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 throughv
.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 setU
andm
be the number of nodes in the node setV
, then nodes inU
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:
- Ggraph
A bipartite graph
- nodeslist or container
Container with all nodes in one bipartite node set.
- Returns:
- betweennessdictionary
Dictionary keyed by node with bipartite betweenness centrality as the value.
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. See
bipartite documentation
for further details on how bipartite graphs are handled in NetworkX.References
[1]Borgatti, S.P. and Halgin, D. In press. â€śAnalyzing Affiliation Networksâ€ť. In Carrington, P. and Scott, J. (eds) The Sage Handbook of Social Network Analysis. Sage Publications. https://dx.doi.org/10.4135/9781446294413.n28
Examples
>>> G = nx.cycle_graph(4) >>> top_nodes = {1, 2} >>> nx.bipartite.betweenness_centrality(G, nodes=top_nodes) {0: 0.25, 1: 0.25, 2: 0.25, 3: 0.25}