closeness_centrality

closeness_centrality(G, u=None, distance=None, normalized=True)

Compute closeness centrality for nodes.

Closeness centrality [R171] of a node \(u\) is the reciprocal of the sum of the shortest path distances from \(u\) to all \(n-1\) other nodes. Since the sum of distances depends on the number of nodes in the graph, closeness is normalized by the sum of minimum possible distances \(n-1\).

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

where \(d(v, u)\) is the shortest-path distance between \(v\) and \(u\), and \(n\) is the number of nodes in the graph.

Notice that higher values of closeness indicate higher centrality.

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

normalized : bool, optional

If True (default) normalize by the number of nodes in the connected part of the graph.

Returns :

nodes : dictionary

Dictionary of nodes with closeness centrality as the value.

Notes

The closeness centrality is normalized to \((n-1)/(|G|-1)\) where \(n\) is the number of nodes in the connected part of graph containing the node. If the graph is not completely connected, this algorithm computes the closeness centrality for each connected part separately.

If the ‘distance’ keyword is set to an edge attribute key then the shortest-path length will be computed using Dijkstra’s algorithm with that edge attribute as the edge weight.

References

[R171](1, 2) Freeman, L.C., 1979. Centrality in networks: I. Conceptual clarification. Social Networks 1, 215–239. http://www.soc.ucsb.edu/faculty/friedkin/Syllabi/Soc146/Freeman78.PDF