betweenness_centrality¶

betweenness_centrality(G, nodes)[source]¶

Compute betweenness centrality for nodes in a bipartite network.

Betweenness centrality of a node \(v\) is the sum of the fraction of all-pairs shortest paths that pass through \(v\).

Values of betweenness are normalized by the maximum possible value which for bipartite graphs is limited by the relative size of the two node sets [R154].

Let \(n\) be the number of nodes in the node set \(U\) and \(m\) be the number of nodes in the node set \(V\), then nodes in \(U\) are normalized by dividing by

\[\frac{1}{2} [m^2 (s + 1)^2 + m (s + 1)(2t - s - 1) - t (2s - t + 3)] ,\]

where

\[s = (n - 1) \div m , t = (n - 1) \mod m ,\]

and nodes in \(V\) are normalized by dividing by

\[\frac{1}{2} [n^2 (p + 1)^2 + n (p + 1)(2r - p - 1) - r (2p - r + 3)] ,\]

where,

\[p = (m - 1) \div n , r = (m - 1) \mod n .\]

Parameters :

G : graph

A bipartite graph

nodes : list or container

Container with all nodes in one bipartite node set.

Returns :

betweenness : dictionary

Dictionary keyed by node with bipartite betweenness centrality as the value.