Warning

This documents an unmaintained version of NetworkX. Please upgrade to a maintained version and see the current NetworkX documentation.

# Source code for networkx.algorithms.centrality.closeness

"""
Closeness centrality measures.
"""
#    Aric Hagberg <hagberg@lanl.gov>
#    Dan Schult <dschult@colgate.edu>
#    Pieter Swart <swart@lanl.gov>
import functools
import networkx as nx
__author__ = "\n".join(['Aric Hagberg <aric.hagberg@gmail.com>',
'Pieter Swart (swart@lanl.gov)',
'Sasha Gutfraind (ag362@cornell.edu)'])
__all__ = ['closeness_centrality']

[docs]def closeness_centrality(G, u=None, distance=None, normalized=True):
r"""Compute closeness centrality for nodes.

Closeness centrality _ 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.

.. 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 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.

--------
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.

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
----------
..  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
"""
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:
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 = 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 normalized:
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