Mercurial > repos > shellac > sam_consensus_v3
annotate env/lib/python3.9/site-packages/networkx/algorithms/centrality/current_flow_betweenness.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 """Current-flow betweenness centrality measures.""" |
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2 import networkx as nx |
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3 from networkx.algorithms.centrality.flow_matrix import ( |
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4 CGInverseLaplacian, |
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5 flow_matrix_row, |
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6 FullInverseLaplacian, |
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7 laplacian_sparse_matrix, |
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8 SuperLUInverseLaplacian, |
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9 ) |
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10 from networkx.utils import ( |
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11 not_implemented_for, |
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12 reverse_cuthill_mckee_ordering, |
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13 py_random_state, |
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14 ) |
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15 |
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16 __all__ = [ |
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17 "current_flow_betweenness_centrality", |
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18 "approximate_current_flow_betweenness_centrality", |
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19 "edge_current_flow_betweenness_centrality", |
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20 ] |
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21 |
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22 |
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23 @py_random_state(7) |
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24 @not_implemented_for("directed") |
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25 def approximate_current_flow_betweenness_centrality( |
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26 G, |
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27 normalized=True, |
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28 weight=None, |
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29 dtype=float, |
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30 solver="full", |
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31 epsilon=0.5, |
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32 kmax=10000, |
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33 seed=None, |
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34 ): |
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35 r"""Compute the approximate current-flow betweenness centrality for nodes. |
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36 |
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37 Approximates the current-flow betweenness centrality within absolute |
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38 error of epsilon with high probability [1]_. |
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39 |
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40 |
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41 Parameters |
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42 ---------- |
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43 G : graph |
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44 A NetworkX graph |
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45 |
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46 normalized : bool, optional (default=True) |
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47 If True the betweenness values are normalized by 2/[(n-1)(n-2)] where |
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48 n is the number of nodes in G. |
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49 |
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50 weight : string or None, optional (default=None) |
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51 Key for edge data used as the edge weight. |
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52 If None, then use 1 as each edge weight. |
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53 |
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54 dtype : data type (float) |
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55 Default data type for internal matrices. |
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56 Set to np.float32 for lower memory consumption. |
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57 |
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58 solver : string (default='full') |
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59 Type of linear solver to use for computing the flow matrix. |
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60 Options are "full" (uses most memory), "lu" (recommended), and |
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61 "cg" (uses least memory). |
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62 |
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63 epsilon: float |
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64 Absolute error tolerance. |
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65 |
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66 kmax: int |
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67 Maximum number of sample node pairs to use for approximation. |
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68 |
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69 seed : integer, random_state, or None (default) |
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70 Indicator of random number generation state. |
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71 See :ref:`Randomness<randomness>`. |
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72 |
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73 Returns |
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74 ------- |
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75 nodes : dictionary |
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76 Dictionary of nodes with betweenness centrality as the value. |
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77 |
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78 See Also |
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79 -------- |
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80 current_flow_betweenness_centrality |
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81 |
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82 Notes |
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83 ----- |
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84 The running time is $O((1/\epsilon^2)m{\sqrt k} \log n)$ |
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85 and the space required is $O(m)$ for $n$ nodes and $m$ edges. |
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86 |
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87 If the edges have a 'weight' attribute they will be used as |
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88 weights in this algorithm. Unspecified weights are set to 1. |
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89 |
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90 References |
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91 ---------- |
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92 .. [1] Ulrik Brandes and Daniel Fleischer: |
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93 Centrality Measures Based on Current Flow. |
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94 Proc. 22nd Symp. Theoretical Aspects of Computer Science (STACS '05). |
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95 LNCS 3404, pp. 533-544. Springer-Verlag, 2005. |
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96 http://algo.uni-konstanz.de/publications/bf-cmbcf-05.pdf |
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97 """ |
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98 try: |
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99 import numpy as np |
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100 except ImportError as e: |
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101 raise ImportError( |
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102 "current_flow_betweenness_centrality requires NumPy " "http://numpy.org/" |
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103 ) from e |
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104 if not nx.is_connected(G): |
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105 raise nx.NetworkXError("Graph not connected.") |
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106 solvername = { |
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107 "full": FullInverseLaplacian, |
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108 "lu": SuperLUInverseLaplacian, |
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109 "cg": CGInverseLaplacian, |
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110 } |
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111 n = G.number_of_nodes() |
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112 ordering = list(reverse_cuthill_mckee_ordering(G)) |
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113 # make a copy with integer labels according to rcm ordering |
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114 # this could be done without a copy if we really wanted to |
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115 H = nx.relabel_nodes(G, dict(zip(ordering, range(n)))) |
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116 L = laplacian_sparse_matrix( |
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117 H, nodelist=range(n), weight=weight, dtype=dtype, format="csc" |
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118 ) |
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119 C = solvername[solver](L, dtype=dtype) # initialize solver |
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120 betweenness = dict.fromkeys(H, 0.0) |
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121 nb = (n - 1.0) * (n - 2.0) # normalization factor |
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122 cstar = n * (n - 1) / nb |
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123 l = 1 # parameter in approximation, adjustable |
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124 k = l * int(np.ceil((cstar / epsilon) ** 2 * np.log(n))) |
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125 if k > kmax: |
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126 msg = f"Number random pairs k>kmax ({k}>{kmax}) " |
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127 raise nx.NetworkXError(msg, "Increase kmax or epsilon") |
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128 cstar2k = cstar / (2 * k) |
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129 for i in range(k): |
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130 s, t = seed.sample(range(n), 2) |
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131 b = np.zeros(n, dtype=dtype) |
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132 b[s] = 1 |
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133 b[t] = -1 |
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134 p = C.solve(b) |
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135 for v in H: |
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136 if v == s or v == t: |
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137 continue |
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138 for nbr in H[v]: |
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139 w = H[v][nbr].get(weight, 1.0) |
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140 betweenness[v] += w * np.abs(p[v] - p[nbr]) * cstar2k |
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141 if normalized: |
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142 factor = 1.0 |
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143 else: |
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144 factor = nb / 2.0 |
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145 # remap to original node names and "unnormalize" if required |
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146 return {ordering[k]: float(v * factor) for k, v in betweenness.items()} |
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147 |
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148 |
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149 @not_implemented_for("directed") |
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150 def current_flow_betweenness_centrality( |
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151 G, normalized=True, weight=None, dtype=float, solver="full" |
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152 ): |
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153 r"""Compute current-flow betweenness centrality for nodes. |
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154 |
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155 Current-flow betweenness centrality uses an electrical current |
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156 model for information spreading in contrast to betweenness |
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157 centrality which uses shortest paths. |
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158 |
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159 Current-flow betweenness centrality is also known as |
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160 random-walk betweenness centrality [2]_. |
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161 |
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162 Parameters |
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163 ---------- |
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164 G : graph |
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165 A NetworkX graph |
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166 |
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167 normalized : bool, optional (default=True) |
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168 If True the betweenness values are normalized by 2/[(n-1)(n-2)] where |
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169 n is the number of nodes in G. |
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170 |
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171 weight : string or None, optional (default=None) |
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172 Key for edge data used as the edge weight. |
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173 If None, then use 1 as each edge weight. |
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174 |
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175 dtype : data type (float) |
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176 Default data type for internal matrices. |
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177 Set to np.float32 for lower memory consumption. |
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178 |
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179 solver : string (default='full') |
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180 Type of linear solver to use for computing the flow matrix. |
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181 Options are "full" (uses most memory), "lu" (recommended), and |
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182 "cg" (uses least memory). |
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183 |
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184 Returns |
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185 ------- |
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186 nodes : dictionary |
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187 Dictionary of nodes with betweenness centrality as the value. |
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188 |
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189 See Also |
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190 -------- |
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191 approximate_current_flow_betweenness_centrality |
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192 betweenness_centrality |
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193 edge_betweenness_centrality |
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194 edge_current_flow_betweenness_centrality |
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195 |
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196 Notes |
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197 ----- |
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198 Current-flow betweenness can be computed in $O(I(n-1)+mn \log n)$ |
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199 time [1]_, where $I(n-1)$ is the time needed to compute the |
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200 inverse Laplacian. For a full matrix this is $O(n^3)$ but using |
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201 sparse methods you can achieve $O(nm{\sqrt k})$ where $k$ is the |
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202 Laplacian matrix condition number. |
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203 |
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204 The space required is $O(nw)$ where $w$ is the width of the sparse |
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205 Laplacian matrix. Worse case is $w=n$ for $O(n^2)$. |
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206 |
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207 If the edges have a 'weight' attribute they will be used as |
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208 weights in this algorithm. Unspecified weights are set to 1. |
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209 |
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210 References |
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211 ---------- |
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212 .. [1] Centrality Measures Based on Current Flow. |
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213 Ulrik Brandes and Daniel Fleischer, |
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214 Proc. 22nd Symp. Theoretical Aspects of Computer Science (STACS '05). |
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215 LNCS 3404, pp. 533-544. Springer-Verlag, 2005. |
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216 http://algo.uni-konstanz.de/publications/bf-cmbcf-05.pdf |
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217 |
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218 .. [2] A measure of betweenness centrality based on random walks, |
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219 M. E. J. Newman, Social Networks 27, 39-54 (2005). |
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220 """ |
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221 if not nx.is_connected(G): |
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222 raise nx.NetworkXError("Graph not connected.") |
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223 n = G.number_of_nodes() |
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224 ordering = list(reverse_cuthill_mckee_ordering(G)) |
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225 # make a copy with integer labels according to rcm ordering |
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226 # this could be done without a copy if we really wanted to |
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227 H = nx.relabel_nodes(G, dict(zip(ordering, range(n)))) |
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228 betweenness = dict.fromkeys(H, 0.0) # b[v]=0 for v in H |
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229 for row, (s, t) in flow_matrix_row(H, weight=weight, dtype=dtype, solver=solver): |
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230 pos = dict(zip(row.argsort()[::-1], range(n))) |
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231 for i in range(n): |
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232 betweenness[s] += (i - pos[i]) * row[i] |
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233 betweenness[t] += (n - i - 1 - pos[i]) * row[i] |
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234 if normalized: |
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235 nb = (n - 1.0) * (n - 2.0) # normalization factor |
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236 else: |
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237 nb = 2.0 |
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238 for v in H: |
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239 betweenness[v] = float((betweenness[v] - v) * 2.0 / nb) |
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240 return {ordering[k]: v for k, v in betweenness.items()} |
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241 |
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242 |
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243 @not_implemented_for("directed") |
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244 def edge_current_flow_betweenness_centrality( |
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245 G, normalized=True, weight=None, dtype=float, solver="full" |
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246 ): |
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247 r"""Compute current-flow betweenness centrality for edges. |
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248 |
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249 Current-flow betweenness centrality uses an electrical current |
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250 model for information spreading in contrast to betweenness |
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251 centrality which uses shortest paths. |
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252 |
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253 Current-flow betweenness centrality is also known as |
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254 random-walk betweenness centrality [2]_. |
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255 |
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256 Parameters |
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257 ---------- |
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258 G : graph |
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259 A NetworkX graph |
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260 |
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261 normalized : bool, optional (default=True) |
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262 If True the betweenness values are normalized by 2/[(n-1)(n-2)] where |
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263 n is the number of nodes in G. |
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264 |
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265 weight : string or None, optional (default=None) |
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266 Key for edge data used as the edge weight. |
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267 If None, then use 1 as each edge weight. |
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268 |
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269 dtype : data type (default=float) |
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270 Default data type for internal matrices. |
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271 Set to np.float32 for lower memory consumption. |
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272 |
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273 solver : string (default='full') |
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274 Type of linear solver to use for computing the flow matrix. |
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275 Options are "full" (uses most memory), "lu" (recommended), and |
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276 "cg" (uses least memory). |
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277 |
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278 Returns |
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279 ------- |
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280 nodes : dictionary |
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281 Dictionary of edge tuples with betweenness centrality as the value. |
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282 |
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283 Raises |
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284 ------ |
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285 NetworkXError |
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286 The algorithm does not support DiGraphs. |
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287 If the input graph is an instance of DiGraph class, NetworkXError |
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288 is raised. |
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289 |
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290 See Also |
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291 -------- |
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292 betweenness_centrality |
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293 edge_betweenness_centrality |
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294 current_flow_betweenness_centrality |
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295 |
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296 Notes |
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297 ----- |
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298 Current-flow betweenness can be computed in $O(I(n-1)+mn \log n)$ |
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299 time [1]_, where $I(n-1)$ is the time needed to compute the |
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300 inverse Laplacian. For a full matrix this is $O(n^3)$ but using |
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301 sparse methods you can achieve $O(nm{\sqrt k})$ where $k$ is the |
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302 Laplacian matrix condition number. |
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303 |
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304 The space required is $O(nw)$ where $w$ is the width of the sparse |
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305 Laplacian matrix. Worse case is $w=n$ for $O(n^2)$. |
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306 |
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307 If the edges have a 'weight' attribute they will be used as |
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308 weights in this algorithm. Unspecified weights are set to 1. |
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309 |
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310 References |
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311 ---------- |
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312 .. [1] Centrality Measures Based on Current Flow. |
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313 Ulrik Brandes and Daniel Fleischer, |
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314 Proc. 22nd Symp. Theoretical Aspects of Computer Science (STACS '05). |
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315 LNCS 3404, pp. 533-544. Springer-Verlag, 2005. |
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316 http://algo.uni-konstanz.de/publications/bf-cmbcf-05.pdf |
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317 |
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318 .. [2] A measure of betweenness centrality based on random walks, |
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319 M. E. J. Newman, Social Networks 27, 39-54 (2005). |
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320 """ |
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321 from networkx.utils import reverse_cuthill_mckee_ordering |
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322 |
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323 if not nx.is_connected(G): |
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324 raise nx.NetworkXError("Graph not connected.") |
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325 n = G.number_of_nodes() |
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326 ordering = list(reverse_cuthill_mckee_ordering(G)) |
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327 # make a copy with integer labels according to rcm ordering |
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328 # this could be done without a copy if we really wanted to |
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329 H = nx.relabel_nodes(G, dict(zip(ordering, range(n)))) |
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330 edges = (tuple(sorted((u, v))) for u, v in H.edges()) |
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331 betweenness = dict.fromkeys(edges, 0.0) |
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332 if normalized: |
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333 nb = (n - 1.0) * (n - 2.0) # normalization factor |
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334 else: |
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335 nb = 2.0 |
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336 for row, (e) in flow_matrix_row(H, weight=weight, dtype=dtype, solver=solver): |
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337 pos = dict(zip(row.argsort()[::-1], range(1, n + 1))) |
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338 for i in range(n): |
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339 betweenness[e] += (i + 1 - pos[i]) * row[i] |
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340 betweenness[e] += (n - i - pos[i]) * row[i] |
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341 betweenness[e] /= nb |
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342 return {(ordering[s], ordering[t]): float(v) for (s, t), v in betweenness.items()} |