209 lines
7.0 KiB
Python
209 lines
7.0 KiB
Python
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# -*- encoding: utf-8 -*-
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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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"""Functions for computing reaching centrality of a node or a graph."""
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import networkx as nx
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from networkx.utils import pairwise
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__all__ = ['global_reaching_centrality', 'local_reaching_centrality']
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def _average_weight(G, path, weight=None):
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"""Returns the average weight of an edge in a weighted path.
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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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path: list
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A list of vertices that define the path.
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weight : None or string, optional (default=None)
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If None, edge weights are ignored. Then the average weight of an edge
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is assumed to be the multiplicative inverse of the length of the path.
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Otherwise holds the name of the edge attribute used as weight.
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"""
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path_length = len(path) - 1
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if path_length <= 0:
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return 0
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if weight is None:
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return 1 / path_length
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total_weight = sum(G.edges[i, j][weight] for i, j in pairwise(path))
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return total_weight / path_length
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def global_reaching_centrality(G, weight=None, normalized=True):
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"""Returns the global reaching centrality of a directed graph.
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The *global reaching centrality* of a weighted directed graph is the
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average over all nodes of the difference between the local reaching
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centrality of the node and the greatest local reaching centrality of
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any node in the graph [1]_. For more information on the local
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reaching centrality, see :func:`local_reaching_centrality`.
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Informally, the local reaching centrality is the proportion of the
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graph that is reachable from the neighbors of the node.
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Parameters
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----------
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G : DiGraph
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A networkx DiGraph.
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weight : None or string, optional (default=None)
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Attribute to use for edge weights. If ``None``, each edge weight
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is assumed to be one. A higher weight implies a stronger
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connection between nodes and a *shorter* path length.
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normalized : bool, optional (default=True)
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Whether to normalize the edge weights by the total sum of edge
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weights.
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Returns
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-------
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h : float
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The global reaching centrality of the graph.
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Examples
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--------
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>>> import networkx as nx
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>>> G = nx.DiGraph()
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>>> G.add_edge(1, 2)
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>>> G.add_edge(1, 3)
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>>> nx.global_reaching_centrality(G)
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1.0
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>>> G.add_edge(3, 2)
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>>> nx.global_reaching_centrality(G)
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0.75
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See also
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--------
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local_reaching_centrality
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References
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----------
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.. [1] Mones, Enys, Lilla Vicsek, and Tamás Vicsek.
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"Hierarchy Measure for Complex Networks."
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*PLoS ONE* 7.3 (2012): e33799.
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https://doi.org/10.1371/journal.pone.0033799
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"""
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if nx.is_negatively_weighted(G, weight=weight):
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raise nx.NetworkXError('edge weights must be positive')
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total_weight = G.size(weight=weight)
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if total_weight <= 0:
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raise nx.NetworkXError('Size of G must be positive')
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# If provided, weights must be interpreted as connection strength
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# (so higher weights are more likely to be chosen). However, the
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# shortest path algorithms in NetworkX assume the provided "weight"
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# is actually a distance (so edges with higher weight are less
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# likely to be chosen). Therefore we need to invert the weights when
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# computing shortest paths.
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#
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# If weight is None, we leave it as-is so that the shortest path
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# algorithm can use a faster, unweighted algorithm.
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if weight is not None:
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def as_distance(u, v, d): return total_weight / d.get(weight, 1)
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shortest_paths = nx.shortest_path(G, weight=as_distance)
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else:
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shortest_paths = nx.shortest_path(G)
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centrality = local_reaching_centrality
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# TODO This can be trivially parallelized.
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lrc = [centrality(G, node, paths=paths, weight=weight,
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normalized=normalized)
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for node, paths in shortest_paths.items()]
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max_lrc = max(lrc)
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return sum(max_lrc - c for c in lrc) / (len(G) - 1)
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def local_reaching_centrality(G, v, paths=None, weight=None, normalized=True):
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"""Returns the local reaching centrality of a node in a directed
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graph.
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The *local reaching centrality* of a node in a directed graph is the
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proportion of other nodes reachable from that node [1]_.
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Parameters
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----------
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G : DiGraph
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A NetworkX DiGraph.
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v : node
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A node in the directed graph `G`.
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paths : dictionary (default=None)
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If this is not `None` it must be a dictionary representation
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of single-source shortest paths, as computed by, for example,
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:func:`networkx.shortest_path` with source node `v`. Use this
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keyword argument if you intend to invoke this function many
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times but don't want the paths to be recomputed each time.
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weight : None or string, optional (default=None)
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Attribute to use for edge weights. If `None`, each edge weight
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is assumed to be one. A higher weight implies a stronger
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connection between nodes and a *shorter* path length.
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normalized : bool, optional (default=True)
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Whether to normalize the edge weights by the total sum of edge
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weights.
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Returns
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-------
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h : float
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The local reaching centrality of the node ``v`` in the graph
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``G``.
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Examples
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--------
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>>> import networkx as nx
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>>> G = nx.DiGraph()
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>>> G.add_edges_from([(1, 2), (1, 3)])
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>>> nx.local_reaching_centrality(G, 3)
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0.0
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>>> G.add_edge(3, 2)
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>>> nx.local_reaching_centrality(G, 3)
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0.5
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See also
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--------
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global_reaching_centrality
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References
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----------
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.. [1] Mones, Enys, Lilla Vicsek, and Tamás Vicsek.
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"Hierarchy Measure for Complex Networks."
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*PLoS ONE* 7.3 (2012): e33799.
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https://doi.org/10.1371/journal.pone.0033799
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"""
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if paths is None:
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if nx.is_negatively_weighted(G, weight=weight):
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raise nx.NetworkXError('edge weights must be positive')
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total_weight = G.size(weight=weight)
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if total_weight <= 0:
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raise nx.NetworkXError('Size of G must be positive')
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if weight is not None:
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# Interpret weights as lengths.
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def as_distance(u, v, d): return total_weight / d.get(weight, 1)
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paths = nx.shortest_path(G, source=v, weight=as_distance)
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else:
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paths = nx.shortest_path(G, source=v)
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# If the graph is unweighted, simply return the proportion of nodes
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# reachable from the source node ``v``.
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if weight is None and G.is_directed():
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return (len(paths) - 1) / (len(G) - 1)
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if normalized and weight is not None:
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norm = G.size(weight=weight) / G.size()
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else:
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norm = 1
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# TODO This can be trivially parallelized.
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avgw = (_average_weight(G, path, weight=weight) for path in paths.values())
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sum_avg_weight = sum(avgw) / norm
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return sum_avg_weight / (len(G) - 1)
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