# coding=utf8
# Copyright (C) 2004-2018 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)
"""Betweenness centrality measures."""
from __future__ import division
from heapq import heappush, heappop
from itertools import count
import networkx as nx
from networkx.utils import py_random_state
__all__ = ['betweenness_centrality', 'edge_betweenness_centrality',
'edge_betweenness']
[docs]@py_random_state(5)
def betweenness_centrality(G, k=None, normalized=True, weight=None,
endpoints=False, seed=None):
r"""Compute the shortest-path betweenness centrality for nodes.
Betweenness centrality of a node $v$ is the sum of the
fraction of all-pairs shortest paths that pass through $v$
.. math::
c_B(v) =\sum_{s,t \in V} \frac{\sigma(s, t|v)}{\sigma(s, t)}
where $V$ is the set of nodes, $\sigma(s, t)$ is the number of
shortest $(s, t)$-paths, and $\sigma(s, t|v)$ is the number of
those paths passing through some node $v$ other than $s, t$.
If $s = t$, $\sigma(s, t) = 1$, and if $v \in {s, t}$,
$\sigma(s, t|v) = 0$ [2]_.
Parameters
----------
G : graph
A NetworkX graph.
k : int, optional (default=None)
If k is not None use k node samples to estimate betweenness.
The value of k <= n where n is the number of nodes in the graph.
Higher values give better approximation.
normalized : bool, optional
If True the betweenness values are normalized by `2/((n-1)(n-2))`
for graphs, and `1/((n-1)(n-2))` for directed graphs where `n`
is the number of nodes in G.
weight : None or string, optional (default=None)
If None, all edge weights are considered equal.
Otherwise holds the name of the edge attribute used as weight.
endpoints : bool, optional
If True include the endpoints in the shortest path counts.
seed : integer, random_state, or None (default)
Indicator of random number generation state.
See :ref:`Randomness<randomness>`.
Note that this is only used if k is not None.
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]_.
See [4]_ for the original first published version and [2]_ for details on
algorithms for variations and related metrics.
For approximate betweenness calculations set k=#samples to use
k nodes ("pivots") to estimate the betweenness values. For an estimate
of the number of pivots needed see [3]_.
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] Ulrik Brandes:
A Faster Algorithm for Betweenness Centrality.
Journal of Mathematical Sociology 25(2):163-177, 2001.
http://www.inf.uni-konstanz.de/algo/publications/b-fabc-01.pdf
.. [2] Ulrik Brandes:
On Variants of Shortest-Path Betweenness
Centrality and their Generic Computation.
Social Networks 30(2):136-145, 2008.
http://www.inf.uni-konstanz.de/algo/publications/b-vspbc-08.pdf
.. [3] Ulrik Brandes and Christian Pich:
Centrality Estimation in Large Networks.
International Journal of Bifurcation and Chaos 17(7):2303-2318, 2007.
http://www.inf.uni-konstanz.de/algo/publications/bp-celn-06.pdf
.. [4] Linton C. Freeman:
A set of measures of centrality based on betweenness.
Sociometry 40: 35–41, 1977
http://moreno.ss.uci.edu/23.pdf
"""
betweenness = dict.fromkeys(G, 0.0) # b[v]=0 for v in G
if k is None:
nodes = G
else:
nodes = seed.sample(G.nodes(), k)
for s in nodes:
# single source shortest paths
if weight is None: # use BFS
S, P, sigma = _single_source_shortest_path_basic(G, s)
else: # use Dijkstra's algorithm
S, P, sigma = _single_source_dijkstra_path_basic(G, s, weight)
# accumulation
if endpoints:
betweenness = _accumulate_endpoints(betweenness, S, P, sigma, s)
else:
betweenness = _accumulate_basic(betweenness, S, P, sigma, s)
# rescaling
betweenness = _rescale(betweenness, len(G), normalized=normalized,
directed=G.is_directed(), k=k, endpoints=endpoints)
return betweenness
[docs]@py_random_state(4)
def edge_betweenness_centrality(G, k=None, normalized=True, weight=None,
seed=None):
r"""Compute betweenness centrality for edges.
Betweenness centrality of an edge $e$ is the sum of the
fraction of all-pairs shortest paths that pass through $e$
.. math::
c_B(e) =\sum_{s,t \in V} \frac{\sigma(s, t|e)}{\sigma(s, t)}
where $V$ is the set of nodes, $\sigma(s, t)$ is the number of
shortest $(s, t)$-paths, and $\sigma(s, t|e)$ is the number of
those paths passing through edge $e$ [2]_.
Parameters
----------
G : graph
A NetworkX graph.
k : int, optional (default=None)
If k is not None use k node samples to estimate betweenness.
The value of k <= n where n is the number of nodes in the graph.
Higher values give better approximation.
normalized : bool, optional
If True the betweenness values are normalized by $2/(n(n-1))$
for graphs, and $1/(n(n-1))$ for directed graphs where $n$
is the number of nodes in G.
weight : None or string, optional (default=None)
If None, all edge weights are considered equal.
Otherwise holds the name of the edge attribute used as weight.
seed : integer, random_state, or None (default)
Indicator of random number generation state.
See :ref:`Randomness<randomness>`.
Note that this is only used if k is not None.
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
.. [2] Ulrik Brandes: On Variants of Shortest-Path Betweenness
Centrality and their Generic Computation.
Social Networks 30(2):136-145, 2008.
http://www.inf.uni-konstanz.de/algo/publications/b-vspbc-08.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))
if k is None:
nodes = G
else:
nodes = seed.sample(G.nodes(), k)
for s in nodes:
# single source shortest paths
if weight is None: # use BFS
S, P, sigma = _single_source_shortest_path_basic(G, s)
else: # use Dijkstra's algorithm
S, P, sigma = _single_source_dijkstra_path_basic(G, s, weight)
# 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_e(betweenness, len(G), normalized=normalized,
directed=G.is_directed())
return betweenness
# obsolete name
def edge_betweenness(G, k=None, normalized=True, weight=None, seed=None):
return edge_betweenness_centrality(G, k, normalized, weight, seed)
# 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, weight):
# 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 = heappush
pop = heappop
seen = {s: 0}
c = count()
Q = [] # use Q as heap with (distance,node id) tuples
push(Q, (0, next(c), 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].items():
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, next(c), 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 + 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 + 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 + 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, n, normalized,
directed=False, k=None, endpoints=False):
if normalized:
if endpoints:
if n < 2:
scale = None # no normalization
else:
# Scale factor should include endpoint nodes
scale = 1 / (n * (n - 1))
elif n <= 2:
scale = None # no normalization b=0 for all nodes
else:
scale = 1 / ((n - 1) * (n - 2))
else: # rescale by 2 for undirected graphs
if not directed:
scale = 0.5
else:
scale = None
if scale is not None:
if k is not None:
scale = scale * n / k
for v in betweenness:
betweenness[v] *= scale
return betweenness
def _rescale_e(betweenness, n, normalized, directed=False, k=None):
if normalized:
if n <= 1:
scale = None # no normalization b=0 for all nodes
else:
scale = 1 / (n * (n - 1))
else: # rescale by 2 for undirected graphs
if not directed:
scale = 0.5
else:
scale = None
if scale is not None:
if k is not None:
scale = scale * n / k
for v in betweenness:
betweenness[v] *= scale
return betweenness