Mercurial > repos > shellac > sam_consensus_v3
annotate env/lib/python3.9/sitepackages/networkx/algorithms/centrality/katz.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 """Katz centrality.""" 
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2 from math import sqrt 
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3 
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4 import networkx as nx 
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5 from networkx.utils import not_implemented_for 
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6 
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7 __all__ = ["katz_centrality", "katz_centrality_numpy"] 
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8 
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9 
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10 @not_implemented_for("multigraph") 
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11 def katz_centrality( 
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12 G, 
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13 alpha=0.1, 
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14 beta=1.0, 
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15 max_iter=1000, 
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16 tol=1.0e6, 
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17 nstart=None, 
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18 normalized=True, 
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19 weight=None, 
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20 ): 
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21 r"""Compute the Katz centrality for the nodes of the graph G. 
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22 
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23 Katz centrality computes the centrality for a node based on the centrality 
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24 of its neighbors. It is a generalization of the eigenvector centrality. The 
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25 Katz centrality for node $i$ is 
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26 
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27 .. math:: 
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28 
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29 x_i = \alpha \sum_{j} A_{ij} x_j + \beta, 
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30 
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31 where $A$ is the adjacency matrix of graph G with eigenvalues $\lambda$. 
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32 
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33 The parameter $\beta$ controls the initial centrality and 
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34 
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35 .. math:: 
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36 
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37 \alpha < \frac{1}{\lambda_{\max}}. 
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38 
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39 Katz centrality computes the relative influence of a node within a 
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40 network by measuring the number of the immediate neighbors (first 
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41 degree nodes) and also all other nodes in the network that connect 
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42 to the node under consideration through these immediate neighbors. 
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43 
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44 Extra weight can be provided to immediate neighbors through the 
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45 parameter $\beta$. Connections made with distant neighbors 
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46 are, however, penalized by an attenuation factor $\alpha$ which 
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47 should be strictly less than the inverse largest eigenvalue of the 
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48 adjacency matrix in order for the Katz centrality to be computed 
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49 correctly. More information is provided in [1]_. 
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50 
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51 Parameters 
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52  
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53 G : graph 
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54 A NetworkX graph. 
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55 
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56 alpha : float 
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57 Attenuation factor 
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58 
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59 beta : scalar or dictionary, optional (default=1.0) 
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60 Weight attributed to the immediate neighborhood. If not a scalar, the 
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61 dictionary must have an value for every node. 
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62 
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63 max_iter : integer, optional (default=1000) 
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64 Maximum number of iterations in power method. 
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65 
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66 tol : float, optional (default=1.0e6) 
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67 Error tolerance used to check convergence in power method iteration. 
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68 
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69 nstart : dictionary, optional 
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70 Starting value of Katz iteration for each node. 
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71 
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72 normalized : bool, optional (default=True) 
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73 If True normalize the resulting values. 
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74 
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75 weight : None or string, optional (default=None) 
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76 If None, all edge weights are considered equal. 
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77 Otherwise holds the name of the edge attribute used as weight. 
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78 
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79 Returns 
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80  
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81 nodes : dictionary 
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82 Dictionary of nodes with Katz centrality as the value. 
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83 
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84 Raises 
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85  
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86 NetworkXError 
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87 If the parameter `beta` is not a scalar but lacks a value for at least 
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88 one node 
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89 
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90 PowerIterationFailedConvergence 
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91 If the algorithm fails to converge to the specified tolerance 
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92 within the specified number of iterations of the power iteration 
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93 method. 
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94 
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95 Examples 
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96  
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97 >>> import math 
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98 >>> G = nx.path_graph(4) 
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99 >>> phi = (1 + math.sqrt(5)) / 2.0 # largest eigenvalue of adj matrix 
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100 >>> centrality = nx.katz_centrality(G, 1 / phi  0.01) 
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101 >>> for n, c in sorted(centrality.items()): 
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102 ... print(f"{n} {c:.2f}") 
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103 0 0.37 
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104 1 0.60 
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105 2 0.60 
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106 3 0.37 
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107 
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108 See Also 
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109  
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110 katz_centrality_numpy 
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111 eigenvector_centrality 
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112 eigenvector_centrality_numpy 
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113 pagerank 
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114 hits 
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115 
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116 Notes 
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117  
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118 Katz centrality was introduced by [2]_. 
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119 
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120 This algorithm it uses the power method to find the eigenvector 
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121 corresponding to the largest eigenvalue of the adjacency matrix of ``G``. 
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122 The parameter ``alpha`` should be strictly less than the inverse of largest 
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123 eigenvalue of the adjacency matrix for the algorithm to converge. 
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124 You can use ``max(nx.adjacency_spectrum(G))`` to get $\lambda_{\max}$ the largest 
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125 eigenvalue of the adjacency matrix. 
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126 The iteration will stop after ``max_iter`` iterations or an error tolerance of 
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127 ``number_of_nodes(G) * tol`` has been reached. 
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128 
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129 When $\alpha = 1/\lambda_{\max}$ and $\beta=0$, Katz centrality is the same 
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130 as eigenvector centrality. 
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131 
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132 For directed graphs this finds "left" eigenvectors which corresponds 
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133 to the inedges in the graph. For outedges Katz centrality 
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134 first reverse the graph with ``G.reverse()``. 
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135 
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136 References 
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137  
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138 .. [1] Mark E. J. Newman: 
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139 Networks: An Introduction. 
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140 Oxford University Press, USA, 2010, p. 720. 
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141 .. [2] Leo Katz: 
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142 A New Status Index Derived from Sociometric Index. 
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143 Psychometrika 18(1):39–43, 1953 
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144 http://phya.snu.ac.kr/~dkim/PRL87278701.pdf 
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145 """ 
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146 if len(G) == 0: 
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147 return {} 
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148 
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149 nnodes = G.number_of_nodes() 
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150 
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151 if nstart is None: 
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152 # choose starting vector with entries of 0 
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153 x = {n: 0 for n in G} 
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154 else: 
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155 x = nstart 
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156 
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157 try: 
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158 b = dict.fromkeys(G, float(beta)) 
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159 except (TypeError, ValueError, AttributeError) as e: 
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160 b = beta 
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161 if set(beta) != set(G): 
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162 raise nx.NetworkXError( 
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163 "beta dictionary " "must have a value for every node" 
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164 ) from e 
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165 
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166 # make up to max_iter iterations 
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167 for i in range(max_iter): 
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168 xlast = x 
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169 x = dict.fromkeys(xlast, 0) 
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170 # do the multiplication y^T = Alpha * x^T A  Beta 
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171 for n in x: 
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172 for nbr in G[n]: 
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173 x[nbr] += xlast[n] * G[n][nbr].get(weight, 1) 
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174 for n in x: 
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175 x[n] = alpha * x[n] + b[n] 
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176 
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177 # check convergence 
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178 err = sum([abs(x[n]  xlast[n]) for n in x]) 
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179 if err < nnodes * tol: 
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180 if normalized: 
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181 # normalize vector 
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182 try: 
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183 s = 1.0 / sqrt(sum(v ** 2 for v in x.values())) 
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184 # this should never be zero? 
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185 except ZeroDivisionError: 
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186 s = 1.0 
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187 else: 
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188 s = 1 
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189 for n in x: 
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190 x[n] *= s 
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191 return x 
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192 raise nx.PowerIterationFailedConvergence(max_iter) 
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193 
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194 
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195 @not_implemented_for("multigraph") 
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196 def katz_centrality_numpy(G, alpha=0.1, beta=1.0, normalized=True, weight=None): 
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197 r"""Compute the Katz centrality for the graph G. 
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198 
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199 Katz centrality computes the centrality for a node based on the centrality 
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200 of its neighbors. It is a generalization of the eigenvector centrality. The 
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201 Katz centrality for node $i$ is 
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202 
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203 .. math:: 
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204 
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205 x_i = \alpha \sum_{j} A_{ij} x_j + \beta, 
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206 
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207 where $A$ is the adjacency matrix of graph G with eigenvalues $\lambda$. 
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208 
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209 The parameter $\beta$ controls the initial centrality and 
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210 
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211 .. math:: 
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212 
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213 \alpha < \frac{1}{\lambda_{\max}}. 
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214 
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215 Katz centrality computes the relative influence of a node within a 
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216 network by measuring the number of the immediate neighbors (first 
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217 degree nodes) and also all other nodes in the network that connect 
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218 to the node under consideration through these immediate neighbors. 
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219 
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220 Extra weight can be provided to immediate neighbors through the 
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221 parameter $\beta$. Connections made with distant neighbors 
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222 are, however, penalized by an attenuation factor $\alpha$ which 
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223 should be strictly less than the inverse largest eigenvalue of the 
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224 adjacency matrix in order for the Katz centrality to be computed 
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225 correctly. More information is provided in [1]_. 
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226 
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227 Parameters 
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228  
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229 G : graph 
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230 A NetworkX graph 
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231 
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232 alpha : float 
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233 Attenuation factor 
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234 
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235 beta : scalar or dictionary, optional (default=1.0) 
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236 Weight attributed to the immediate neighborhood. If not a scalar the 
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237 dictionary must have an value for every node. 
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238 
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239 normalized : bool 
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240 If True normalize the resulting values. 
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241 
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242 weight : None or string, optional 
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243 If None, all edge weights are considered equal. 
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244 Otherwise holds the name of the edge attribute used as weight. 
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245 
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246 Returns 
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247  
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248 nodes : dictionary 
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249 Dictionary of nodes with Katz centrality as the value. 
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250 
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251 Raises 
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252  
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253 NetworkXError 
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254 If the parameter `beta` is not a scalar but lacks a value for at least 
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255 one node 
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256 
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257 Examples 
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258  
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259 >>> import math 
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260 >>> G = nx.path_graph(4) 
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261 >>> phi = (1 + math.sqrt(5)) / 2.0 # largest eigenvalue of adj matrix 
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262 >>> centrality = nx.katz_centrality_numpy(G, 1 / phi) 
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263 >>> for n, c in sorted(centrality.items()): 
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264 ... print(f"{n} {c:.2f}") 
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265 0 0.37 
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266 1 0.60 
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267 2 0.60 
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268 3 0.37 
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269 
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270 See Also 
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271  
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272 katz_centrality 
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273 eigenvector_centrality_numpy 
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274 eigenvector_centrality 
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275 pagerank 
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276 hits 
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277 
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278 Notes 
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279  
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280 Katz centrality was introduced by [2]_. 
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281 
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282 This algorithm uses a direct linear solver to solve the above equation. 
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283 The parameter ``alpha`` should be strictly less than the inverse of largest 
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284 eigenvalue of the adjacency matrix for there to be a solution. 
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285 You can use ``max(nx.adjacency_spectrum(G))`` to get $\lambda_{\max}$ the largest 
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286 eigenvalue of the adjacency matrix. 
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287 
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288 When $\alpha = 1/\lambda_{\max}$ and $\beta=0$, Katz centrality is the same 
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289 as eigenvector centrality. 
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290 
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291 For directed graphs this finds "left" eigenvectors which corresponds 
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292 to the inedges in the graph. For outedges Katz centrality 
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293 first reverse the graph with ``G.reverse()``. 
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294 
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295 References 
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296  
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297 .. [1] Mark E. J. Newman: 
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298 Networks: An Introduction. 
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299 Oxford University Press, USA, 2010, p. 720. 
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300 .. [2] Leo Katz: 
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301 A New Status Index Derived from Sociometric Index. 
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302 Psychometrika 18(1):39–43, 1953 
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303 http://phya.snu.ac.kr/~dkim/PRL87278701.pdf 
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304 """ 
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305 try: 
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306 import numpy as np 
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307 except ImportError as e: 
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308 raise ImportError("Requires NumPy: http://numpy.org/") from e 
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309 if len(G) == 0: 
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310 return {} 
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311 try: 
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312 nodelist = beta.keys() 
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313 if set(nodelist) != set(G): 
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314 raise nx.NetworkXError( 
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315 "beta dictionary " "must have a value for every node" 
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316 ) 
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317 b = np.array(list(beta.values()), dtype=float) 
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318 except AttributeError: 
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319 nodelist = list(G) 
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320 try: 
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321 b = np.ones((len(nodelist), 1)) * float(beta) 
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322 except (TypeError, ValueError, AttributeError) as e: 
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323 raise nx.NetworkXError("beta must be a number") from e 
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324 
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325 A = nx.adj_matrix(G, nodelist=nodelist, weight=weight).todense().T 
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326 n = A.shape[0] 
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327 centrality = np.linalg.solve(np.eye(n, n)  (alpha * A), b) 
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328 if normalized: 
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329 norm = np.sign(sum(centrality)) * np.linalg.norm(centrality) 
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330 else: 
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331 norm = 1.0 
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332 centrality = dict(zip(nodelist, map(float, centrality / norm))) 
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333 return centrality 