Mercurial > repos > shellac > sam_consensus_v3
annotate env/lib/python3.9/sitepackages/networkx/algorithms/centrality/eigenvector.py @ 0:4f3585e2f14b draft default tip
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author  shellac 

date  Mon, 22 Mar 2021 18:12:50 +0000 
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1 """Functions for computing eigenvector centrality.""" 
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2 
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3 from math import sqrt 
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4 
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5 import networkx as nx 
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6 from networkx.utils import not_implemented_for 
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7 
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8 __all__ = ["eigenvector_centrality", "eigenvector_centrality_numpy"] 
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9 
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10 
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11 @not_implemented_for("multigraph") 
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12 def eigenvector_centrality(G, max_iter=100, tol=1.0e6, nstart=None, weight=None): 
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13 r"""Compute the eigenvector centrality for the graph `G`. 
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14 
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15 Eigenvector centrality computes the centrality for a node based on the 
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16 centrality of its neighbors. The eigenvector centrality for node $i$ is 
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17 the $i$th element of the vector $x$ defined by the equation 
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18 
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19 .. math:: 
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20 
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21 Ax = \lambda x 
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22 
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23 where $A$ is the adjacency matrix of the graph `G` with eigenvalue 
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24 $\lambda$. By virtue of the Perron–Frobenius theorem, there is a unique 
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25 solution $x$, all of whose entries are positive, if $\lambda$ is the 
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26 largest eigenvalue of the adjacency matrix $A$ ([2]_). 
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27 
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28 Parameters 
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29  
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30 G : graph 
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31 A networkx graph 
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32 
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33 max_iter : integer, optional (default=100) 
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34 Maximum number of iterations in power method. 
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35 
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36 tol : float, optional (default=1.0e6) 
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37 Error tolerance used to check convergence in power method iteration. 
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38 
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39 nstart : dictionary, optional (default=None) 
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40 Starting value of eigenvector iteration for each node. 
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41 
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42 weight : None or string, optional (default=None) 
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43 If None, all edge weights are considered equal. 
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44 Otherwise holds the name of the edge attribute used as weight. 
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45 
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46 Returns 
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47  
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48 nodes : dictionary 
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49 Dictionary of nodes with eigenvector centrality as the value. 
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50 
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51 Examples 
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52  
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53 >>> G = nx.path_graph(4) 
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54 >>> centrality = nx.eigenvector_centrality(G) 
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55 >>> sorted((v, f"{c:0.2f}") for v, c in centrality.items()) 
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56 [(0, '0.37'), (1, '0.60'), (2, '0.60'), (3, '0.37')] 
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57 
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58 Raises 
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59  
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60 NetworkXPointlessConcept 
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61 If the graph `G` is the null graph. 
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62 
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63 NetworkXError 
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64 If each value in `nstart` is zero. 
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65 
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66 PowerIterationFailedConvergence 
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67 If the algorithm fails to converge to the specified tolerance 
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68 within the specified number of iterations of the power iteration 
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69 method. 
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70 
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71 See Also 
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72  
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73 eigenvector_centrality_numpy 
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74 pagerank 
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75 hits 
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76 
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77 Notes 
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78  
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79 The measure was introduced by [1]_ and is discussed in [2]_. 
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80 
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81 The power iteration method is used to compute the eigenvector and 
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82 convergence is **not** guaranteed. Our method stops after ``max_iter`` 
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83 iterations or when the change in the computed vector between two 
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84 iterations is smaller than an error tolerance of 
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85 ``G.number_of_nodes() * tol``. This implementation uses ($A + I$) 
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86 rather than the adjacency matrix $A$ because it shifts the spectrum 
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87 to enable discerning the correct eigenvector even for networks with 
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88 multiple dominant eigenvalues. 
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89 
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90 For directed graphs this is "left" eigenvector centrality which corresponds 
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91 to the inedges in the graph. For outedges eigenvector centrality 
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92 first reverse the graph with ``G.reverse()``. 
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93 
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94 References 
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95  
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96 .. [1] Phillip Bonacich. 
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97 "Power and Centrality: A Family of Measures." 
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98 *American Journal of Sociology* 92(5):1170–1182, 1986 
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99 <http://www.leonidzhukov.net/hse/2014/socialnetworks/papers/BonacichCentrality.pdf> 
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100 .. [2] Mark E. J. Newman. 
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101 *Networks: An Introduction.* 
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102 Oxford University Press, USA, 2010, pp. 169. 
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103 
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104 """ 
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105 if len(G) == 0: 
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106 raise nx.NetworkXPointlessConcept( 
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107 "cannot compute centrality for the null graph" 
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108 ) 
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109 # If no initial vector is provided, start with the allones vector. 
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110 if nstart is None: 
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111 nstart = {v: 1 for v in G} 
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112 if all(v == 0 for v in nstart.values()): 
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113 raise nx.NetworkXError("initial vector cannot have all zero values") 
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114 # Normalize the initial vector so that each entry is in [0, 1]. This is 
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115 # guaranteed to never have a dividebyzero error by the previous line. 
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116 nstart_sum = sum(nstart.values()) 
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117 x = {k: v / nstart_sum for k, v in nstart.items()} 
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118 nnodes = G.number_of_nodes() 
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119 # make up to max_iter iterations 
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120 for i in range(max_iter): 
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121 xlast = x 
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122 x = xlast.copy() # Start with xlast times I to iterate with (A+I) 
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123 # do the multiplication y^T = x^T A (left eigenvector) 
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124 for n in x: 
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125 for nbr in G[n]: 
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126 w = G[n][nbr].get(weight, 1) if weight else 1 
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127 x[nbr] += xlast[n] * w 
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128 # Normalize the vector. The normalization denominator `norm` 
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129 # should never be zero by the PerronFrobenius 
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130 # theorem. However, in case it is due to numerical error, we 
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131 # assume the norm to be one instead. 
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132 norm = sqrt(sum(z ** 2 for z in x.values())) or 1 
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133 x = {k: v / norm for k, v in x.items()} 
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134 # Check for convergence (in the L_1 norm). 
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135 if sum(abs(x[n]  xlast[n]) for n in x) < nnodes * tol: 
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136 return x 
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137 raise nx.PowerIterationFailedConvergence(max_iter) 
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138 
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139 
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140 def eigenvector_centrality_numpy(G, weight=None, max_iter=50, tol=0): 
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141 r"""Compute the eigenvector centrality for the graph G. 
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142 
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143 Eigenvector centrality computes the centrality for a node based on the 
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144 centrality of its neighbors. The eigenvector centrality for node $i$ is 
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145 
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146 .. math:: 
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147 
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148 Ax = \lambda x 
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149 
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150 where $A$ is the adjacency matrix of the graph G with eigenvalue $\lambda$. 
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151 By virtue of the Perron–Frobenius theorem, there is a unique and positive 
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152 solution if $\lambda$ is the largest eigenvalue associated with the 
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153 eigenvector of the adjacency matrix $A$ ([2]_). 
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154 
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155 Parameters 
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156  
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157 G : graph 
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158 A networkx graph 
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159 
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160 weight : None or string, optional (default=None) 
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161 The name of the edge attribute used as weight. 
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162 If None, all edge weights are considered equal. 
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163 
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164 max_iter : integer, optional (default=100) 
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165 Maximum number of iterations in power method. 
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166 
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167 tol : float, optional (default=1.0e6) 
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168 Relative accuracy for eigenvalues (stopping criterion). 
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169 The default value of 0 implies machine precision. 
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170 
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171 Returns 
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172  
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173 nodes : dictionary 
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174 Dictionary of nodes with eigenvector centrality as the value. 
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175 
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176 Examples 
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177  
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178 >>> G = nx.path_graph(4) 
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179 >>> centrality = nx.eigenvector_centrality_numpy(G) 
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180 >>> print([f"{node} {centrality[node]:0.2f}" for node in centrality]) 
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181 ['0 0.37', '1 0.60', '2 0.60', '3 0.37'] 
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182 
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183 See Also 
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184  
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185 eigenvector_centrality 
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186 pagerank 
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187 hits 
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188 
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189 Notes 
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190  
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191 The measure was introduced by [1]_. 
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192 
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193 This algorithm uses the SciPy sparse eigenvalue solver (ARPACK) to 
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194 find the largest eigenvalue/eigenvector pair. 
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195 
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196 For directed graphs this is "left" eigenvector centrality which corresponds 
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197 to the inedges in the graph. For outedges eigenvector centrality 
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198 first reverse the graph with ``G.reverse()``. 
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199 
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200 Raises 
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201  
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202 NetworkXPointlessConcept 
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203 If the graph ``G`` is the null graph. 
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204 
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205 References 
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206  
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207 .. [1] Phillip Bonacich: 
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208 Power and Centrality: A Family of Measures. 
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209 American Journal of Sociology 92(5):1170–1182, 1986 
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210 http://www.leonidzhukov.net/hse/2014/socialnetworks/papers/BonacichCentrality.pdf 
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211 .. [2] Mark E. J. Newman: 
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212 Networks: An Introduction. 
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213 Oxford University Press, USA, 2010, pp. 169. 
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214 """ 
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215 import numpy as np 
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216 import scipy as sp 
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217 from scipy.sparse import linalg 
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218 
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219 if len(G) == 0: 
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220 raise nx.NetworkXPointlessConcept( 
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221 "cannot compute centrality for the null graph" 
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222 ) 
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223 M = nx.to_scipy_sparse_matrix(G, nodelist=list(G), weight=weight, dtype=float) 
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224 eigenvalue, eigenvector = linalg.eigs( 
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225 M.T, k=1, which="LR", maxiter=max_iter, tol=tol 
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226 ) 
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227 largest = eigenvector.flatten().real 
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228 norm = np.sign(largest.sum()) * sp.linalg.norm(largest) 
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229 return dict(zip(G, largest / norm)) 