# Copyright (C) 2004-2017 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