`E` such that each successive pair of edges are s-edge-adjacent. The length of the
s-edge-walk is the number of adjacent pairs in the sequence.

s-walk
Either an s-node-walk or an s-edge-walk. The length of the
s-walk is the number of adjacent pairs in the sequence.

s-connected component, s-node-connected component, s-edge-connected component
For a hypergraph, :math:`H = (V, E, \mathcal{I})`, and positive integer s, an s-connected component is a :term:`subhypergraph` induced by a subset of :math:`V` with the property that there exists an s-walk between every pair of nodes in this subset.
An s-connected component is the maximal such subset in the sense that it is not properly contained in any other subset satisfying this property.

An s-node-connected component is an s-connected component. An
s-edge-connected component is an s-connected component of the dual
of :math:`H`.

s-connected, s-node-connected, s-edge-connected
A hypergraph is s-connected if it has one s-connected component.
Similarly for s-node-connected and s-edge-connected.

s-degree
For a hypergraph, :math:`H = (V, E, \mathcal{I})`, and positive integer s, the s-degree of a node, :math:`v \in V` is the number of edges in :math:`E` of size at least s to which :math:`v` belongs. See also: :term:`degree`

s-distance, s-edge-distance
For a hypergraph, :math:`H = (V, E, \mathcal{I})`, and positive integer s, the s-distances between two nodes in :math:`V` is the length of the shortest :term:`s-node-walk` between them. If no s-node-walk between the pair of nodes exists, the s-distance between them is infinite. The s-edge-distance
between edges is the length of the shortest :term:`s-edge-walk` between them. If no s-edge-walk between the pair of edges exists, then s-distance between them is infinite.

s-diameter
For a hypergraph, :math:`H = (V, E, \mathcal{I})`, and positive integer s, the s-diameter is the maximum s-distance over all pairs of nodes in Nodes.


s-edge
For a hypergraph, :math:`H = (V, E, \mathcal{I})`, and positive integer s, an s-edge is any edge :math:`e \in E` of size at least s, where the
size of :math:`e` equals the number of nodes in :math:`V` belonging to :math:`e`.

s-linegraph
For a hypergraph, :math:`H = (V, E, \mathcal{I})`, and positive integer s, an s-linegraph :math:`G` is a graph representing
the node to node or edge to edge connections defined by the s-adjacency matrices.
The node s-linegraph, :math:`G_V` is a graph on the set :math:`V`. Two nodes in :math:`V` are incident in :math:`G_V` if they are s-adjacent.
The edge s-linegraph, :math:`G_E` is a graph on the set :math:`E`. Two edges in :math:`E` are incident in :math:`G_E` if they are s-edge-adjacent.
PropertyStore
Class in property_store.py. Each of the basic sets in a hypergraph, (Nodes, Edges, Incidences), have metadata stored in a
PropertyStore. By storing the data and metadata in a single place, updates and references have a single source of
truth.

IncidenceStore
Class in icidence_store.py. The minimal amount of data required to instantiate a hypergraph is a set of Incidences, :math:`\mathcal{I}`. The
Edges and Nodes can be inferred from the pairs :math:`(e,v)` in the Incidences.

hypergraph
A hypergraph is a tuple of three sets, :math:`H = (V, E, \mathcal{I})`.

- :math:`V`, a set of *nodes* (aka hypernodes, vertices), distinguished by unique identifiers
- :math:`E` a set of *edges* (aka hyperedges), distinguished by unique identifiers
- :math:`\mathcal{I}`, a set of *incidences*, which form a subset of :math:`E \times V`, distinguished by the pairing of unique identifiers of edges in :math:`E` and nodes in :math:`V`

multihypergraph
HNX hypergraphs may be multihypergraphs. A multihypergraph is a hypergraph that allows distinct edges to contain the same set of *elements* and distinct nodes to belong to the same set of edges (aka *memberships*). When collapsing a hypergraph,
edges incident with the same set of nodes or nodes incident with the same set of edges are collapsed to single objects.

IncidenceStore
A set of ordered pairs of Edges and Nodes.
A subset of Edges :math:`\times` Nodes.
Each ordered pair uniquely identifies a single
incidence. Each incidence has metadata assigned to it. Incidences
in a hypergraph are assigned a weight either by default or specified by a user.

elements
The set of nodes incident to the edge in the Hypergraph.

memberships
The set of edges incident to the node in the Hypergraph.

incidence matrix
A rectangular matrix constructed from a hypergraph, :math:`H = (V, E, \mathcal{I})`. The rows of the matrix are indexed and ordering of :math:`V`. The columns of the matrix are indexed by an ordering of :math:`E`. An entry in the matrix at
position :math:`(v,e)` for some :math:`v \in V` and :math:`e \in E :math:` is nonzero if and only if :math:`(e,v) \in I`.
A *weighted* incidence matrix uses the incidence weight associated with :math:`(e,v)` for the nonzero entry. An *unweighted* incidence
matrix has the integer :math:`1` in all nonzero entries.
If :math:`(e,v) \in \mathcal{I}` then :math:`e` *contains* :math:`v`, :math:`v` is an
`element` of :math:`e`, and :math:`v` has membership in :math:`e`.

edges
hyperedges
A set of objects distinguished by unique identifiers (uids). Each edge has
metadata associated with it. Edges are assigned a weight either by default or
specified by the user.

nodes
vertices
hypernodes
A set of objects distinguished by unique identifiers (uids). Each node has
metadata associated with it. Nodes are assigned a weight either by default or
specified by the user.

subhypergraph
A subhypergraph of a hypergraph, :math:`H = (V, E, \mathcal{I})`, is a hypergraph, :math:`H' = (V', E', \mathcal{I'})` such that :math:`(e',v') \in \mathcal{I'}` if and only if :math:`e' \in E' \subset E`, :math:`v' \in V' \subset V` and :math:`(e,v) \in \mathcal{I}`.

degree
Given a hypergraph (Nodes, Edges, Incidence), the degree of a node in Nodes is the number of edges in Edges to which the node is incident.
See also: :term:`s-degree`

dual
The dual of a hypergraph exchanges the roles of the edges and nodes in the hypergraph.
For a hypergraph :math:`H = (V, E, \mathcal{I})` the dual is
`H_D = (E, V, \mathcal{I}^T)` where the ordered pairs in :math:`\mathcal{I}^T)` are the transposes of the ordered pairs in :math:`\mathcal{I}`. The :term:`incidence matrix` of :math:`H_D` is the transpose of the incidence matrix of :math:`H`.

toplex
A toplex in a hypergraph, :math:`H = (V, E, \mathcal{I})`, is an edge :math:`e \in E` whose set of elements is not properly contained in any other edge in :math:`E`. That is, if :math:`f \in E` and the elements of :math:`e` are all elements of :math:`f` then the elements of :math:`f` are all elements of :math:`e`.

simple hypergraph
A hypergraph for which no edge is completely contained in another.

s-adjacent
s-edge-adjacent
For a hypergraph, :math:`H = (V, E, \mathcal{I})`, and positive integer s,
two nodes in :math:`V` are s-adjacent if there are at least s edges in :math:`E`, which contain both of them. Two edges are s-edge-adjacent if
they there are at least s nodes in :math:`V` belonging to both of them.
Another way of saying this is two edges are s-edge-adjacent if
they are s-adjacent in the dual of :math:`H`.

s-adjacency matrix
s-edge-adjacency matrix
For a positive integer s, a square matrix for a hypergraph, :math:`H = (V, E, \mathcal{I})`, indexed by :math:`V` such that an
entry :math:`(v_1,v_2)` is nonzero if only if :math:`v_1, v_2 \in V` are s-adjacent. An s-adjacency matrix can be weighted or unweighted, in which case all entries are 0's and 1's.

An s-edge-adjacency matrix is the s-adjacency matrix for the dual
of :math:`H`.

s-auxiliary matrix
s-edge-auxiliary matrix
For a hypergraph, :math:`H = (V, E, \mathcal{I})`, and positive integer s, the submatrix of the :term:`s-adjacency matrix` or the :term:`s-edge-adjacency matrix` obtained by removing all 0-rows and 0-columns.

s-node-walk
For a hypergraph, :math:`H = (V, E, \mathcal{I})`, and positive integer s, a sequence of nodes in :math:`V` such that each successive pair of nodes are s-adjacent. The length of the
s-node-walk is the number of adjacent pairs in the sequence.

s-edge-walk
For a hypergraph, :math:`H = (V, E, \mathcal{I})`, and positive integer s, a sequence of edges in :math:`E` such that each successive pair of edges are s-edge-adjacent. The length of the
s-edge-walk is the number of adjacent pairs in the sequence.

s-walk
Either an s-node-walk or an s-edge-walk. The length of the
s-walk is the number of adjacent pairs in the sequence.

s-connected component
s-node-connected component
s-edge-connected component
For a hypergraph, :math:`H = (V, E, \mathcal{I})`, and positive integer s, an s-connected component is a :term:`subhypergraph` induced by a subset of :math:`V` with the property that there exists an s-walk between every pair of nodes in this subset.
An s-connected component is the maximal such subset in the sense that it is not properly contained in any other subset satisfying this property.

An s-node-connected component is an s-connected component. An
s-edge-connected component is an s-connected component of the dual
of :math:`H`.

s-connected
s-node-connected
s-edge-connected
A hypergraph is s-connected if it has one s-connected component.
Similarly for s-node-connected and s-edge-connected.

s-degree
For a hypergraph, :math:`H = (V, E, \mathcal{I})`, and positive integer s, the s-degree of a node, :math:`v \in V` is the number of edges in :math:`E` of size at least s to which :math:`v` belongs. See also: :term:`degree`

s-distance
s-edge-distance
For a hypergraph, :math:`H = (V, E, \mathcal{I})`, and positive integer s, the s-distances between two nodes in :math:`V` is the length of the shortest :term:`s-node-walk` between them. If no s-node-walk between the pair of nodes exists, the s-distance between them is infinite. The s-edge-distance
between edges is the length of the shortest :term:`s-edge-walk` between them. If no s-edge-walk between the pair of edges exists, then s-distance between them is infinite.

s-diameter
For a hypergraph, :math:`H = (V, E, \mathcal{I})`, and positive integer s, the s-diameter is the maximum s-distance over all pairs of nodes in Nodes.


s-edge
For a hypergraph, :math:`H = (V, E, \mathcal{I})`, and positive integer s, an s-edge is any edge :math:`e \in E` of size at least s, where the
size of :math:`e` equals the number of nodes in :math:`V` belonging to :math:`e`.

s-linegraph
For a hypergraph, :math:`H = (V, E, \mathcal{I})`, and positive integer s, an s-linegraph :math:`G` is a graph representing
the node to node or edge to edge connections defined by the :term:`s-adjacency matrices<s-adjacency matrix>`.

The node s-linegraph, :math:`G_V` is a graph on the set :math:`V`. Two nodes in :math:`V` are incident in :math:`G_V` if they are :term:`s-adjacent`.

The edge s-linegraph, :math:`G_E` is a graph on the set :math:`E`. Two edges in :math:`E` are incident in :math:`G_E` if they are :term:`s-edge-adjacent`.



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