Mercurial > repos > shellac > sam_consensus_v3
annotate env/lib/python3.9/site-packages/networkx/algorithms/core.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 """ |
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2 Find the k-cores of a graph. |
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3 |
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4 The k-core is found by recursively pruning nodes with degrees less than k. |
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5 |
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6 See the following references for details: |
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7 |
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8 An O(m) Algorithm for Cores Decomposition of Networks |
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9 Vladimir Batagelj and Matjaz Zaversnik, 2003. |
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10 https://arxiv.org/abs/cs.DS/0310049 |
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11 |
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12 Generalized Cores |
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13 Vladimir Batagelj and Matjaz Zaversnik, 2002. |
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14 https://arxiv.org/pdf/cs/0202039 |
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15 |
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16 For directed graphs a more general notion is that of D-cores which |
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17 looks at (k, l) restrictions on (in, out) degree. The (k, k) D-core |
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18 is the k-core. |
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19 |
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20 D-cores: Measuring Collaboration of Directed Graphs Based on Degeneracy |
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21 Christos Giatsidis, Dimitrios M. Thilikos, Michalis Vazirgiannis, ICDM 2011. |
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22 http://www.graphdegeneracy.org/dcores_ICDM_2011.pdf |
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23 |
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24 Multi-scale structure and topological anomaly detection via a new network \ |
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25 statistic: The onion decomposition |
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26 L. Hébert-Dufresne, J. A. Grochow, and A. Allard |
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27 Scientific Reports 6, 31708 (2016) |
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28 http://doi.org/10.1038/srep31708 |
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29 |
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30 """ |
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31 import networkx as nx |
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32 from networkx.exception import NetworkXError |
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33 from networkx.utils import not_implemented_for |
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34 |
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35 __all__ = [ |
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36 "core_number", |
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37 "find_cores", |
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38 "k_core", |
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39 "k_shell", |
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40 "k_crust", |
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41 "k_corona", |
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42 "k_truss", |
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43 "onion_layers", |
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44 ] |
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45 |
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46 |
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47 @not_implemented_for("multigraph") |
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48 def core_number(G): |
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49 """Returns the core number for each vertex. |
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50 |
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51 A k-core is a maximal subgraph that contains nodes of degree k or more. |
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52 |
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53 The core number of a node is the largest value k of a k-core containing |
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54 that node. |
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55 |
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56 Parameters |
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57 ---------- |
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58 G : NetworkX graph |
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59 A graph or directed graph |
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60 |
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61 Returns |
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62 ------- |
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63 core_number : dictionary |
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64 A dictionary keyed by node to the core number. |
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65 |
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66 Raises |
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67 ------ |
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68 NetworkXError |
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69 The k-core is not implemented for graphs with self loops |
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70 or parallel edges. |
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71 |
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72 Notes |
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73 ----- |
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74 Not implemented for graphs with parallel edges or self loops. |
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75 |
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76 For directed graphs the node degree is defined to be the |
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77 in-degree + out-degree. |
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78 |
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79 References |
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80 ---------- |
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81 .. [1] An O(m) Algorithm for Cores Decomposition of Networks |
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82 Vladimir Batagelj and Matjaz Zaversnik, 2003. |
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83 https://arxiv.org/abs/cs.DS/0310049 |
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84 """ |
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85 if nx.number_of_selfloops(G) > 0: |
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86 msg = ( |
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87 "Input graph has self loops which is not permitted; " |
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88 "Consider using G.remove_edges_from(nx.selfloop_edges(G))." |
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89 ) |
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90 raise NetworkXError(msg) |
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91 degrees = dict(G.degree()) |
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92 # Sort nodes by degree. |
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93 nodes = sorted(degrees, key=degrees.get) |
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94 bin_boundaries = [0] |
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95 curr_degree = 0 |
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96 for i, v in enumerate(nodes): |
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97 if degrees[v] > curr_degree: |
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98 bin_boundaries.extend([i] * (degrees[v] - curr_degree)) |
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99 curr_degree = degrees[v] |
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100 node_pos = {v: pos for pos, v in enumerate(nodes)} |
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101 # The initial guess for the core number of a node is its degree. |
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102 core = degrees |
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103 nbrs = {v: list(nx.all_neighbors(G, v)) for v in G} |
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104 for v in nodes: |
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105 for u in nbrs[v]: |
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106 if core[u] > core[v]: |
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107 nbrs[u].remove(v) |
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108 pos = node_pos[u] |
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109 bin_start = bin_boundaries[core[u]] |
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110 node_pos[u] = bin_start |
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111 node_pos[nodes[bin_start]] = pos |
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112 nodes[bin_start], nodes[pos] = nodes[pos], nodes[bin_start] |
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113 bin_boundaries[core[u]] += 1 |
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114 core[u] -= 1 |
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115 return core |
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116 |
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117 |
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118 find_cores = core_number |
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119 |
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120 |
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121 def _core_subgraph(G, k_filter, k=None, core=None): |
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122 """Returns the subgraph induced by nodes passing filter `k_filter`. |
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123 |
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124 Parameters |
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125 ---------- |
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126 G : NetworkX graph |
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127 The graph or directed graph to process |
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128 k_filter : filter function |
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129 This function filters the nodes chosen. It takes three inputs: |
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130 A node of G, the filter's cutoff, and the core dict of the graph. |
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131 The function should return a Boolean value. |
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132 k : int, optional |
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133 The order of the core. If not specified use the max core number. |
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134 This value is used as the cutoff for the filter. |
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135 core : dict, optional |
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136 Precomputed core numbers keyed by node for the graph `G`. |
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137 If not specified, the core numbers will be computed from `G`. |
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138 |
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139 """ |
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140 if core is None: |
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141 core = core_number(G) |
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142 if k is None: |
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143 k = max(core.values()) |
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144 nodes = (v for v in core if k_filter(v, k, core)) |
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145 return G.subgraph(nodes).copy() |
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146 |
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147 |
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148 def k_core(G, k=None, core_number=None): |
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149 """Returns the k-core of G. |
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150 |
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151 A k-core is a maximal subgraph that contains nodes of degree k or more. |
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152 |
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153 Parameters |
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154 ---------- |
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155 G : NetworkX graph |
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156 A graph or directed graph |
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157 k : int, optional |
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158 The order of the core. If not specified return the main core. |
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159 core_number : dictionary, optional |
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160 Precomputed core numbers for the graph G. |
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161 |
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162 Returns |
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163 ------- |
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164 G : NetworkX graph |
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165 The k-core subgraph |
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166 |
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167 Raises |
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168 ------ |
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169 NetworkXError |
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170 The k-core is not defined for graphs with self loops or parallel edges. |
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171 |
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172 Notes |
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173 ----- |
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174 The main core is the core with the largest degree. |
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175 |
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176 Not implemented for graphs with parallel edges or self loops. |
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177 |
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178 For directed graphs the node degree is defined to be the |
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179 in-degree + out-degree. |
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180 |
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181 Graph, node, and edge attributes are copied to the subgraph. |
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182 |
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183 See Also |
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184 -------- |
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185 core_number |
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186 |
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187 References |
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188 ---------- |
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189 .. [1] An O(m) Algorithm for Cores Decomposition of Networks |
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190 Vladimir Batagelj and Matjaz Zaversnik, 2003. |
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191 https://arxiv.org/abs/cs.DS/0310049 |
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192 """ |
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193 |
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194 def k_filter(v, k, c): |
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195 return c[v] >= k |
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196 |
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197 return _core_subgraph(G, k_filter, k, core_number) |
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198 |
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199 |
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200 def k_shell(G, k=None, core_number=None): |
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201 """Returns the k-shell of G. |
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202 |
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203 The k-shell is the subgraph induced by nodes with core number k. |
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204 That is, nodes in the k-core that are not in the (k+1)-core. |
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205 |
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206 Parameters |
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207 ---------- |
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208 G : NetworkX graph |
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209 A graph or directed graph. |
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210 k : int, optional |
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211 The order of the shell. If not specified return the outer shell. |
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212 core_number : dictionary, optional |
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213 Precomputed core numbers for the graph G. |
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214 |
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215 |
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216 Returns |
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217 ------- |
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218 G : NetworkX graph |
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219 The k-shell subgraph |
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220 |
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221 Raises |
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222 ------ |
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223 NetworkXError |
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224 The k-shell is not implemented for graphs with self loops |
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225 or parallel edges. |
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226 |
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227 Notes |
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228 ----- |
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229 This is similar to k_corona but in that case only neighbors in the |
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230 k-core are considered. |
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231 |
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232 Not implemented for graphs with parallel edges or self loops. |
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233 |
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234 For directed graphs the node degree is defined to be the |
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235 in-degree + out-degree. |
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236 |
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237 Graph, node, and edge attributes are copied to the subgraph. |
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238 |
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239 See Also |
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240 -------- |
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241 core_number |
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242 k_corona |
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243 |
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244 |
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245 References |
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246 ---------- |
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247 .. [1] A model of Internet topology using k-shell decomposition |
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248 Shai Carmi, Shlomo Havlin, Scott Kirkpatrick, Yuval Shavitt, |
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249 and Eran Shir, PNAS July 3, 2007 vol. 104 no. 27 11150-11154 |
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250 http://www.pnas.org/content/104/27/11150.full |
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251 """ |
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252 |
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253 def k_filter(v, k, c): |
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254 return c[v] == k |
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255 |
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256 return _core_subgraph(G, k_filter, k, core_number) |
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257 |
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258 |
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259 def k_crust(G, k=None, core_number=None): |
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260 """Returns the k-crust of G. |
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261 |
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262 The k-crust is the graph G with the k-core removed. |
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263 |
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264 Parameters |
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265 ---------- |
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266 G : NetworkX graph |
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267 A graph or directed graph. |
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268 k : int, optional |
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269 The order of the shell. If not specified return the main crust. |
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270 core_number : dictionary, optional |
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271 Precomputed core numbers for the graph G. |
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272 |
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273 Returns |
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274 ------- |
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275 G : NetworkX graph |
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276 The k-crust subgraph |
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277 |
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278 Raises |
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279 ------ |
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280 NetworkXError |
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281 The k-crust is not implemented for graphs with self loops |
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282 or parallel edges. |
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283 |
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284 Notes |
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285 ----- |
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286 This definition of k-crust is different than the definition in [1]_. |
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287 The k-crust in [1]_ is equivalent to the k+1 crust of this algorithm. |
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288 |
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289 Not implemented for graphs with parallel edges or self loops. |
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290 |
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291 For directed graphs the node degree is defined to be the |
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292 in-degree + out-degree. |
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293 |
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294 Graph, node, and edge attributes are copied to the subgraph. |
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295 |
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296 See Also |
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297 -------- |
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298 core_number |
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299 |
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300 References |
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301 ---------- |
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302 .. [1] A model of Internet topology using k-shell decomposition |
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303 Shai Carmi, Shlomo Havlin, Scott Kirkpatrick, Yuval Shavitt, |
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304 and Eran Shir, PNAS July 3, 2007 vol. 104 no. 27 11150-11154 |
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305 http://www.pnas.org/content/104/27/11150.full |
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306 """ |
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307 # Default for k is one less than in _core_subgraph, so just inline. |
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308 # Filter is c[v] <= k |
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309 if core_number is None: |
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310 core_number = find_cores(G) |
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311 if k is None: |
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312 k = max(core_number.values()) - 1 |
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313 nodes = (v for v in core_number if core_number[v] <= k) |
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314 return G.subgraph(nodes).copy() |
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315 |
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316 |
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317 def k_corona(G, k, core_number=None): |
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318 """Returns the k-corona of G. |
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319 |
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320 The k-corona is the subgraph of nodes in the k-core which have |
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321 exactly k neighbours in the k-core. |
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322 |
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323 Parameters |
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324 ---------- |
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325 G : NetworkX graph |
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326 A graph or directed graph |
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327 k : int |
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328 The order of the corona. |
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329 core_number : dictionary, optional |
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330 Precomputed core numbers for the graph G. |
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331 |
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332 Returns |
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333 ------- |
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334 G : NetworkX graph |
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335 The k-corona subgraph |
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336 |
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337 Raises |
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338 ------ |
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339 NetworkXError |
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340 The k-cornoa is not defined for graphs with self loops or |
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341 parallel edges. |
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342 |
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343 Notes |
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344 ----- |
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345 Not implemented for graphs with parallel edges or self loops. |
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346 |
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347 For directed graphs the node degree is defined to be the |
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348 in-degree + out-degree. |
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349 |
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350 Graph, node, and edge attributes are copied to the subgraph. |
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351 |
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352 See Also |
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353 -------- |
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354 core_number |
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355 |
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356 References |
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357 ---------- |
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358 .. [1] k -core (bootstrap) percolation on complex networks: |
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359 Critical phenomena and nonlocal effects, |
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360 A. V. Goltsev, S. N. Dorogovtsev, and J. F. F. Mendes, |
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361 Phys. Rev. E 73, 056101 (2006) |
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362 http://link.aps.org/doi/10.1103/PhysRevE.73.056101 |
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363 """ |
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364 |
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365 def func(v, k, c): |
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366 return c[v] == k and k == sum(1 for w in G[v] if c[w] >= k) |
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367 |
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368 return _core_subgraph(G, func, k, core_number) |
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369 |
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370 |
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371 @not_implemented_for("directed") |
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372 @not_implemented_for("multigraph") |
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373 def k_truss(G, k): |
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374 """Returns the k-truss of `G`. |
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375 |
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376 The k-truss is the maximal induced subgraph of `G` which contains at least |
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377 three vertices where every edge is incident to at least `k-2` triangles. |
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378 |
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379 Parameters |
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380 ---------- |
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381 G : NetworkX graph |
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382 An undirected graph |
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383 k : int |
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384 The order of the truss |
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385 |
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386 Returns |
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387 ------- |
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388 H : NetworkX graph |
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389 The k-truss subgraph |
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390 |
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391 Raises |
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392 ------ |
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393 NetworkXError |
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394 |
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395 The k-truss is not defined for graphs with self loops or parallel edges |
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396 or directed graphs. |
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397 |
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398 Notes |
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399 ----- |
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400 A k-clique is a (k-2)-truss and a k-truss is a (k+1)-core. |
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401 |
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402 Not implemented for digraphs or graphs with parallel edges or self loops. |
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403 |
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404 Graph, node, and edge attributes are copied to the subgraph. |
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405 |
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406 K-trusses were originally defined in [2] which states that the k-truss |
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407 is the maximal induced subgraph where each edge belongs to at least |
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408 `k-2` triangles. A more recent paper, [1], uses a slightly different |
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409 definition requiring that each edge belong to at least `k` triangles. |
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410 This implementation uses the original definition of `k-2` triangles. |
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411 |
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412 References |
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413 ---------- |
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414 .. [1] Bounds and Algorithms for k-truss. Paul Burkhardt, Vance Faber, |
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415 David G. Harris, 2018. https://arxiv.org/abs/1806.05523v2 |
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416 .. [2] Trusses: Cohesive Subgraphs for Social Network Analysis. Jonathan |
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417 Cohen, 2005. |
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418 """ |
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419 H = G.copy() |
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420 |
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421 n_dropped = 1 |
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422 while n_dropped > 0: |
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423 n_dropped = 0 |
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424 to_drop = [] |
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425 seen = set() |
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426 for u in H: |
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427 nbrs_u = set(H[u]) |
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428 seen.add(u) |
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429 new_nbrs = [v for v in nbrs_u if v not in seen] |
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430 for v in new_nbrs: |
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431 if len(nbrs_u & set(H[v])) < (k - 2): |
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432 to_drop.append((u, v)) |
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433 H.remove_edges_from(to_drop) |
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434 n_dropped = len(to_drop) |
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435 H.remove_nodes_from(list(nx.isolates(H))) |
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436 |
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437 return H |
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438 |
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439 |
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440 @not_implemented_for("multigraph") |
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441 @not_implemented_for("directed") |
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442 def onion_layers(G): |
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443 """Returns the layer of each vertex in an onion decomposition of the graph. |
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444 |
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445 The onion decomposition refines the k-core decomposition by providing |
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446 information on the internal organization of each k-shell. It is usually |
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447 used alongside the `core numbers`. |
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448 |
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449 Parameters |
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450 ---------- |
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451 G : NetworkX graph |
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452 A simple graph without self loops or parallel edges |
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453 |
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454 Returns |
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455 ------- |
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456 od_layers : dictionary |
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457 A dictionary keyed by vertex to the onion layer. The layers are |
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458 contiguous integers starting at 1. |
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459 |
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460 Raises |
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461 ------ |
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462 NetworkXError |
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463 The onion decomposition is not implemented for graphs with self loops |
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464 or parallel edges or for directed graphs. |
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465 |
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466 Notes |
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467 ----- |
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468 Not implemented for graphs with parallel edges or self loops. |
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469 |
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470 Not implemented for directed graphs. |
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471 |
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472 See Also |
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473 -------- |
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474 core_number |
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475 |
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476 References |
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477 ---------- |
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478 .. [1] Multi-scale structure and topological anomaly detection via a new |
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479 network statistic: The onion decomposition |
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480 L. Hébert-Dufresne, J. A. Grochow, and A. Allard |
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481 Scientific Reports 6, 31708 (2016) |
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482 http://doi.org/10.1038/srep31708 |
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483 .. [2] Percolation and the effective structure of complex networks |
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484 A. Allard and L. Hébert-Dufresne |
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485 Physical Review X 9, 011023 (2019) |
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486 http://doi.org/10.1103/PhysRevX.9.011023 |
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487 """ |
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488 if nx.number_of_selfloops(G) > 0: |
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489 msg = ( |
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490 "Input graph contains self loops which is not permitted; " |
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491 "Consider using G.remove_edges_from(nx.selfloop_edges(G))." |
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492 ) |
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493 raise NetworkXError(msg) |
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494 # Dictionaries to register the k-core/onion decompositions. |
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495 od_layers = {} |
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496 # Adjacency list |
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497 neighbors = {v: list(nx.all_neighbors(G, v)) for v in G} |
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498 # Effective degree of nodes. |
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499 degrees = dict(G.degree()) |
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500 # Performs the onion decomposition. |
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501 current_core = 1 |
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502 current_layer = 1 |
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503 # Sets vertices of degree 0 to layer 1, if any. |
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504 isolated_nodes = [v for v in nx.isolates(G)] |
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505 if len(isolated_nodes) > 0: |
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506 for v in isolated_nodes: |
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507 od_layers[v] = current_layer |
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508 degrees.pop(v) |
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509 current_layer = 2 |
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510 # Finds the layer for the remaining nodes. |
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511 while len(degrees) > 0: |
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512 # Sets the order for looking at nodes. |
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513 nodes = sorted(degrees, key=degrees.get) |
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514 # Sets properly the current core. |
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515 min_degree = degrees[nodes[0]] |
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516 if min_degree > current_core: |
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517 current_core = min_degree |
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518 # Identifies vertices in the current layer. |
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519 this_layer = [] |
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520 for n in nodes: |
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521 if degrees[n] > current_core: |
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522 break |
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523 this_layer.append(n) |
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524 # Identifies the core/layer of the vertices in the current layer. |
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525 for v in this_layer: |
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526 od_layers[v] = current_layer |
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527 for n in neighbors[v]: |
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528 neighbors[n].remove(v) |
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529 degrees[n] = degrees[n] - 1 |
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530 degrees.pop(v) |
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531 # Updates the layer count. |
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532 current_layer = current_layer + 1 |
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533 # Returns the dictionaries containing the onion layer of each vertices. |
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534 return od_layers |