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annotate planemo/lib/python3.7/site-packages/networkx/algorithms/richclub.py @ 1:56ad4e20f292 draft

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"planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
author guerler
date Fri, 31 Jul 2020 00:32:28 -0400
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1 # -*- coding: utf-8 -*-
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2 # Copyright (C) 2004-2019 by
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3 # Aric Hagberg <hagberg@lanl.gov>
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4 # Dan Schult <dschult@colgate.edu>
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5 # Pieter Swart <swart@lanl.gov>
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6 # All rights reserved.
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7 # BSD license.
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8 #
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9 # Authors: Ben Edwards (bedwards@cs.unm.edu)
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10 # Aric Hagberg (hagberg@lanl.gov)
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11 """Functions for computing rich-club coefficients."""
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12
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13 import networkx as nx
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14 from networkx.utils import accumulate
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15 from networkx.utils import not_implemented_for
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16
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17 __all__ = ['rich_club_coefficient']
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18
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19
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20 @not_implemented_for('directed')
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21 @not_implemented_for('multigraph')
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22 def rich_club_coefficient(G, normalized=True, Q=100, seed=None):
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23 r"""Returns the rich-club coefficient of the graph `G`.
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24
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25 For each degree *k*, the *rich-club coefficient* is the ratio of the
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26 number of actual to the number of potential edges for nodes with
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27 degree greater than *k*:
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28
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29 .. math::
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30
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31 \phi(k) = \frac{2 E_k}{N_k (N_k - 1)}
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32
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33 where `N_k` is the number of nodes with degree larger than *k*, and
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34 `E_k` is the number of edges among those nodes.
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35
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36 Parameters
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37 ----------
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38 G : NetworkX graph
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39 Undirected graph with neither parallel edges nor self-loops.
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40 normalized : bool (optional)
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41 Normalize using randomized network as in [1]_
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42 Q : float (optional, default=100)
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43 If `normalized` is True, perform `Q * m` double-edge
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44 swaps, where `m` is the number of edges in `G`, to use as a
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45 null-model for normalization.
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46 seed : integer, random_state, or None (default)
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47 Indicator of random number generation state.
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48 See :ref:`Randomness<randomness>`.
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49
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50 Returns
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51 -------
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52 rc : dictionary
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53 A dictionary, keyed by degree, with rich-club coefficient values.
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54
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55 Examples
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56 --------
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57 >>> G = nx.Graph([(0, 1), (0, 2), (1, 2), (1, 3), (1, 4), (4, 5)])
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58 >>> rc = nx.rich_club_coefficient(G, normalized=False)
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59 >>> rc[0] # doctest: +SKIP
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60 0.4
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61
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62 Notes
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63 -----
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64 The rich club definition and algorithm are found in [1]_. This
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65 algorithm ignores any edge weights and is not defined for directed
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66 graphs or graphs with parallel edges or self loops.
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67
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68 Estimates for appropriate values of `Q` are found in [2]_.
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69
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70 References
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71 ----------
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72 .. [1] Julian J. McAuley, Luciano da Fontoura Costa,
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73 and Tibério S. Caetano,
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74 "The rich-club phenomenon across complex network hierarchies",
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75 Applied Physics Letters Vol 91 Issue 8, August 2007.
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76 https://arxiv.org/abs/physics/0701290
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77 .. [2] R. Milo, N. Kashtan, S. Itzkovitz, M. E. J. Newman, U. Alon,
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78 "Uniform generation of random graphs with arbitrary degree
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79 sequences", 2006. https://arxiv.org/abs/cond-mat/0312028
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80 """
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81 if nx.number_of_selfloops(G) > 0:
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82 raise Exception('rich_club_coefficient is not implemented for '
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83 'graphs with self loops.')
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84 rc = _compute_rc(G)
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85 if normalized:
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86 # make R a copy of G, randomize with Q*|E| double edge swaps
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87 # and use rich_club coefficient of R to normalize
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88 R = G.copy()
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89 E = R.number_of_edges()
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90 nx.double_edge_swap(R, Q * E, max_tries=Q * E * 10, seed=seed)
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91 rcran = _compute_rc(R)
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92 rc = {k: v / rcran[k] for k, v in rc.items()}
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93 return rc
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94
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95
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96 def _compute_rc(G):
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97 """Returns the rich-club coefficient for each degree in the graph
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98 `G`.
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99
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100 `G` is an undirected graph without multiedges.
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101
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102 Returns a dictionary mapping degree to rich-club coefficient for
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103 that degree.
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104
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105 """
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106 deghist = nx.degree_histogram(G)
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107 total = sum(deghist)
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108 # Compute the number of nodes with degree greater than `k`, for each
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109 # degree `k` (omitting the last entry, which is zero).
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110 nks = (total - cs for cs in accumulate(deghist) if total - cs > 1)
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111 # Create a sorted list of pairs of edge endpoint degrees.
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112 #
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113 # The list is sorted in reverse order so that we can pop from the
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114 # right side of the list later, instead of popping from the left
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115 # side of the list, which would have a linear time cost.
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116 edge_degrees = sorted((sorted(map(G.degree, e)) for e in G.edges()),
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117 reverse=True)
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118 ek = G.number_of_edges()
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119 k1, k2 = edge_degrees.pop()
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120 rc = {}
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121 for d, nk in enumerate(nks):
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122 while k1 <= d:
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123 if len(edge_degrees) == 0:
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124 ek = 0
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125 break
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
126 k1, k2 = edge_degrees.pop()
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
127 ek -= 1
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
128 rc[d] = 2 * ek / (nk * (nk - 1))
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
129 return rc