Mercurial > repos > shellac > sam_consensus_v3
annotate env/lib/python3.9/site-packages/networkx/algorithms/distance_measures.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 """Graph diameter, radius, eccentricity and other properties.""" |
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2 |
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3 import networkx as nx |
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4 from networkx.utils import not_implemented_for |
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5 |
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6 __all__ = [ |
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7 "extrema_bounding", |
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8 "eccentricity", |
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9 "diameter", |
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10 "radius", |
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11 "periphery", |
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12 "center", |
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13 "barycenter", |
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14 "resistance_distance", |
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15 ] |
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16 |
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17 |
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18 def extrema_bounding(G, compute="diameter"): |
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19 """Compute requested extreme distance metric of undirected graph G |
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20 |
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21 Computation is based on smart lower and upper bounds, and in practice |
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22 linear in the number of nodes, rather than quadratic (except for some |
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23 border cases such as complete graphs or circle shaped graphs). |
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24 |
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25 Parameters |
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26 ---------- |
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27 G : NetworkX graph |
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28 An undirected graph |
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29 |
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30 compute : string denoting the requesting metric |
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31 "diameter" for the maximal eccentricity value, |
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32 "radius" for the minimal eccentricity value, |
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33 "periphery" for the set of nodes with eccentricity equal to the diameter |
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34 "center" for the set of nodes with eccentricity equal to the radius |
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35 |
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36 Returns |
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37 ------- |
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38 value : value of the requested metric |
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39 int for "diameter" and "radius" or |
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40 list of nodes for "center" and "periphery" |
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41 |
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42 Raises |
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43 ------ |
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44 NetworkXError |
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45 If the graph consists of multiple components |
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46 |
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47 Notes |
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48 ----- |
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49 This algorithm was proposed in the following papers: |
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50 |
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51 F.W. Takes and W.A. Kosters, Determining the Diameter of Small World |
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52 Networks, in Proceedings of the 20th ACM International Conference on |
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53 Information and Knowledge Management (CIKM 2011), pp. 1191-1196, 2011. |
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54 doi: https://doi.org/10.1145/2063576.2063748 |
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55 |
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56 F.W. Takes and W.A. Kosters, Computing the Eccentricity Distribution of |
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57 Large Graphs, Algorithms 6(1): 100-118, 2013. |
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58 doi: https://doi.org/10.3390/a6010100 |
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59 |
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60 M. Borassi, P. Crescenzi, M. Habib, W.A. Kosters, A. Marino and F.W. Takes, |
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61 Fast Graph Diameter and Radius BFS-Based Computation in (Weakly Connected) |
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62 Real-World Graphs, Theoretical Computer Science 586: 59-80, 2015. |
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63 doi: https://doi.org/10.1016/j.tcs.2015.02.033 |
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64 """ |
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65 |
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66 # init variables |
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67 degrees = dict(G.degree()) # start with the highest degree node |
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68 minlowernode = max(degrees, key=degrees.get) |
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69 N = len(degrees) # number of nodes |
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70 # alternate between smallest lower and largest upper bound |
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71 high = False |
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72 # status variables |
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73 ecc_lower = dict.fromkeys(G, 0) |
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74 ecc_upper = dict.fromkeys(G, N) |
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75 candidates = set(G) |
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76 |
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77 # (re)set bound extremes |
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78 minlower = N |
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79 maxlower = 0 |
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80 minupper = N |
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81 maxupper = 0 |
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82 |
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83 # repeat the following until there are no more candidates |
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84 while candidates: |
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85 if high: |
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86 current = maxuppernode # select node with largest upper bound |
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87 else: |
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88 current = minlowernode # select node with smallest lower bound |
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89 high = not high |
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90 |
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91 # get distances from/to current node and derive eccentricity |
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92 dist = dict(nx.single_source_shortest_path_length(G, current)) |
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93 if len(dist) != N: |
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94 msg = "Cannot compute metric because graph is not connected." |
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95 raise nx.NetworkXError(msg) |
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96 current_ecc = max(dist.values()) |
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97 |
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98 # print status update |
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99 # print ("ecc of " + str(current) + " (" + str(ecc_lower[current]) + "/" |
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100 # + str(ecc_upper[current]) + ", deg: " + str(dist[current]) + ") is " |
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101 # + str(current_ecc)) |
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102 # print(ecc_upper) |
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103 |
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104 # (re)set bound extremes |
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105 maxuppernode = None |
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106 minlowernode = None |
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107 |
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108 # update node bounds |
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109 for i in candidates: |
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110 # update eccentricity bounds |
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111 d = dist[i] |
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112 ecc_lower[i] = low = max(ecc_lower[i], max(d, (current_ecc - d))) |
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113 ecc_upper[i] = upp = min(ecc_upper[i], current_ecc + d) |
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114 |
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115 # update min/max values of lower and upper bounds |
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116 minlower = min(ecc_lower[i], minlower) |
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117 maxlower = max(ecc_lower[i], maxlower) |
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118 minupper = min(ecc_upper[i], minupper) |
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119 maxupper = max(ecc_upper[i], maxupper) |
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120 |
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121 # update candidate set |
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122 if compute == "diameter": |
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123 ruled_out = { |
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124 i |
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125 for i in candidates |
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126 if ecc_upper[i] <= maxlower and 2 * ecc_lower[i] >= maxupper |
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127 } |
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128 |
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129 elif compute == "radius": |
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130 ruled_out = { |
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131 i |
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132 for i in candidates |
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133 if ecc_lower[i] >= minupper and ecc_upper[i] + 1 <= 2 * minlower |
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134 } |
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135 |
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136 elif compute == "periphery": |
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137 ruled_out = { |
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138 i |
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139 for i in candidates |
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140 if ecc_upper[i] < maxlower |
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141 and (maxlower == maxupper or ecc_lower[i] > maxupper) |
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142 } |
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143 |
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144 elif compute == "center": |
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145 ruled_out = { |
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146 i |
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147 for i in candidates |
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148 if ecc_lower[i] > minupper |
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149 and (minlower == minupper or ecc_upper[i] + 1 < 2 * minlower) |
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150 } |
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151 |
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152 elif compute == "eccentricities": |
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153 ruled_out = {} |
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154 |
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155 ruled_out.update(i for i in candidates if ecc_lower[i] == ecc_upper[i]) |
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156 candidates -= ruled_out |
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157 |
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158 # for i in ruled_out: |
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159 # print("removing %g: ecc_u: %g maxl: %g ecc_l: %g maxu: %g"% |
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160 # (i,ecc_upper[i],maxlower,ecc_lower[i],maxupper)) |
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161 # print("node %g: ecc_u: %g maxl: %g ecc_l: %g maxu: %g"% |
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162 # (4,ecc_upper[4],maxlower,ecc_lower[4],maxupper)) |
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163 # print("NODE 4: %g"%(ecc_upper[4] <= maxlower)) |
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164 # print("NODE 4: %g"%(2 * ecc_lower[4] >= maxupper)) |
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165 # print("NODE 4: %g"%(ecc_upper[4] <= maxlower |
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166 # and 2 * ecc_lower[4] >= maxupper)) |
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167 |
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168 # updating maxuppernode and minlowernode for selection in next round |
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169 for i in candidates: |
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170 if ( |
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171 minlowernode is None |
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172 or ( |
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173 ecc_lower[i] == ecc_lower[minlowernode] |
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174 and degrees[i] > degrees[minlowernode] |
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175 ) |
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176 or (ecc_lower[i] < ecc_lower[minlowernode]) |
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177 ): |
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178 minlowernode = i |
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179 |
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180 if ( |
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181 maxuppernode is None |
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182 or ( |
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183 ecc_upper[i] == ecc_upper[maxuppernode] |
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184 and degrees[i] > degrees[maxuppernode] |
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185 ) |
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186 or (ecc_upper[i] > ecc_upper[maxuppernode]) |
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187 ): |
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188 maxuppernode = i |
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189 |
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190 # print status update |
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191 # print (" min=" + str(minlower) + "/" + str(minupper) + |
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192 # " max=" + str(maxlower) + "/" + str(maxupper) + |
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193 # " candidates: " + str(len(candidates))) |
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194 # print("cand:",candidates) |
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195 # print("ecc_l",ecc_lower) |
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196 # print("ecc_u",ecc_upper) |
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197 # wait = input("press Enter to continue") |
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198 |
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199 # return the correct value of the requested metric |
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200 if compute == "diameter": |
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201 return maxlower |
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202 elif compute == "radius": |
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203 return minupper |
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204 elif compute == "periphery": |
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205 p = [v for v in G if ecc_lower[v] == maxlower] |
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206 return p |
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207 elif compute == "center": |
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208 c = [v for v in G if ecc_upper[v] == minupper] |
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209 return c |
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210 elif compute == "eccentricities": |
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211 return ecc_lower |
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212 return None |
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213 |
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214 |
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215 def eccentricity(G, v=None, sp=None): |
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216 """Returns the eccentricity of nodes in G. |
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217 |
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218 The eccentricity of a node v is the maximum distance from v to |
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219 all other nodes in G. |
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220 |
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221 Parameters |
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222 ---------- |
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223 G : NetworkX graph |
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224 A graph |
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225 |
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226 v : node, optional |
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227 Return value of specified node |
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228 |
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229 sp : dict of dicts, optional |
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230 All pairs shortest path lengths as a dictionary of dictionaries |
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231 |
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232 Returns |
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233 ------- |
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234 ecc : dictionary |
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235 A dictionary of eccentricity values keyed by node. |
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236 """ |
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237 # if v is None: # none, use entire graph |
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238 # nodes=G.nodes() |
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239 # elif v in G: # is v a single node |
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240 # nodes=[v] |
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241 # else: # assume v is a container of nodes |
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242 # nodes=v |
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243 order = G.order() |
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244 |
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245 e = {} |
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246 for n in G.nbunch_iter(v): |
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247 if sp is None: |
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248 length = nx.single_source_shortest_path_length(G, n) |
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249 L = len(length) |
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250 else: |
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251 try: |
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252 length = sp[n] |
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253 L = len(length) |
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254 except TypeError as e: |
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255 raise nx.NetworkXError('Format of "sp" is invalid.') from e |
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256 if L != order: |
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257 if G.is_directed(): |
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258 msg = ( |
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259 "Found infinite path length because the digraph is not" |
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260 " strongly connected" |
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261 ) |
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262 else: |
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263 msg = "Found infinite path length because the graph is not" " connected" |
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264 raise nx.NetworkXError(msg) |
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265 |
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266 e[n] = max(length.values()) |
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267 |
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268 if v in G: |
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269 return e[v] # return single value |
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270 else: |
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271 return e |
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272 |
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273 |
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274 def diameter(G, e=None, usebounds=False): |
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275 """Returns the diameter of the graph G. |
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276 |
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277 The diameter is the maximum eccentricity. |
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278 |
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279 Parameters |
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280 ---------- |
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281 G : NetworkX graph |
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282 A graph |
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283 |
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284 e : eccentricity dictionary, optional |
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285 A precomputed dictionary of eccentricities. |
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286 |
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287 Returns |
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288 ------- |
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289 d : integer |
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290 Diameter of graph |
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291 |
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292 See Also |
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293 -------- |
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294 eccentricity |
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295 """ |
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296 if usebounds is True and e is None and not G.is_directed(): |
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297 return extrema_bounding(G, compute="diameter") |
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298 if e is None: |
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299 e = eccentricity(G) |
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300 return max(e.values()) |
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301 |
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302 |
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303 def periphery(G, e=None, usebounds=False): |
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304 """Returns the periphery of the graph G. |
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305 |
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306 The periphery is the set of nodes with eccentricity equal to the diameter. |
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307 |
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308 Parameters |
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309 ---------- |
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310 G : NetworkX graph |
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311 A graph |
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312 |
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313 e : eccentricity dictionary, optional |
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314 A precomputed dictionary of eccentricities. |
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315 |
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316 Returns |
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317 ------- |
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318 p : list |
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319 List of nodes in periphery |
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320 |
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321 See Also |
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322 -------- |
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323 barycenter |
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324 center |
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325 """ |
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326 if usebounds is True and e is None and not G.is_directed(): |
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327 return extrema_bounding(G, compute="periphery") |
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328 if e is None: |
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329 e = eccentricity(G) |
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330 diameter = max(e.values()) |
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331 p = [v for v in e if e[v] == diameter] |
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332 return p |
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333 |
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334 |
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335 def radius(G, e=None, usebounds=False): |
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336 """Returns the radius of the graph G. |
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337 |
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338 The radius is the minimum eccentricity. |
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339 |
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340 Parameters |
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341 ---------- |
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342 G : NetworkX graph |
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343 A graph |
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344 |
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345 e : eccentricity dictionary, optional |
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346 A precomputed dictionary of eccentricities. |
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347 |
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348 Returns |
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349 ------- |
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350 r : integer |
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351 Radius of graph |
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352 """ |
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353 if usebounds is True and e is None and not G.is_directed(): |
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354 return extrema_bounding(G, compute="radius") |
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355 if e is None: |
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356 e = eccentricity(G) |
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357 return min(e.values()) |
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358 |
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359 |
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360 def center(G, e=None, usebounds=False): |
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361 """Returns the center of the graph G. |
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362 |
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363 The center is the set of nodes with eccentricity equal to radius. |
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364 |
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365 Parameters |
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366 ---------- |
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367 G : NetworkX graph |
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368 A graph |
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369 |
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370 e : eccentricity dictionary, optional |
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371 A precomputed dictionary of eccentricities. |
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372 |
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373 Returns |
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374 ------- |
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375 c : list |
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376 List of nodes in center |
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377 |
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378 See Also |
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379 -------- |
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380 barycenter |
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381 periphery |
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382 """ |
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383 if usebounds is True and e is None and not G.is_directed(): |
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384 return extrema_bounding(G, compute="center") |
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385 if e is None: |
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386 e = eccentricity(G) |
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387 radius = min(e.values()) |
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388 p = [v for v in e if e[v] == radius] |
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389 return p |
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390 |
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391 |
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392 def barycenter(G, weight=None, attr=None, sp=None): |
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393 r"""Calculate barycenter of a connected graph, optionally with edge weights. |
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394 |
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395 The :dfn:`barycenter` a |
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396 :func:`connected <networkx.algorithms.components.is_connected>` graph |
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397 :math:`G` is the subgraph induced by the set of its nodes :math:`v` |
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398 minimizing the objective function |
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399 |
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400 .. math:: |
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401 |
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402 \sum_{u \in V(G)} d_G(u, v), |
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403 |
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404 where :math:`d_G` is the (possibly weighted) :func:`path length |
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405 <networkx.algorithms.shortest_paths.generic.shortest_path_length>`. |
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406 The barycenter is also called the :dfn:`median`. See [West01]_, p. 78. |
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407 |
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408 Parameters |
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409 ---------- |
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410 G : :class:`networkx.Graph` |
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411 The connected graph :math:`G`. |
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412 weight : :class:`str`, optional |
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413 Passed through to |
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414 :func:`~networkx.algorithms.shortest_paths.generic.shortest_path_length`. |
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415 attr : :class:`str`, optional |
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416 If given, write the value of the objective function to each node's |
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417 `attr` attribute. Otherwise do not store the value. |
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418 sp : dict of dicts, optional |
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419 All pairs shortest path lengths as a dictionary of dictionaries |
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420 |
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421 Returns |
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422 ------- |
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423 list |
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424 Nodes of `G` that induce the barycenter of `G`. |
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425 |
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426 Raises |
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427 ------ |
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428 NetworkXNoPath |
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429 If `G` is disconnected. `G` may appear disconnected to |
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430 :func:`barycenter` if `sp` is given but is missing shortest path |
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431 lengths for any pairs. |
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432 ValueError |
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433 If `sp` and `weight` are both given. |
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434 |
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435 See Also |
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436 -------- |
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437 center |
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438 periphery |
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439 """ |
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440 if sp is None: |
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441 sp = nx.shortest_path_length(G, weight=weight) |
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442 else: |
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443 sp = sp.items() |
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444 if weight is not None: |
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445 raise ValueError("Cannot use both sp, weight arguments together") |
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446 smallest, barycenter_vertices, n = float("inf"), [], len(G) |
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447 for v, dists in sp: |
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448 if len(dists) < n: |
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449 raise nx.NetworkXNoPath( |
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450 f"Input graph {G} is disconnected, so every induced subgraph " |
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451 "has infinite barycentricity." |
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452 ) |
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453 barycentricity = sum(dists.values()) |
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454 if attr is not None: |
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455 G.nodes[v][attr] = barycentricity |
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456 if barycentricity < smallest: |
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457 smallest = barycentricity |
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458 barycenter_vertices = [v] |
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459 elif barycentricity == smallest: |
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460 barycenter_vertices.append(v) |
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461 return barycenter_vertices |
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462 |
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463 |
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464 def _laplacian_submatrix(node, mat, node_list): |
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465 """Removes row/col from a sparse matrix and returns the submatrix |
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466 """ |
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467 j = node_list.index(node) |
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468 n = list(range(len(node_list))) |
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469 n.pop(j) |
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470 |
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471 if mat.shape[0] != mat.shape[1]: |
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472 raise nx.NetworkXError("Matrix must be square") |
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473 elif len(node_list) != mat.shape[0]: |
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474 msg = "Node list length does not match matrix dimentions" |
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475 raise nx.NetworkXError(msg) |
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476 |
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477 mat = mat.tocsr() |
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478 mat = mat[n, :] |
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479 |
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480 mat = mat.tocsc() |
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481 mat = mat[:, n] |
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482 |
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483 node_list.pop(j) |
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484 |
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485 return mat, node_list |
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486 |
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487 |
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488 def _count_lu_permutations(perm_array): |
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489 """Counts the number of permutations in SuperLU perm_c or perm_r |
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490 """ |
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491 perm_cnt = 0 |
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492 arr = perm_array.tolist() |
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493 for i in range(len(arr)): |
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494 if i != arr[i]: |
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495 perm_cnt += 1 |
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496 n = arr.index(i) |
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497 arr[n] = arr[i] |
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498 arr[i] = i |
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499 |
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500 return perm_cnt |
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501 |
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502 |
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503 @not_implemented_for("directed") |
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504 def resistance_distance(G, nodeA, nodeB, weight=None, invert_weight=True): |
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505 """Returns the resistance distance between node A and node B on graph G. |
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506 |
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507 The resistance distance between two nodes of a graph is akin to treating |
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508 the graph as a grid of resistorses with a resistance equal to the provided |
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509 weight. |
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510 |
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511 If weight is not provided, then a weight of 1 is used for all edges. |
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512 |
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513 Parameters |
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514 ---------- |
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515 G : NetworkX graph |
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516 A graph |
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517 |
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518 nodeA : node |
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519 A node within graph G. |
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520 |
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521 nodeB : node |
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522 A node within graph G, exclusive of Node A. |
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523 |
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524 weight : string or None, optional (default=None) |
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525 The edge data key used to compute the resistance distance. |
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526 If None, then each edge has weight 1. |
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527 |
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528 invert_weight : boolean (default=True) |
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529 Proper calculation of resistance distance requires building the |
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530 Laplacian matrix with the reciprocal of the weight. Not required |
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531 if the weight is already inverted. Weight cannot be zero. |
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532 |
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533 Returns |
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534 ------- |
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535 rd : float |
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536 Value of effective resistance distance |
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537 |
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538 Notes |
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539 ----- |
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540 Overview discussion: |
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541 * https://en.wikipedia.org/wiki/Resistance_distance |
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542 * http://mathworld.wolfram.com/ResistanceDistance.html |
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543 |
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544 Additional details: |
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545 Vaya Sapobi Samui Vos, “Methods for determining the effective resistance,” M.S., |
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546 Mathematisch Instituut, Universiteit Leiden, Leiden, Netherlands, 2016 |
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547 Available: `Link to thesis <https://www.universiteitleiden.nl/binaries/content/assets/science/mi/scripties/master/vos_vaya_master.pdf>`_ |
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548 """ |
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549 import numpy as np |
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550 import scipy.sparse |
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551 |
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552 if not nx.is_connected(G): |
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553 msg = "Graph G must be strongly connected." |
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554 raise nx.NetworkXError(msg) |
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555 elif nodeA not in G: |
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556 msg = "Node A is not in graph G." |
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557 raise nx.NetworkXError(msg) |
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558 elif nodeB not in G: |
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559 msg = "Node B is not in graph G." |
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560 raise nx.NetworkXError(msg) |
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561 elif nodeA == nodeB: |
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562 msg = "Node A and Node B cannot be the same." |
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563 raise nx.NetworkXError(msg) |
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564 |
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565 G = G.copy() |
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566 node_list = list(G) |
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567 |
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568 if invert_weight and weight is not None: |
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569 if G.is_multigraph(): |
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570 for (u, v, k, d) in G.edges(keys=True, data=True): |
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571 d[weight] = 1 / d[weight] |
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572 else: |
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573 for (u, v, d) in G.edges(data=True): |
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574 d[weight] = 1 / d[weight] |
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575 # Replace with collapsing topology or approximated zero? |
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576 |
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577 # Using determinants to compute the effective resistance is more memory |
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578 # efficent than directly calculating the psuedo-inverse |
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579 L = nx.laplacian_matrix(G, node_list, weight=weight) |
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580 |
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581 Lsub_a, node_list_a = _laplacian_submatrix(nodeA, L.copy(), node_list[:]) |
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582 Lsub_ab, node_list_ab = _laplacian_submatrix(nodeB, Lsub_a.copy(), node_list_a[:]) |
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583 |
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584 # Factorize Laplacian submatrixes and extract diagonals |
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585 # Order the diagonals to minimize the likelihood over overflows |
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586 # during computing the determinant |
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587 lu_a = scipy.sparse.linalg.splu(Lsub_a, options=dict(SymmetricMode=True)) |
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588 LdiagA = lu_a.U.diagonal() |
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589 LdiagA_s = np.product(np.sign(LdiagA)) * np.product(lu_a.L.diagonal()) |
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590 LdiagA_s *= (-1) ** _count_lu_permutations(lu_a.perm_r) |
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591 LdiagA_s *= (-1) ** _count_lu_permutations(lu_a.perm_c) |
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592 LdiagA = np.absolute(LdiagA) |
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593 LdiagA = np.sort(LdiagA) |
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594 |
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595 lu_ab = scipy.sparse.linalg.splu(Lsub_ab, options=dict(SymmetricMode=True)) |
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596 LdiagAB = lu_ab.U.diagonal() |
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597 LdiagAB_s = np.product(np.sign(LdiagAB)) * np.product(lu_ab.L.diagonal()) |
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598 LdiagAB_s *= (-1) ** _count_lu_permutations(lu_ab.perm_r) |
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599 LdiagAB_s *= (-1) ** _count_lu_permutations(lu_ab.perm_c) |
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600 LdiagAB = np.absolute(LdiagAB) |
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601 LdiagAB = np.sort(LdiagAB) |
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602 |
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603 # Calculate the ratio of determinant, rd = det(Lsub_ab)/det(Lsub_a) |
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604 Ldet = np.product(np.divide(np.append(LdiagAB, [1]), LdiagA)) |
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605 rd = Ldet * LdiagAB_s / LdiagA_s |
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606 |
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607 return rd |