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"""
Betweenness centrality measures.
"""
# Copyright (C) 2004-2010 by
# Aric Hagberg <hagberg@lanl.gov>
# Dan Schult <dschult@colgate.edu>
# Pieter Swart <swart@lanl.gov>
# All rights reserved.
# BSD license.
__author__ = """Aric Hagberg (hagberg@lanl.gov)"""
__all__ = ['betweenness_centrality',
'edge_betweenness_centrality',
'edge_betweenness'
]
import heapq
import networkx as nx
def betweenness_centrality(G,normalized=True,
weighted_edges=False,
endpoints=False):
"""Compute betweenness centrality for nodes.
Betweenness 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.
weighted_edges : bool, optional
Consider the edge weights in determining the shortest paths.
The edge weights must be greater than zero.
If False, all edge weights are considered equal.
Returns
-------
nodes : dictionary
Dictionary of nodes with betweenness centrality as the value.
See Also
--------
edge_betweenness_centrality
load_centrality
Notes
-----
The algorithm is from Ulrik Brandes [1]_.
For weighted graphs the edge weights must be greater than zero.
Zero edge weights can produce an infinite number of equal length
paths between pairs of nodes.
References
----------
.. [1] A Faster Algorithm for Betweenness Centrality.
Ulrik Brandes,
Journal of Mathematical Sociology 25(2):163-177, 2001.
http://www.inf.uni-konstanz.de/algo/publications/b-fabc-01.pdf
"""
betweenness=dict.fromkeys(G,0.0) # b[v]=0 for v in G
for s in G:
# single source shortest paths
if weighted_edges: # use Dijkstra's algorithm
S,P,sigma=_single_source_dijkstra_path_basic(G,s)
else: # use BFS
S,P,sigma=_single_source_shortest_path_basic(G,s)
# accumulation
if endpoints:
betweenness=_accumulate_endpoints(betweenness,S,P,sigma,s)
else:
betweenness=_accumulate_basic(betweenness,S,P,sigma,s)
# rescaling
betweenness=_rescale(betweenness,
normalized=normalized,
directed=G.is_directed())
return betweenness
def edge_betweenness_centrality(G,normalized=True,
weighted_edges=False):
"""Compute betweenness centrality for edges.
Betweenness centrality of an edge is the fraction of all shortest
paths that pass through that edge.
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.
weighted_edges : bool, optional
Consider the edge weights in determining the shortest paths.
The edge weights must be greater than zero.
If False, all edge weights are considered equal.
Returns
-------
edges : dictionary
Dictionary of edges with betweenness centrality as the value.
See Also
--------
betweenness_centrality
edge_load
Notes
-----
The algorithm is from Ulrik Brandes [1]_.
For weighted graphs the edge weights must be greater than zero.
Zero edge weights can produce an infinite number of equal length
paths between pairs of nodes.
References
----------
.. [1] A Faster Algorithm for Betweenness Centrality.
Ulrik Brandes,
Journal of Mathematical Sociology 25(2):163-177, 2001.
http://www.inf.uni-konstanz.de/algo/publications/b-fabc-01.pdf
"""
betweenness=dict.fromkeys(G,0.0) # b[v]=0 for v in G
# b[e]=0 for e in G.edges()
betweenness.update(dict.fromkeys(G.edges(),0.0))
for s in G:
# single source shortest paths
if weighted_edges: # use Dijkstra's algorithm
S,P,sigma=_single_source_dijkstra_path_basic(G,s)
else: # use BFS
S,P,sigma=_single_source_shortest_path_basic(G,s)
# accumulation
betweenness=_accumulate_edges(betweenness,S,P,sigma,s)
# rescaling
for n in G: # remove nodes to only return edges
del betweenness[n]
betweenness=_rescale(betweenness,
normalized=normalized,
directed=G.is_directed())
return betweenness
# obsolete name
def edge_betweenness(G,normalized=True,weighted_edges=False):
import warnings
warnings.warn("""edge_betweenness() is deprecated,
use edge_betweenness_centrality()""",
DeprecationWarning)
return edge_betweenness_centrality(G,
normalized=normalized,
weighted_edges=weighted_edges)
# helpers for betweenness centrality
def _single_source_shortest_path_basic(G,s):
S=[]
P={}
for v in G:
P[v]=[]
sigma=dict.fromkeys(G,0.0) # sigma[v]=0 for v in G
D={}
sigma[s]=1.0
D[s]=0
Q=[s]
while Q: # use BFS to find shortest paths
v=Q.pop(0)
S.append(v)
Dv=D[v]
sigmav=sigma[v]
for w in G[v]:
if w not in D:
Q.append(w)
D[w]=Dv+1
if D[w]==Dv+1: # this is a shortest path, count paths
sigma[w] += sigmav
P[w].append(v) # predecessors
return S,P,sigma
def _single_source_dijkstra_path_basic(G,s):
# modified from Eppstein
S=[]
P={}
for v in G:
P[v]=[]
sigma=dict.fromkeys(G,0.0) # sigma[v]=0 for v in G
D={}
sigma[s]=1.0
push=heapq.heappush
pop=heapq.heappop
seen = {s:0}
Q=[] # use Q as heap with (distance,node id) tuples
push(Q,(0,s,s))
while Q:
(dist,pred,v)=pop(Q)
if v in D:
continue # already searched this node.
sigma[v] += sigma[pred] # count paths
S.append(v)
D[v] = dist
for w,edgedata in G[v].iteritems():
vw_dist = dist + edgedata.get('weight',1)
if w not in D and (w not in seen or vw_dist < seen[w]):
seen[w] = vw_dist
push(Q,(vw_dist,v,w))
sigma[w]=0.0
P[w]=[v]
elif vw_dist==seen[w]: # handle equal paths
sigma[w] += sigma[v]
P[w].append(v)
return S,P,sigma
def _accumulate_basic(betweenness,S,P,sigma,s):
delta=dict.fromkeys(S,0)
while S:
w=S.pop()
coeff=(1.0+delta[w])/sigma[w]
for v in P[w]:
delta[v] += sigma[v]*coeff
if w != s:
betweenness[w]+=delta[w]
return betweenness
def _accumulate_endpoints(betweenness,S,P,sigma,s):
betweenness[s]+=len(S)-1
delta=dict.fromkeys(S,0)
while S:
w=S.pop()
coeff=(1.0+delta[w])/sigma[w]
for v in P[w]:
delta[v] += sigma[v]*coeff
if w != s:
betweenness[w] += delta[w]+1
return betweenness
def _accumulate_edges(betweenness,S,P,sigma,s):
delta=dict.fromkeys(S,0)
while S:
w=S.pop()
coeff=(1.0+delta[w])/sigma[w]
for v in P[w]:
c=sigma[v]*coeff
if (v,w) not in betweenness:
betweenness[(w,v)]+=c
else:
betweenness[(v,w)]+=c
delta[v]+=c
if w != s:
betweenness[w]+=delta[w]
return betweenness
def _rescale(betweenness,normalized,directed=False):
if normalized is True:
order=len(betweenness)
if order <=2:
scale=None # no normalization b=0 for all nodes
else:
scale=1.0/((order-1)*(order-2))
else: # rescale by 2 for undirected graphs
if not directed:
scale=1.0/2.0
else:
scale=None
if scale is not None:
for v in betweenness:
betweenness[v] *= scale
return betweenness
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