prominent_group

prominent_group(G, k, weight=None, C=None, endpoints=False, normalized=True, greedy=False)

Find the prominent group of size \(k\) in graph \(G\). The prominence of the group is evaluated by the group betweenness centrality.

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.

kint

The number of nodes in the group.

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.

Clist or set, optional (default=None)

list of nodes which won’t be candidates of the prominent group.

greedybool, optional (default=False)

Using a naive greedy algorithm in order to find non-optimal prominent group. For scale free networks the results are negligibly below the optimal results.

Returns:

max_GBCfloat

The group betweenness centrality of the prominent group.

max_grouplist

The list of nodes in the prominent group.

Raises:

NodeNotFound

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

See also

betweenness_centrality, group_betweenness_centrality

Notes

Group betweenness centrality is described in [1] and its importance discussed in [3]. The algorithm is described in [2] and is based on techniques mentioned 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]

Rami Puzis, Yuval Elovici, and Shlomi Dolev: “Finding the Most Prominent Group in Complex Networks” AI communications 20(4): 287-296, 2007. https://www.researchgate.net/profile/Rami_Puzis2/publication/220308855

[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