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Source code for networkx.algorithms.centrality.closeness

#    Copyright (C) 2004-2018 by
#    Aric Hagberg <hagberg@lanl.gov>
#    Dan Schult <dschult@colgate.edu>
#    Pieter Swart <swart@lanl.gov>
#    All rights reserved.
#    BSD license.
#
# Authors:  Aric Hagberg <aric.hagberg@gmail.com>
#           Pieter Swart <swart@lanl.gov>
#           Sasha Gutfraind <ag362@cornell.edu>
#           Dan Schult <dschult@colgate.edu>
"""
Closeness centrality measures.
"""
import functools
import networkx as nx

__all__ = ['closeness_centrality']


[docs]def closeness_centrality(G, u=None, distance=None, wf_improved=True, reverse=False): r"""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. .. math:: 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 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, .. math:: 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. reverse : bool, optional (default=False) If True and G is a digraph, reverse the edges of G, using successors instead of predecessors. Returns ------- nodes : dictionary Dictionary of nodes with closeness centrality as the value. See Also -------- betweenness_centrality, load_centrality, eigenvector_centrality, degree_centrality 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 scaled by that parts size. 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 ---------- .. [1] Linton C. Freeman: Centrality in networks: I. Conceptual clarification. Social Networks 1:215-239, 1979. http://leonidzhukov.ru/hse/2013/socialnetworks/papers/freeman79-centrality.pdf .. [2] pg. 201 of Wasserman, S. and Faust, K., Social Network Analysis: Methods and Applications, 1994, Cambridge University Press. """ if distance is not None: # use Dijkstra's algorithm with specified attribute as edge weight path_length = functools.partial(nx.single_source_dijkstra_path_length, weight=distance) else: # handle either directed or undirected if G.is_directed() and not reverse: path_length = nx.single_target_shortest_path_length else: path_length = nx.single_source_shortest_path_length if u is None: nodes = G.nodes() else: nodes = [u] closeness_centrality = {} for n in nodes: sp = dict(path_length(G, n)) totsp = sum(sp.values()) if totsp > 0.0 and len(G) > 1: closeness_centrality[n] = (len(sp) - 1.0) / totsp # normalize to number of nodes-1 in connected part if wf_improved: s = (len(sp) - 1.0) / (len(G) - 1) closeness_centrality[n] *= s else: closeness_centrality[n] = 0.0 if u is not None: return closeness_centrality[u] else: return closeness_centrality