Mercurial > repos > shellac > sam_consensus_v3
annotate env/lib/python3.9/site-packages/networkx/algorithms/link_analysis/pagerank_alg.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 """PageRank analysis of graph structure. """ |
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2 import networkx as nx |
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3 from networkx.utils import not_implemented_for |
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4 |
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5 __all__ = ["pagerank", "pagerank_numpy", "pagerank_scipy", "google_matrix"] |
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6 |
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7 |
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8 @not_implemented_for("multigraph") |
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9 def pagerank( |
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10 G, |
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11 alpha=0.85, |
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12 personalization=None, |
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13 max_iter=100, |
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14 tol=1.0e-6, |
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15 nstart=None, |
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16 weight="weight", |
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17 dangling=None, |
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18 ): |
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19 """Returns the PageRank of the nodes in the graph. |
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20 |
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21 PageRank computes a ranking of the nodes in the graph G based on |
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22 the structure of the incoming links. It was originally designed as |
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23 an algorithm to rank web pages. |
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24 |
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25 Parameters |
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26 ---------- |
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27 G : graph |
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28 A NetworkX graph. Undirected graphs will be converted to a directed |
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29 graph with two directed edges for each undirected edge. |
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30 |
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31 alpha : float, optional |
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32 Damping parameter for PageRank, default=0.85. |
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33 |
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34 personalization: dict, optional |
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35 The "personalization vector" consisting of a dictionary with a |
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36 key some subset of graph nodes and personalization value each of those. |
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37 At least one personalization value must be non-zero. |
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38 If not specfiied, a nodes personalization value will be zero. |
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39 By default, a uniform distribution is used. |
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40 |
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41 max_iter : integer, optional |
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42 Maximum number of iterations in power method eigenvalue solver. |
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43 |
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44 tol : float, optional |
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45 Error tolerance used to check convergence in power method solver. |
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46 |
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47 nstart : dictionary, optional |
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48 Starting value of PageRank iteration for each node. |
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49 |
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50 weight : key, optional |
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51 Edge data key to use as weight. If None weights are set to 1. |
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52 |
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53 dangling: dict, optional |
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54 The outedges to be assigned to any "dangling" nodes, i.e., nodes without |
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55 any outedges. The dict key is the node the outedge points to and the dict |
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56 value is the weight of that outedge. By default, dangling nodes are given |
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57 outedges according to the personalization vector (uniform if not |
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58 specified). This must be selected to result in an irreducible transition |
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59 matrix (see notes under google_matrix). It may be common to have the |
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60 dangling dict to be the same as the personalization dict. |
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61 |
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62 Returns |
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63 ------- |
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64 pagerank : dictionary |
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65 Dictionary of nodes with PageRank as value |
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66 |
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67 Examples |
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68 -------- |
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69 >>> G = nx.DiGraph(nx.path_graph(4)) |
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70 >>> pr = nx.pagerank(G, alpha=0.9) |
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71 |
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72 Notes |
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73 ----- |
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74 The eigenvector calculation is done by the power iteration method |
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75 and has no guarantee of convergence. The iteration will stop after |
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76 an error tolerance of ``len(G) * tol`` has been reached. If the |
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77 number of iterations exceed `max_iter`, a |
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78 :exc:`networkx.exception.PowerIterationFailedConvergence` exception |
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79 is raised. |
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80 |
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81 The PageRank algorithm was designed for directed graphs but this |
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82 algorithm does not check if the input graph is directed and will |
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83 execute on undirected graphs by converting each edge in the |
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84 directed graph to two edges. |
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85 |
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86 See Also |
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87 -------- |
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88 pagerank_numpy, pagerank_scipy, google_matrix |
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89 |
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90 Raises |
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91 ------ |
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92 PowerIterationFailedConvergence |
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93 If the algorithm fails to converge to the specified tolerance |
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94 within the specified number of iterations of the power iteration |
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95 method. |
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96 |
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97 References |
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98 ---------- |
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99 .. [1] A. Langville and C. Meyer, |
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100 "A survey of eigenvector methods of web information retrieval." |
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101 http://citeseer.ist.psu.edu/713792.html |
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102 .. [2] Page, Lawrence; Brin, Sergey; Motwani, Rajeev and Winograd, Terry, |
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103 The PageRank citation ranking: Bringing order to the Web. 1999 |
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104 http://dbpubs.stanford.edu:8090/pub/showDoc.Fulltext?lang=en&doc=1999-66&format=pdf |
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105 |
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106 """ |
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107 if len(G) == 0: |
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108 return {} |
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109 |
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110 if not G.is_directed(): |
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111 D = G.to_directed() |
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112 else: |
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113 D = G |
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114 |
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115 # Create a copy in (right) stochastic form |
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116 W = nx.stochastic_graph(D, weight=weight) |
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117 N = W.number_of_nodes() |
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118 |
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119 # Choose fixed starting vector if not given |
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120 if nstart is None: |
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121 x = dict.fromkeys(W, 1.0 / N) |
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122 else: |
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123 # Normalized nstart vector |
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124 s = float(sum(nstart.values())) |
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125 x = {k: v / s for k, v in nstart.items()} |
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126 |
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127 if personalization is None: |
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128 # Assign uniform personalization vector if not given |
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129 p = dict.fromkeys(W, 1.0 / N) |
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130 else: |
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131 s = float(sum(personalization.values())) |
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132 p = {k: v / s for k, v in personalization.items()} |
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133 |
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134 if dangling is None: |
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135 # Use personalization vector if dangling vector not specified |
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136 dangling_weights = p |
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137 else: |
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138 s = float(sum(dangling.values())) |
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139 dangling_weights = {k: v / s for k, v in dangling.items()} |
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140 dangling_nodes = [n for n in W if W.out_degree(n, weight=weight) == 0.0] |
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141 |
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142 # power iteration: make up to max_iter iterations |
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143 for _ in range(max_iter): |
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144 xlast = x |
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145 x = dict.fromkeys(xlast.keys(), 0) |
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146 danglesum = alpha * sum(xlast[n] for n in dangling_nodes) |
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147 for n in x: |
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148 # this matrix multiply looks odd because it is |
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149 # doing a left multiply x^T=xlast^T*W |
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150 for nbr in W[n]: |
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151 x[nbr] += alpha * xlast[n] * W[n][nbr][weight] |
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152 x[n] += danglesum * dangling_weights.get(n, 0) + (1.0 - alpha) * p.get(n, 0) |
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153 # check convergence, l1 norm |
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154 err = sum([abs(x[n] - xlast[n]) for n in x]) |
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155 if err < N * tol: |
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156 return x |
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157 raise nx.PowerIterationFailedConvergence(max_iter) |
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158 |
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159 |
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160 def google_matrix( |
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161 G, alpha=0.85, personalization=None, nodelist=None, weight="weight", dangling=None |
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162 ): |
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163 """Returns the Google matrix of the graph. |
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164 |
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165 Parameters |
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166 ---------- |
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167 G : graph |
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168 A NetworkX graph. Undirected graphs will be converted to a directed |
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169 graph with two directed edges for each undirected edge. |
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170 |
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171 alpha : float |
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172 The damping factor. |
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173 |
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174 personalization: dict, optional |
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175 The "personalization vector" consisting of a dictionary with a |
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176 key some subset of graph nodes and personalization value each of those. |
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177 At least one personalization value must be non-zero. |
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178 If not specfiied, a nodes personalization value will be zero. |
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179 By default, a uniform distribution is used. |
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180 |
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181 nodelist : list, optional |
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182 The rows and columns are ordered according to the nodes in nodelist. |
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183 If nodelist is None, then the ordering is produced by G.nodes(). |
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184 |
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185 weight : key, optional |
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186 Edge data key to use as weight. If None weights are set to 1. |
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187 |
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188 dangling: dict, optional |
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189 The outedges to be assigned to any "dangling" nodes, i.e., nodes without |
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190 any outedges. The dict key is the node the outedge points to and the dict |
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191 value is the weight of that outedge. By default, dangling nodes are given |
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192 outedges according to the personalization vector (uniform if not |
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193 specified) This must be selected to result in an irreducible transition |
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194 matrix (see notes below). It may be common to have the dangling dict to |
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195 be the same as the personalization dict. |
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196 |
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197 Returns |
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198 ------- |
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199 A : NumPy matrix |
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200 Google matrix of the graph |
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201 |
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202 Notes |
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203 ----- |
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204 The matrix returned represents the transition matrix that describes the |
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205 Markov chain used in PageRank. For PageRank to converge to a unique |
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206 solution (i.e., a unique stationary distribution in a Markov chain), the |
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207 transition matrix must be irreducible. In other words, it must be that |
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208 there exists a path between every pair of nodes in the graph, or else there |
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209 is the potential of "rank sinks." |
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210 |
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211 This implementation works with Multi(Di)Graphs. For multigraphs the |
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212 weight between two nodes is set to be the sum of all edge weights |
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213 between those nodes. |
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214 |
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215 See Also |
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216 -------- |
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217 pagerank, pagerank_numpy, pagerank_scipy |
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218 """ |
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219 import numpy as np |
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220 |
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221 if nodelist is None: |
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222 nodelist = list(G) |
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223 |
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224 M = nx.to_numpy_matrix(G, nodelist=nodelist, weight=weight) |
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225 N = len(G) |
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226 if N == 0: |
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227 return M |
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228 |
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229 # Personalization vector |
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230 if personalization is None: |
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231 p = np.repeat(1.0 / N, N) |
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232 else: |
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233 p = np.array([personalization.get(n, 0) for n in nodelist], dtype=float) |
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234 p /= p.sum() |
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235 |
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236 # Dangling nodes |
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237 if dangling is None: |
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238 dangling_weights = p |
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239 else: |
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240 # Convert the dangling dictionary into an array in nodelist order |
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241 dangling_weights = np.array([dangling.get(n, 0) for n in nodelist], dtype=float) |
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242 dangling_weights /= dangling_weights.sum() |
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243 dangling_nodes = np.where(M.sum(axis=1) == 0)[0] |
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244 |
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245 # Assign dangling_weights to any dangling nodes (nodes with no out links) |
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246 for node in dangling_nodes: |
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247 M[node] = dangling_weights |
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248 |
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249 M /= M.sum(axis=1) # Normalize rows to sum to 1 |
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250 |
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251 return alpha * M + (1 - alpha) * p |
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252 |
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253 |
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254 def pagerank_numpy(G, alpha=0.85, personalization=None, weight="weight", dangling=None): |
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255 """Returns the PageRank of the nodes in the graph. |
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256 |
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257 PageRank computes a ranking of the nodes in the graph G based on |
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258 the structure of the incoming links. It was originally designed as |
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259 an algorithm to rank web pages. |
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260 |
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261 Parameters |
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262 ---------- |
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263 G : graph |
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264 A NetworkX graph. Undirected graphs will be converted to a directed |
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265 graph with two directed edges for each undirected edge. |
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266 |
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267 alpha : float, optional |
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268 Damping parameter for PageRank, default=0.85. |
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269 |
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270 personalization: dict, optional |
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271 The "personalization vector" consisting of a dictionary with a |
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272 key some subset of graph nodes and personalization value each of those. |
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273 At least one personalization value must be non-zero. |
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274 If not specfiied, a nodes personalization value will be zero. |
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275 By default, a uniform distribution is used. |
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276 |
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277 weight : key, optional |
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278 Edge data key to use as weight. If None weights are set to 1. |
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279 |
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280 dangling: dict, optional |
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281 The outedges to be assigned to any "dangling" nodes, i.e., nodes without |
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282 any outedges. The dict key is the node the outedge points to and the dict |
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283 value is the weight of that outedge. By default, dangling nodes are given |
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284 outedges according to the personalization vector (uniform if not |
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285 specified) This must be selected to result in an irreducible transition |
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286 matrix (see notes under google_matrix). It may be common to have the |
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287 dangling dict to be the same as the personalization dict. |
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288 |
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289 Returns |
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290 ------- |
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291 pagerank : dictionary |
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292 Dictionary of nodes with PageRank as value. |
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293 |
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294 Examples |
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295 -------- |
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296 >>> G = nx.DiGraph(nx.path_graph(4)) |
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297 >>> pr = nx.pagerank_numpy(G, alpha=0.9) |
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298 |
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299 Notes |
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300 ----- |
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301 The eigenvector calculation uses NumPy's interface to the LAPACK |
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302 eigenvalue solvers. This will be the fastest and most accurate |
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303 for small graphs. |
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304 |
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305 This implementation works with Multi(Di)Graphs. For multigraphs the |
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306 weight between two nodes is set to be the sum of all edge weights |
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307 between those nodes. |
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308 |
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309 See Also |
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310 -------- |
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311 pagerank, pagerank_scipy, google_matrix |
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312 |
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313 References |
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314 ---------- |
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315 .. [1] A. Langville and C. Meyer, |
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316 "A survey of eigenvector methods of web information retrieval." |
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317 http://citeseer.ist.psu.edu/713792.html |
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318 .. [2] Page, Lawrence; Brin, Sergey; Motwani, Rajeev and Winograd, Terry, |
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319 The PageRank citation ranking: Bringing order to the Web. 1999 |
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320 http://dbpubs.stanford.edu:8090/pub/showDoc.Fulltext?lang=en&doc=1999-66&format=pdf |
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321 """ |
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322 import numpy as np |
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323 |
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324 if len(G) == 0: |
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325 return {} |
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326 M = google_matrix( |
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327 G, alpha, personalization=personalization, weight=weight, dangling=dangling |
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328 ) |
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329 # use numpy LAPACK solver |
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330 eigenvalues, eigenvectors = np.linalg.eig(M.T) |
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331 ind = np.argmax(eigenvalues) |
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332 # eigenvector of largest eigenvalue is at ind, normalized |
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333 largest = np.array(eigenvectors[:, ind]).flatten().real |
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334 norm = float(largest.sum()) |
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335 return dict(zip(G, map(float, largest / norm))) |
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336 |
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337 |
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338 def pagerank_scipy( |
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339 G, |
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340 alpha=0.85, |
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341 personalization=None, |
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342 max_iter=100, |
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343 tol=1.0e-6, |
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344 nstart=None, |
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345 weight="weight", |
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346 dangling=None, |
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347 ): |
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348 """Returns the PageRank of the nodes in the graph. |
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349 |
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350 PageRank computes a ranking of the nodes in the graph G based on |
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351 the structure of the incoming links. It was originally designed as |
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352 an algorithm to rank web pages. |
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353 |
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354 Parameters |
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355 ---------- |
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356 G : graph |
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357 A NetworkX graph. Undirected graphs will be converted to a directed |
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358 graph with two directed edges for each undirected edge. |
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359 |
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360 alpha : float, optional |
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361 Damping parameter for PageRank, default=0.85. |
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362 |
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363 personalization: dict, optional |
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364 The "personalization vector" consisting of a dictionary with a |
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365 key some subset of graph nodes and personalization value each of those. |
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366 At least one personalization value must be non-zero. |
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367 If not specfiied, a nodes personalization value will be zero. |
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368 By default, a uniform distribution is used. |
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369 |
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370 max_iter : integer, optional |
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371 Maximum number of iterations in power method eigenvalue solver. |
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372 |
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373 tol : float, optional |
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374 Error tolerance used to check convergence in power method solver. |
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375 |
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376 nstart : dictionary, optional |
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377 Starting value of PageRank iteration for each node. |
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378 |
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379 weight : key, optional |
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380 Edge data key to use as weight. If None weights are set to 1. |
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381 |
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382 dangling: dict, optional |
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383 The outedges to be assigned to any "dangling" nodes, i.e., nodes without |
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384 any outedges. The dict key is the node the outedge points to and the dict |
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385 value is the weight of that outedge. By default, dangling nodes are given |
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386 outedges according to the personalization vector (uniform if not |
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387 specified) This must be selected to result in an irreducible transition |
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388 matrix (see notes under google_matrix). It may be common to have the |
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389 dangling dict to be the same as the personalization dict. |
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390 |
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391 Returns |
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392 ------- |
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393 pagerank : dictionary |
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394 Dictionary of nodes with PageRank as value |
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395 |
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396 Examples |
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397 -------- |
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398 >>> G = nx.DiGraph(nx.path_graph(4)) |
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399 >>> pr = nx.pagerank_scipy(G, alpha=0.9) |
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400 |
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401 Notes |
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402 ----- |
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403 The eigenvector calculation uses power iteration with a SciPy |
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404 sparse matrix representation. |
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405 |
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406 This implementation works with Multi(Di)Graphs. For multigraphs the |
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407 weight between two nodes is set to be the sum of all edge weights |
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408 between those nodes. |
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409 |
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410 See Also |
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411 -------- |
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412 pagerank, pagerank_numpy, google_matrix |
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413 |
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414 Raises |
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415 ------ |
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416 PowerIterationFailedConvergence |
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417 If the algorithm fails to converge to the specified tolerance |
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418 within the specified number of iterations of the power iteration |
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419 method. |
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420 |
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421 References |
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422 ---------- |
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423 .. [1] A. Langville and C. Meyer, |
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424 "A survey of eigenvector methods of web information retrieval." |
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425 http://citeseer.ist.psu.edu/713792.html |
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426 .. [2] Page, Lawrence; Brin, Sergey; Motwani, Rajeev and Winograd, Terry, |
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427 The PageRank citation ranking: Bringing order to the Web. 1999 |
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428 http://dbpubs.stanford.edu:8090/pub/showDoc.Fulltext?lang=en&doc=1999-66&format=pdf |
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429 """ |
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430 import numpy as np |
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431 import scipy.sparse |
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432 |
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433 N = len(G) |
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434 if N == 0: |
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435 return {} |
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436 |
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437 nodelist = list(G) |
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438 M = nx.to_scipy_sparse_matrix(G, nodelist=nodelist, weight=weight, dtype=float) |
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439 S = np.array(M.sum(axis=1)).flatten() |
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440 S[S != 0] = 1.0 / S[S != 0] |
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441 Q = scipy.sparse.spdiags(S.T, 0, *M.shape, format="csr") |
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442 M = Q * M |
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443 |
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444 # initial vector |
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445 if nstart is None: |
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446 x = np.repeat(1.0 / N, N) |
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447 else: |
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448 x = np.array([nstart.get(n, 0) for n in nodelist], dtype=float) |
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449 x = x / x.sum() |
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450 |
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451 # Personalization vector |
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452 if personalization is None: |
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453 p = np.repeat(1.0 / N, N) |
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454 else: |
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455 p = np.array([personalization.get(n, 0) for n in nodelist], dtype=float) |
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456 p = p / p.sum() |
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457 |
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458 # Dangling nodes |
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459 if dangling is None: |
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460 dangling_weights = p |
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461 else: |
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462 # Convert the dangling dictionary into an array in nodelist order |
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463 dangling_weights = np.array([dangling.get(n, 0) for n in nodelist], dtype=float) |
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464 dangling_weights /= dangling_weights.sum() |
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465 is_dangling = np.where(S == 0)[0] |
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466 |
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467 # power iteration: make up to max_iter iterations |
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468 for _ in range(max_iter): |
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469 xlast = x |
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470 x = alpha * (x * M + sum(x[is_dangling]) * dangling_weights) + (1 - alpha) * p |
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471 # check convergence, l1 norm |
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472 err = np.absolute(x - xlast).sum() |
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473 if err < N * tol: |
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474 return dict(zip(nodelist, map(float, x))) |
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475 raise nx.PowerIterationFailedConvergence(max_iter) |