group_betweenness_centrality#

group_betweenness_centrality(G, C, normalized=True, weight=None, endpoints=False)[source]#

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 \(1/(N_{out}(N_{out}-1))\) where \(N_{out}\) is the number of nodes in G that are not in C. This ensures the reported value is between 0 and 1. If endpoints is True, the normalization uses all nodes in G. The reported value is then between \(2N_{in}/(N-1)\) and 1 where \(N_{in}\) is the number of nodes in C and \(N\) the number of nodes in G.

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)

By default, only node-pairs that are both not in C are counted for group betweenness centrality. The count is how many non-C node-pairs have nodes from C “between” them on a shortest path.

When endpoints=True, we also count node-pairs with one or both nodes in C while considering endpoint nodes as being between the node-pairs. So we count paths that start in C whether or not they pass through any other nodes in C. This adds centrality to large groups without any reference to the connectivity of the group. The minimum normalized score is \(N_{in}(N_{in}-1)/(N(N-1))\) instead of 0. For that reason, this feature is rarely used.

We don’t currently support considering node-pairs with nodes in C without also counting their endpoints. Nor do we support counting endpoints while only considering node-pairs that are both not in C. This keyword indicates both coutning endpoints of paths and allowing node-pairs in C.

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.