279 lines
9.4 KiB
Python
279 lines
9.4 KiB
Python
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# Copyright (C) 2004-2019 by
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# Aric Hagberg <hagberg@lanl.gov>
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# Dan Schult <dschult@colgate.edu>
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# Pieter Swart <swart@lanl.gov>
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# All rights reserved.
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# BSD license.
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#
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# Author: Aric Hagberg (hagberg@lanl.gov)
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"""Betweenness centrality measures for subsets of nodes."""
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import networkx as nx
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from networkx.algorithms.centrality.betweenness import\
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_single_source_dijkstra_path_basic as dijkstra
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from networkx.algorithms.centrality.betweenness import\
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_single_source_shortest_path_basic as shortest_path
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__all__ = ['betweenness_centrality_subset', 'betweenness_centrality_source',
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'edge_betweenness_centrality_subset']
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def betweenness_centrality_subset(G, sources, targets, normalized=False,
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weight=None):
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r"""Compute betweenness centrality for a subset of nodes.
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.. math::
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c_B(v) =\sum_{s\in S, t \in T} \frac{\sigma(s, t|v)}{\sigma(s, t)}
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where $S$ is the set of sources, $T$ is the set of targets,
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$\sigma(s, t)$ is the number of shortest $(s, t)$-paths,
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and $\sigma(s, t|v)$ is the number of those paths
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passing through some node $v$ other than $s, t$.
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If $s = t$, $\sigma(s, t) = 1$,
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and if $v \in {s, t}$, $\sigma(s, t|v) = 0$ [2]_.
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Parameters
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----------
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G : graph
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A NetworkX graph.
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sources: list of nodes
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Nodes to use as sources for shortest paths in betweenness
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targets: list of nodes
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Nodes to use as targets for shortest paths in betweenness
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normalized : bool, optional
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If True the betweenness values are normalized by $2/((n-1)(n-2))$
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for graphs, and $1/((n-1)(n-2))$ for directed graphs where $n$
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is the number of nodes in G.
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weight : None or string, optional (default=None)
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If None, all edge weights are considered equal.
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Otherwise holds the name of the edge attribute used as weight.
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Returns
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-------
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nodes : dictionary
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Dictionary of nodes with betweenness centrality as the value.
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See Also
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--------
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edge_betweenness_centrality
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load_centrality
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Notes
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-----
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The basic algorithm is from [1]_.
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For weighted graphs the edge weights must be greater than zero.
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Zero edge weights can produce an infinite number of equal length
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paths between pairs of nodes.
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The normalization might seem a little strange but it is
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designed to make betweenness_centrality(G) be the same as
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betweenness_centrality_subset(G,sources=G.nodes(),targets=G.nodes()).
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The total number of paths between source and target is counted
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differently for directed and undirected graphs. Directed paths
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are easy to count. Undirected paths are tricky: should a path
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from "u" to "v" count as 1 undirected path or as 2 directed paths?
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For betweenness_centrality we report the number of undirected
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paths when G is undirected.
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For betweenness_centrality_subset the reporting is different.
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If the source and target subsets are the same, then we want
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to count undirected paths. But if the source and target subsets
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differ -- for example, if sources is {0} and targets is {1},
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then we are only counting the paths in one direction. They are
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undirected paths but we are counting them in a directed way.
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To count them as undirected paths, each should count as half a path.
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References
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----------
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.. [1] Ulrik Brandes, A Faster Algorithm for Betweenness Centrality.
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Journal of Mathematical Sociology 25(2):163-177, 2001.
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http://www.inf.uni-konstanz.de/algo/publications/b-fabc-01.pdf
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.. [2] Ulrik Brandes: On Variants of Shortest-Path Betweenness
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Centrality and their Generic Computation.
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Social Networks 30(2):136-145, 2008.
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http://www.inf.uni-konstanz.de/algo/publications/b-vspbc-08.pdf
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"""
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b = dict.fromkeys(G, 0.0) # b[v]=0 for v in G
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for s in sources:
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# single source shortest paths
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if weight is None: # use BFS
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S, P, sigma = shortest_path(G, s)
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else: # use Dijkstra's algorithm
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S, P, sigma = dijkstra(G, s, weight)
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b = _accumulate_subset(b, S, P, sigma, s, targets)
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b = _rescale(b, len(G), normalized=normalized, directed=G.is_directed())
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return b
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def edge_betweenness_centrality_subset(G, sources, targets, normalized=False,
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weight=None):
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r"""Compute betweenness centrality for edges for a subset of nodes.
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.. math::
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c_B(v) =\sum_{s\in S,t \in T} \frac{\sigma(s, t|e)}{\sigma(s, t)}
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where $S$ is the set of sources, $T$ is the set of targets,
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$\sigma(s, t)$ is the number of shortest $(s, t)$-paths,
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and $\sigma(s, t|e)$ is the number of those paths
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passing through edge $e$ [2]_.
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Parameters
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----------
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G : graph
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A networkx graph.
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sources: list of nodes
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Nodes to use as sources for shortest paths in betweenness
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targets: list of nodes
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Nodes to use as targets for shortest paths in betweenness
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normalized : bool, optional
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If True the betweenness values are normalized by `2/(n(n-1))`
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for graphs, and `1/(n(n-1))` for directed graphs where `n`
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is the number of nodes in G.
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weight : None or string, optional (default=None)
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If None, all edge weights are considered equal.
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Otherwise holds the name of the edge attribute used as weight.
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Returns
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-------
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edges : dictionary
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Dictionary of edges with Betweenness centrality as the value.
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See Also
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--------
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betweenness_centrality
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edge_load
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Notes
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-----
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The basic algorithm is from [1]_.
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For weighted graphs the edge weights must be greater than zero.
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Zero edge weights can produce an infinite number of equal length
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paths between pairs of nodes.
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The normalization might seem a little strange but it is the same
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as in edge_betweenness_centrality() and is designed to make
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edge_betweenness_centrality(G) be the same as
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edge_betweenness_centrality_subset(G,sources=G.nodes(),targets=G.nodes()).
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References
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----------
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.. [1] Ulrik Brandes, A Faster Algorithm for Betweenness Centrality.
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Journal of Mathematical Sociology 25(2):163-177, 2001.
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http://www.inf.uni-konstanz.de/algo/publications/b-fabc-01.pdf
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.. [2] Ulrik Brandes: On Variants of Shortest-Path Betweenness
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Centrality and their Generic Computation.
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Social Networks 30(2):136-145, 2008.
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http://www.inf.uni-konstanz.de/algo/publications/b-vspbc-08.pdf
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"""
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b = dict.fromkeys(G, 0.0) # b[v]=0 for v in G
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b.update(dict.fromkeys(G.edges(), 0.0)) # b[e] for e in G.edges()
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for s in sources:
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# single source shortest paths
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if weight is None: # use BFS
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S, P, sigma = shortest_path(G, s)
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else: # use Dijkstra's algorithm
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S, P, sigma = dijkstra(G, s, weight)
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b = _accumulate_edges_subset(b, S, P, sigma, s, targets)
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for n in G: # remove nodes to only return edges
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del b[n]
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b = _rescale_e(b, len(G), normalized=normalized, directed=G.is_directed())
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return b
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# obsolete name
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def betweenness_centrality_source(G, normalized=True, weight=None,
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sources=None):
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if sources is None:
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sources = G.nodes()
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targets = list(G)
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return betweenness_centrality_subset(G, sources, targets, normalized,
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weight)
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def _accumulate_subset(betweenness, S, P, sigma, s, targets):
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delta = dict.fromkeys(S, 0.0)
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target_set = set(targets) - {s}
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while S:
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w = S.pop()
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if w in target_set:
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coeff = (delta[w] + 1.0) / sigma[w]
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else:
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coeff = delta[w] / sigma[w]
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for v in P[w]:
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delta[v] += sigma[v] * coeff
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if w != s:
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betweenness[w] += delta[w]
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return betweenness
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def _accumulate_edges_subset(betweenness, S, P, sigma, s, targets):
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"""edge_betweenness_centrality_subset helper."""
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delta = dict.fromkeys(S, 0)
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target_set = set(targets)
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while S:
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w = S.pop()
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for v in P[w]:
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if w in target_set:
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c = (sigma[v] / sigma[w]) * (1.0 + delta[w])
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else:
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c = delta[w] / len(P[w])
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if (v, w) not in betweenness:
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betweenness[(w, v)] += c
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else:
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betweenness[(v, w)] += c
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delta[v] += c
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if w != s:
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betweenness[w] += delta[w]
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return betweenness
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def _rescale(betweenness, n, normalized, directed=False):
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"""betweenness_centrality_subset helper."""
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if normalized:
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if n <= 2:
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scale = None # no normalization b=0 for all nodes
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else:
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scale = 1.0 / ((n - 1) * (n - 2))
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else: # rescale by 2 for undirected graphs
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if not directed:
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scale = 0.5
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else:
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scale = None
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if scale is not None:
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for v in betweenness:
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betweenness[v] *= scale
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return betweenness
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def _rescale_e(betweenness, n, normalized, directed=False):
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"""edge_betweenness_centrality_subset helper."""
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if normalized:
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if n <= 1:
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scale = None # no normalization b=0 for all nodes
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else:
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scale = 1.0 / (n * (n - 1))
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else: # rescale by 2 for undirected graphs
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if not directed:
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scale = 0.5
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else:
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scale = None
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if scale is not None:
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for v in betweenness:
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betweenness[v] *= scale
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return betweenness
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