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Source code for networkx.algorithms.centrality.load
# coding=utf8
"""
Load centrality.
"""
# Copyright (C) 2004-2015 by
# Aric Hagberg <hagberg@lanl.gov>
# Dan Schult <dschult@colgate.edu>
# Pieter Swart <swart@lanl.gov>
# All rights reserved.
# BSD license.
__author__ = "\n".join(['Aric Hagberg (hagberg@lanl.gov)',
'Pieter Swart (swart@lanl.gov)',
'Sasha Gutfraind (ag362@cornell.edu)'])
__all__ = ['load_centrality',
'edge_load']
import networkx as nx
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 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.
In this case a cutoff is not implemented and so is ignored.
"""
# 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,weight=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=1.0/float((l-1)*(l-2)) # 1/the number of possible paths
for v in between:
between[v] *= scale
return between
load_centrality=newman_betweenness_centrality
[docs]def edge_load(G,nodes=None,cutoff=False):
"""Compute edge load.
WARNING:
This module is for demonstration and testing purposes.
"""
betweenness={}
if not nodes: # find betweenness for every node in graph
nodes=G.nodes() # that probably is what you want...
for source in nodes:
ubetween=_edge_betweenness(G,source,nodes,cutoff=cutoff)
for v in ubetween.keys():
b=betweenness.setdefault(v,0) # get or set default
betweenness[v]=ubetween[v]+b # cumulative total
return betweenness
def _edge_betweenness(G,source,nodes,cutoff=False):
"""
Edge betweenness helper.
"""
between={}
# get the predecessor data
#(pred,length)=_fast_predecessor(G,source,cutoff=cutoff)
(pred,length)=nx.predecessor(G,source,cutoff=cutoff,return_seen=True)
# order the nodes by path length
onodes = [ nn for dd,nn in sorted( (dist,n) for n,dist in length.items() )]
# intialize betweenness, doesn't account for any edge weights
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:
num_paths=len(pred[v]) # Discount betweenness if more than
for w in pred[v]: # one shortest path.
if w in pred:
num_paths=len(pred[w]) # Discount betweenness, mult path
for x in pred[w]:
between[(w,x)]+=between[(v,w)]/num_paths
between[(x,w)]+=between[(w,v)]/num_paths
return between