communicability_betweenness_centrality¶
- communicability_betweenness_centrality(G)[source]¶
Returns subgraph communicability for all pairs of nodes in G.
Communicability betweenness measure makes use of the number of walks connecting every pair of nodes as the basis of a betweenness centrality measure.
- Parameters
- G: graph
- Returns
- nodesdictionary
Dictionary of nodes with communicability betweenness as the value.
- Raises
- NetworkXError
If the graph is not undirected and simple.
Notes
Let
G=(V,E)
be a simple undirected graph withn
nodes andm
edges, andA
denote the adjacency matrix ofG
.Let
G(r)=(V,E(r))
be the graph resulting from removing all edges connected to noder
but not the node itself.The adjacency matrix for
G(r)
isA+E(r)
, whereE(r)
has nonzeros only in row and columnr
.The subraph betweenness of a node
r
is [1]\[\omega_{r} = \frac{1}{C}\sum_{p}\sum_{q}\frac{G_{prq}}{G_{pq}}, p\neq q, q\neq r,\]where
G_{prq}=(e^{A}_{pq} - (e^{A+E(r)})_{pq}
is the number of walks involving node r,G_{pq}=(e^{A})_{pq}
is the number of closed walks starting at nodep
and ending at nodeq
, andC=(n-1)^{2}-(n-1)
is a normalization factor equal to the number of terms in the sum.The resulting
omega_{r}
takes values between zero and one. The lower bound cannot be attained for a connected graph, and the upper bound is attained in the star graph.References
- 1
Ernesto Estrada, Desmond J. Higham, Naomichi Hatano, “Communicability Betweenness in Complex Networks” Physica A 388 (2009) 764-774. https://arxiv.org/abs/0905.4102
Examples
>>> G = nx.Graph([(0, 1), (1, 2), (1, 5), (5, 4), (2, 4), (2, 3), (4, 3), (3, 6)]) >>> cbc = nx.communicability_betweenness_centrality(G) >>> print([f"{node} {cbc[node]:0.2f}" for node in sorted(cbc)]) ['0 0.03', '1 0.45', '2 0.51', '3 0.45', '4 0.40', '5 0.19', '6 0.03']