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Note

This documents the development version of NetworkX. Documentation for the current release can be found here.

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networkx.algorithms.centrality.closeness_centrality¶

closeness_centrality(G, u=None, distance=None, wf_improved=True)[source]¶

Compute closeness centrality for nodes.

Closeness centrality 1 of a node u is the reciprocal of the average shortest path distance to u over all n-1 reachable nodes.

\[C(u) = \frac{n - 1}{\sum_{v=1}^{n-1} d(v, u)},\]

where d(v, u) is the shortest-path distance between v and u, and n is the number of nodes that can reach u. Notice that the closeness distance function computes the incoming distance to u for directed graphs. To use outward distance, act on G.reverse().

Notice that higher values of closeness indicate higher centrality.

Wasserman and Faust propose an improved formula for graphs with more than one connected component. The result is “a ratio of the fraction of actors in the group who are reachable, to the average distance” from the reachable actors 2. You might think this scale factor is inverted but it is not. As is, nodes from small components receive a smaller closeness value. Letting N denote the number of nodes in the graph,

\[C_{WF}(u) = \frac{n-1}{N-1} \frac{n - 1}{\sum_{v=1}^{n-1} d(v, u)},\]

Parameters

G (graph) – A NetworkX graph
u (node, optional) – Return only the value for node u
distance (edge attribute key, optional (default=None)) – Use the specified edge attribute as the edge distance in shortest path calculations
wf_improved (bool, optional (default=True)) – If True, scale by the fraction of nodes reachable. This gives the Wasserman and Faust improved formula. For single component graphs it is the same as the original formula.

Returns

nodes – Dictionary of nodes with closeness centrality as the value.

Return type

dictionary