"""Betweenness centrality measures for subsets of nodes."""
import networkx as nx
from networkx.algorithms.centrality.betweenness import (
_add_edge_keys,
_rescale,
)
from networkx.algorithms.centrality.betweenness import (
_single_source_dijkstra_path_basic as dijkstra,
)
from networkx.algorithms.centrality.betweenness import (
_single_source_shortest_path_basic as shortest_path,
)
__all__ = [
"betweenness_centrality_subset",
"edge_betweenness_centrality_subset",
]
[docs]
@nx._dispatchable(edge_attrs="weight")
def betweenness_centrality_subset(G, sources, targets, normalized=False, weight=None):
r"""Compute betweenness centrality for a subset of nodes.
.. math::
c_B(v) = \sum_{s \in S, t \in T} \frac{\sigma(s, t | v)}{\sigma(s, t)}
where $S$ is the set of sources, $T$ is the set of targets,
$\sigma(s, t)$ 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 $s$ and $t$.
If $s = t$, $\sigma(s, t) = 1$,
and if $v \in \{s, t\}$, $\sigma(s, t | v) = 0$ [2]_.
The denominator $\sigma(s, t)$ is a normalization factor that can be
turned off to get the raw path counts.
Parameters
----------
G : graph
A NetworkX graph.
sources: list of nodes
Nodes to use as sources for shortest paths in betweenness.
targets: list of nodes
Nodes to use as targets for shortest paths in betweenness.
normalized : bool, optional (default=False)
If `True`, the betweenness values are rescaled by dividing by the number of
possible $(s, t)$-pairs in the graph.
weight : None or string, optional (default=None)
If `None`, all edge weights are 1.
Otherwise holds the name of the edge attribute used as weight.
Weights are used to calculate weighted shortest paths, so they are
interpreted as distances.
Returns
-------
nodes : dict
Dictionary of nodes with betweenness centrality as the value.
See Also
--------
betweenness_centrality
edge_betweenness_centrality
edge_betweenness_centrality_subset
load_centrality
Notes
-----
The basic algorithm is from [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.
The normalization might seem a little strange but it is
designed to make betweenness_centrality(G) be the same as
betweenness_centrality_subset(G,sources=G.nodes(),targets=G.nodes()).
The total number of paths between source and target is counted
differently for directed and undirected graphs. Directed paths
are easy to count. Undirected paths are tricky: should a path
from ``u`` to ``v`` count as 1 undirected path or as 2 directed paths?
We are only counting the paths in one direction. They are
undirected paths but we are counting them in a directed way.
To count them as undirected paths, each should count as half a path.
References
----------
.. [1] Ulrik Brandes, A Faster Algorithm for Betweenness Centrality.
Journal of Mathematical Sociology 25(2):163-177, 2001.
https://doi.org/10.1080/0022250X.2001.9990249
.. [2] Ulrik Brandes: On Variants of Shortest-Path Betweenness
Centrality and their Generic Computation.
Social Networks 30(2):136-145, 2008.
https://doi.org/10.1016/j.socnet.2007.11.001
"""
b = dict.fromkeys(G, 0.0) # b[v]=0 for v in G
for s in sources:
# single source shortest paths
if weight is None: # use BFS
S, P, sigma, _ = shortest_path(G, s)
else: # use Dijkstra's algorithm
S, P, sigma, _ = dijkstra(G, s, weight)
b = _accumulate_subset(b, S, P, sigma, s, targets)
b = _rescale(
b, len(G), normalized=normalized, directed=G.is_directed(), endpoints=False
)
return b
[docs]
@nx._dispatchable(edge_attrs="weight")
def edge_betweenness_centrality_subset(
G, sources, targets, normalized=False, weight=None
):
r"""Compute betweenness centrality for edges for a subset of nodes.
.. math::
c_B(e) = \sum_{s \in S, t \in T} \frac{\sigma(s, t | e)}{\sigma(s, t)}
where $S$ is the set of sources, $T$ is the set of targets,
$\sigma(s, t)$ is the number of shortest $(s, t)$-paths,
and $\sigma(s, t | e)$ is the number of those paths
passing through edge $e$ [1]_.
The denominator $\sigma(s, t)$ is a normalization factor that can be
turned off to get the raw path counts.
Parameters
----------
G : graph
A networkx graph.
sources: list of nodes
Nodes to use as sources for shortest paths in betweenness.
targets: list of nodes
Nodes to use as targets for shortest paths in betweenness.
normalized : bool, optional (default=False)
If `True`, the betweenness values are rescaled by dividing by the number of
possible $(s, t)$-pairs in the graph.
weight : None or string, optional (default=None)
If `None`, all edge weights are 1.
Otherwise holds the name of the edge attribute used as weight.
Weights are used to calculate weighted shortest paths, so they are
interpreted as distances.
Returns
-------
edges : dict
Dictionary of edges with betweenness centrality as the value.
See Also
--------
betweenness_centrality
betweenness_centrality_subset
edge_betweenness_centrality
edge_load
Notes
-----
The basic algorithm is from [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.
The normalization might seem a little strange but it is the same
as in edge_betweenness_centrality() and is designed to make
edge_betweenness_centrality(G) be the same as
edge_betweenness_centrality_subset(G,sources=G.nodes(),targets=G.nodes()).
References
----------
.. [1] Ulrik Brandes: On Variants of Shortest-Path Betweenness
Centrality and their Generic Computation.
Social Networks 30(2):136-145, 2008.
https://doi.org/10.1016/j.socnet.2007.11.001
"""
b = dict.fromkeys(G, 0.0) # b[v]=0 for v in G
b.update(dict.fromkeys(G.edges(), 0.0)) # b[e] for e in G.edges()
for s in sources:
# single source shortest paths
if weight is None: # use BFS
S, P, sigma, _ = shortest_path(G, s)
else: # use Dijkstra's algorithm
S, P, sigma, _ = dijkstra(G, s, weight)
b = _accumulate_edges_subset(b, S, P, sigma, s, targets)
for n in G: # remove nodes to only return edges
del b[n]
b = _rescale(b, len(G), normalized=normalized, directed=G.is_directed())
if G.is_multigraph():
b = _add_edge_keys(G, b, weight=weight)
return b
def _accumulate_subset(betweenness, S, P, sigma, s, targets):
delta = dict.fromkeys(S, 0.0)
target_set = set(targets) - {s}
while S:
w = S.pop()
if w in target_set:
coeff = (delta[w] + 1.0) / sigma[w]
else:
coeff = 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_edges_subset(betweenness, S, P, sigma, s, targets):
"""edge_betweenness_centrality_subset helper."""
delta = dict.fromkeys(S, 0)
target_set = set(targets)
while S:
w = S.pop()
for v in P[w]:
if w in target_set:
c = (sigma[v] / sigma[w]) * (1.0 + delta[w])
else:
c = delta[w] / len(P[w])
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
