networksns.centrality_measures.total_directed_communicability

networksns.centrality_measures.total_directed_communicability(G, t=1)

Computes the total hub communicability and the total authority communicability of a directed graph \(G\).

Denoting with \(A\) the adjacency matrix of \(G\), with \(\mathcal{A}=\begin{pmatrix} 0 & A \\ A^T & 0 \end{pmatrix}\) the adjacency matrix of the associated undirected bipartite graph and with \(\mathbf{0}\) and \(\mathbf{1}\) the vectors of all zeros and ones respectively, total hub communicability and total authority communicability of \(G\) are defined as

\(T_{h}C(G) = \mathbf{1}^T\cosh{\left(\sqrt{A A^T}\right)}\mathbf{1} = \begin{pmatrix} \mathbf{1}^T & \mathbf{0}^T \end{pmatrix} e^{\mathcal{A}}\begin{pmatrix} \mathbf{1} \\ \mathbf{0} \end{pmatrix}\), \(T_{a}C(G) = \mathbf{1}^T\cosh{\left(\sqrt{A^T A}\right)}\mathbf{1} = \begin{pmatrix} \mathbf{0}^T & \mathbf{1}^T \end{pmatrix} e^{\mathcal{A}}\begin{pmatrix} \mathbf{0} \\ \mathbf{1} \end{pmatrix}\).

See [1] for further details.

Parameters:

  • G (DiGraph object) – a directed graph.

  • t (scalar, optional) – when computing the total hub and authority communicabilities multiply the adjacency matrix by \(t\), default: 1.

Returns:

  • thc (float) – total hub communicability.

  • trc (float) – total authority communicability.

Examples

>>>  from networksns import centrality_measures as cm
>>>  import networkx as nx

Create graph \(G\).

>>>    G = nx.DiGraph()
>>>    G.add_edge(1, 2)
>>>    G.add_edge(1, 3)
>>>    G.add_edge(2, 3)
>>>    G.add_edge(3, 1)
       OutEdgeView([(1, 2), (1, 3), (2, 3), (3, 1)])

Compute total hub communicability and total authority communicability.

>>>    thc, tac = cm.total_directed_communicability(G)

References