# -*- coding: utf-8 -*-
# Copyright (C) 2004-2016 by
# Aric Hagberg <hagberg@lanl.gov>
# Dan Schult <dschult@colgate.edu>
# Pieter Swart <swart@lanl.gov>
# All rights reserved.
# BSD license.
#
# Authors: Aric Hagberg (hagberg@lanl.gov)
# Pieter Swart (swart@lanl.gov)
# Sasha Gutfraind (ag362@cornell.edu)
"""
Load centrality.
"""
from __future__ import division
from operator import itemgetter
import networkx as nx
__all__ = ['load_centrality',
'edge_load_centrality']
def newman_betweenness_centrality(G, v=None, cutoff=None,
normalized=True, weight=None):
"""Compute load centrality for nodes.
The load centrality of a node is the fraction of all shortest
paths that pass through that node.
Parameters
----------
G : graph
A networkx graph
normalized : bool, optional
If True the betweenness values are normalized by b=b/(n-1)(n-2) where
n is the number of nodes in G.
weight : None or string, optional
If None, edge weights are ignored.
Otherwise holds the name of the edge attribute used as weight.
cutoff : bool, optional
If specified, only consider paths of length <= cutoff.
Returns
-------
nodes : dictionary
Dictionary of nodes with centrality as the value.
See Also
--------
betweenness_centrality()
Notes
-----
Load centrality is slightly different than betweenness. It was originally
introduced by [2]_. For this load algorithm see [1]_.
References
----------
.. [1] Mark E. J. Newman:
Scientific collaboration networks. II.
Shortest paths, weighted networks, and centrality.
Physical Review E 64, 016132, 2001.
http://journals.aps.org/pre/abstract/10.1103/PhysRevE.64.016132
.. [2] Kwang-Il Goh, Byungnam Kahng and Doochul Kim
Universal behavior of Load Distribution in Scale-Free Networks.
Physical Review Letters 87(27):1–4, 2001.
http://phya.snu.ac.kr/~dkim/PRL87278701.pdf
"""
if v is not None: # only one node
betweenness = 0.0
for source in G:
ubetween = _node_betweenness(G, source, cutoff, False, weight)
betweenness += ubetween[v] if v in ubetween else 0
if normalized:
order = G.order()
if order <= 2:
return betweenness # no normalization b=0 for all nodes
betweenness *= 1.0 / ((order-1) * (order-2))
return betweenness
else:
betweenness = {}.fromkeys(G, 0.0)
for source in betweenness:
ubetween = _node_betweenness(G, source, cutoff, False, weight)
for vk in ubetween:
betweenness[vk] += ubetween[vk]
if normalized:
order = G.order()
if order <= 2:
return betweenness # no normalization b=0 for all nodes
scale = 1.0 / ((order-1) * (order-2))
for v in betweenness:
betweenness[v] *= scale
return betweenness # all nodes
def _node_betweenness(G, source, cutoff=False, normalized=True, weight=None):
"""Node betweenness_centrality helper:
See betweenness_centrality for what you probably want.
This actually computes "load" and not betweenness.
See https://networkx.lanl.gov/ticket/103
This calculates the load of each node for paths from a single source.
(The fraction of number of shortests paths from source that go
through each node.)
To get the load for a node you need to do all-pairs shortest paths.
If weight is not None then use Dijkstra for finding shortest paths.
"""
# get the predecessor and path length data
if weight is None:
(pred, length) = nx.predecessor(G, source, cutoff=cutoff,
return_seen=True)
else:
(pred, length) = nx.dijkstra_predecessor_and_distance(G, source,
cutoff, weight)
# order the nodes by path length
onodes = [(l, vert) for (vert, l) in length.items()]
onodes.sort()
onodes[:] = [vert for (l, vert) in onodes if l > 0]
# intialize betweenness
between = {}.fromkeys(length, 1.0)
while onodes:
v = onodes.pop()
if v in pred:
num_paths = len(pred[v]) # Discount betweenness if more than
for x in pred[v]: # one shortest path.
if x == source: # stop if hit source because all remaining v
break # also have pred[v]==[source]
between[x] += between[v] / float(num_paths)
# remove source
for v in between:
between[v] -= 1
# rescale to be between 0 and 1
if normalized:
l = len(between)
if l > 2:
# scale by 1/the number of possible paths
scale = 1.0 / float((l - 1) * (l - 2))
for v in between:
between[v] *= scale
return between
load_centrality = newman_betweenness_centrality
[docs]def edge_load_centrality(G, cutoff=False):
"""Compute edge load.
WARNING: This concept of edge load has not been analysed
or discussed outside of NetworkX that we know of.
It is based loosely on load_centrality in the sense that
it counts the number of shortest paths which cross each edge.
This function is for demonstration and testing purposes.
Parameters
----------
G : graph
A networkx graph
cutoff : bool, optional
If specified, only consider paths of length <= cutoff.
Returns
-------
A dict keyed by edge 2-tuple to the number of shortest paths
which use that edge. Where more than one path is shortest
the count is divided equally among paths.
"""
betweenness = {}
for u, v in G.edges():
betweenness[(u, v)] = 0.0
betweenness[(v, u)] = 0.0
for source in G:
ubetween = _edge_betweenness(G, source, cutoff=cutoff)
for e, ubetweenv in ubetween.items():
betweenness[e] += ubetweenv # cumulative total
return betweenness
def _edge_betweenness(G, source, nodes=None, cutoff=False):
"""
Edge betweenness helper.
"""
# get the predecessor data
(pred, length) = nx.predecessor(G, source, cutoff=cutoff, return_seen=True)
# order the nodes by path length
onodes = [n for n, d in sorted(length.items(), key=itemgetter(1))]
# intialize betweenness, doesn't account for any edge weights
between = {}
for u, v in G.edges(nodes):
between[(u, v)] = 1.0
between[(v, u)] = 1.0
while onodes: # work through all paths
v = onodes.pop()
if v in pred:
# Discount betweenness if more than one shortest path.
num_paths = len(pred[v])
for w in pred[v]:
if w in pred:
# Discount betweenness, mult path
num_paths = len(pred[w])
for x in pred[w]:
between[(w, x)] += between[(v, w)] / num_paths
between[(x, w)] += between[(w, v)] / num_paths
return between