group_betweenness_centrality

group_betweenness_centrality(G, C, normalized=True, weight=None, endpoints=False)

Compute the group betweenness centrality for a group of nodes.

Group betweenness centrality of a group of nodes \(C\) is the sum of the fraction of all-pairs shortest paths that pass through any vertex in \(C\)

\[c_B(v) =\sum_{s,t \in V} \frac{\sigma(s, t|v)}{\sigma(s, t)}\]

where \(V\) is the set of nodes, \(\sigma(s, t)\) is the number of shortest \((s, t)\)-paths, and \(\sigma(s, t|C)\) is the number of those paths passing through some node in group \(C\). Note that \((s, t)\) are not members of the group (\(V-C\) is the set of nodes in \(V\) that are not in \(C\)).

Parameters:

Ggraph

A NetworkX graph.

Clist or set or list of lists or list of sets

A group or a list of groups containing nodes which belong to G, for which group betweenness centrality is to be calculated.

normalizedbool, optional (default=True)

If True, group betweenness is normalized by 1/((|V|-|C|)(|V|-|C|-1)) where |V| is the number of nodes in G and |C| is the number of nodes in C.

weightNone or string, optional (default=None)

If None, all edge weights are considered equal. Otherwise holds the name of the edge attribute used as weight. The weight of an edge is treated as the length or distance between the two sides.

endpointsbool, optional (default=False)

If True include the endpoints in the shortest path counts.

Returns:

betweennesslist of floats or float

If C is a single group then return a float. If C is a list with several groups then return a list of group betweenness centralities.

Raises:

NodeNotFound

If node(s) in C are not present in G.

See also

betweenness_centrality

Notes

Group betweenness centrality is described in [1] and its importance discussed in [3]. The initial implementation of the algorithm is mentioned in [2]. This function uses an improved algorithm presented in [4].

The number of nodes in the group must be a maximum of n - 2 where n is the total number of nodes in the graph.

For weighted graphs the edge weights must be greater than zero. Zero edge weights can produce an infinite number of equal length paths between pairs of nodes.

The total number of paths between source and target is counted differently for directed and undirected graphs. Directed paths between “u” and “v” are counted as two possible paths (one each direction) while undirected paths between “u” and “v” are counted as one path. Said another way, the sum in the expression above is over all s != t for directed graphs and for s < t for undirected graphs.

References

[1]

M G Everett and S P Borgatti: The Centrality of Groups and Classes. Journal of Mathematical Sociology. 23(3): 181-201. 1999. http://www.analytictech.com/borgatti/group_centrality.htm

[2]

Ulrik Brandes: On Variants of Shortest-Path Betweenness Centrality and their Generic Computation. Social Networks 30(2):136-145, 2008. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.72.9610&rep=rep1&type=pdf

[3]

Sourav Medya et. al.: Group Centrality Maximization via Network Design. SIAM International Conference on Data Mining, SDM 2018, 126–134. https://sites.cs.ucsb.edu/~arlei/pubs/sdm18.pdf

[4]

Rami Puzis, Yuval Elovici, and Shlomi Dolev. “Fast algorithm for successive computation of group betweenness centrality.” https://journals.aps.org/pre/pdf/10.1103/PhysRevE.76.056709