Mercurial > repos > shellac > sam_consensus_v3
annotate env/lib/python3.9/site-packages/networkx/algorithms/centrality/percolation.py @ 0:4f3585e2f14b draft default tip
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author | shellac |
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date | Mon, 22 Mar 2021 18:12:50 +0000 |
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1 """Percolation centrality measures.""" |
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2 |
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3 import networkx as nx |
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4 |
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5 from networkx.algorithms.centrality.betweenness import ( |
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6 _single_source_dijkstra_path_basic as dijkstra, |
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7 ) |
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8 from networkx.algorithms.centrality.betweenness import ( |
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9 _single_source_shortest_path_basic as shortest_path, |
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10 ) |
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11 |
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12 __all__ = ["percolation_centrality"] |
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13 |
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14 |
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15 def percolation_centrality(G, attribute="percolation", states=None, weight=None): |
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16 r"""Compute the percolation centrality for nodes. |
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17 |
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18 Percolation centrality of a node $v$, at a given time, is defined |
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19 as the proportion of ‘percolated paths’ that go through that node. |
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20 |
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21 This measure quantifies relative impact of nodes based on their |
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22 topological connectivity, as well as their percolation states. |
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23 |
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24 Percolation states of nodes are used to depict network percolation |
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25 scenarios (such as during infection transmission in a social network |
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26 of individuals, spreading of computer viruses on computer networks, or |
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27 transmission of disease over a network of towns) over time. In this |
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28 measure usually the percolation state is expressed as a decimal |
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29 between 0.0 and 1.0. |
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30 |
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31 When all nodes are in the same percolated state this measure is |
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32 equivalent to betweenness centrality. |
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33 |
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34 Parameters |
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35 ---------- |
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36 G : graph |
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37 A NetworkX graph. |
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38 |
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39 attribute : None or string, optional (default='percolation') |
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40 Name of the node attribute to use for percolation state, used |
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41 if `states` is None. |
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42 |
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43 states : None or dict, optional (default=None) |
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44 Specify percolation states for the nodes, nodes as keys states |
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45 as values. |
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46 |
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47 weight : None or string, optional (default=None) |
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48 If None, all edge weights are considered equal. |
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49 Otherwise holds the name of the edge attribute used as weight. |
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50 |
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51 Returns |
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52 ------- |
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53 nodes : dictionary |
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54 Dictionary of nodes with percolation centrality as the value. |
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55 |
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56 See Also |
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57 -------- |
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58 betweenness_centrality |
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59 |
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60 Notes |
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61 ----- |
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62 The algorithm is from Mahendra Piraveenan, Mikhail Prokopenko, and |
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63 Liaquat Hossain [1]_ |
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64 Pair dependecies are calculated and accumulated using [2]_ |
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65 |
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66 For weighted graphs the edge weights must be greater than zero. |
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67 Zero edge weights can produce an infinite number of equal length |
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68 paths between pairs of nodes. |
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69 |
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70 References |
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71 ---------- |
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72 .. [1] Mahendra Piraveenan, Mikhail Prokopenko, Liaquat Hossain |
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73 Percolation Centrality: Quantifying Graph-Theoretic Impact of Nodes |
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74 during Percolation in Networks |
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75 http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0053095 |
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76 .. [2] Ulrik Brandes: |
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77 A Faster Algorithm for Betweenness Centrality. |
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78 Journal of Mathematical Sociology 25(2):163-177, 2001. |
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79 http://www.inf.uni-konstanz.de/algo/publications/b-fabc-01.pdf |
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80 """ |
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81 percolation = dict.fromkeys(G, 0.0) # b[v]=0 for v in G |
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82 |
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83 nodes = G |
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84 |
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85 if states is None: |
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86 states = nx.get_node_attributes(nodes, attribute) |
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87 |
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88 # sum of all percolation states |
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89 p_sigma_x_t = 0.0 |
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90 for v in states.values(): |
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91 p_sigma_x_t += v |
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92 |
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93 for s in nodes: |
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94 # single source shortest paths |
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95 if weight is None: # use BFS |
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96 S, P, sigma = shortest_path(G, s) |
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97 else: # use Dijkstra's algorithm |
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98 S, P, sigma = dijkstra(G, s, weight) |
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99 # accumulation |
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100 percolation = _accumulate_percolation( |
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101 percolation, G, S, P, sigma, s, states, p_sigma_x_t |
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102 ) |
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103 |
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104 n = len(G) |
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105 |
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106 for v in percolation: |
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107 percolation[v] *= 1 / (n - 2) |
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108 |
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109 return percolation |
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110 |
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111 |
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112 def _accumulate_percolation(percolation, G, S, P, sigma, s, states, p_sigma_x_t): |
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113 delta = dict.fromkeys(S, 0) |
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114 while S: |
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115 w = S.pop() |
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116 coeff = (1 + delta[w]) / sigma[w] |
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117 for v in P[w]: |
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118 delta[v] += sigma[v] * coeff |
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119 if w != s: |
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120 # percolation weight |
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121 pw_s_w = states[s] / (p_sigma_x_t - states[w]) |
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122 percolation[w] += delta[w] * pw_s_w |
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123 return percolation |