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 [1].

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 keyed by node with bipartite betweenness centrality as the value.
Return type:	dictionary

See also

degree_centrality(), closeness_centrality(), sets(), is_bipartite()

Notes

The nodes input parameter must contain all nodes in one bipartite node set, but the dictionary returned contains all nodes from both node sets.

References

[1]	Borgatti, S.P. and Halgin, D. In press. “Analyzing Affiliation Networks”. In Carrington, P. and Scott, J. (eds) The Sage Handbook of Social Network Analysis. Sage Publications. http://www.steveborgatti.com/papers/bhaffiliations.pdf