Mercurial > repos > shellac > sam_consensus_v3
annotate env/lib/python3.9/site-packages/networkx/algorithms/distance_regular.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 """ |
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2 ======================= |
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3 Distance-regular graphs |
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4 ======================= |
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5 """ |
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6 |
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7 import networkx as nx |
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8 from networkx.utils import not_implemented_for |
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9 from .distance_measures import diameter |
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10 |
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11 __all__ = [ |
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12 "is_distance_regular", |
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13 "is_strongly_regular", |
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14 "intersection_array", |
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15 "global_parameters", |
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16 ] |
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17 |
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18 |
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19 def is_distance_regular(G): |
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20 """Returns True if the graph is distance regular, False otherwise. |
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21 |
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22 A connected graph G is distance-regular if for any nodes x,y |
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23 and any integers i,j=0,1,...,d (where d is the graph |
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24 diameter), the number of vertices at distance i from x and |
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25 distance j from y depends only on i,j and the graph distance |
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26 between x and y, independently of the choice of x and y. |
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27 |
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28 Parameters |
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29 ---------- |
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30 G: Networkx graph (undirected) |
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31 |
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32 Returns |
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33 ------- |
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34 bool |
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35 True if the graph is Distance Regular, False otherwise |
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36 |
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37 Examples |
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38 -------- |
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39 >>> G = nx.hypercube_graph(6) |
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40 >>> nx.is_distance_regular(G) |
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41 True |
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42 |
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43 See Also |
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44 -------- |
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45 intersection_array, global_parameters |
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46 |
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47 Notes |
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48 ----- |
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49 For undirected and simple graphs only |
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50 |
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51 References |
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52 ---------- |
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53 .. [1] Brouwer, A. E.; Cohen, A. M.; and Neumaier, A. |
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54 Distance-Regular Graphs. New York: Springer-Verlag, 1989. |
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55 .. [2] Weisstein, Eric W. "Distance-Regular Graph." |
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56 http://mathworld.wolfram.com/Distance-RegularGraph.html |
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57 |
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58 """ |
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59 try: |
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60 intersection_array(G) |
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61 return True |
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62 except nx.NetworkXError: |
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63 return False |
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64 |
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65 |
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66 def global_parameters(b, c): |
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67 """Returns global parameters for a given intersection array. |
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68 |
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69 Given a distance-regular graph G with integers b_i, c_i,i = 0,....,d |
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70 such that for any 2 vertices x,y in G at a distance i=d(x,y), there |
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71 are exactly c_i neighbors of y at a distance of i-1 from x and b_i |
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72 neighbors of y at a distance of i+1 from x. |
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73 |
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74 Thus, a distance regular graph has the global parameters, |
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75 [[c_0,a_0,b_0],[c_1,a_1,b_1],......,[c_d,a_d,b_d]] for the |
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76 intersection array [b_0,b_1,.....b_{d-1};c_1,c_2,.....c_d] |
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77 where a_i+b_i+c_i=k , k= degree of every vertex. |
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78 |
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79 Parameters |
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80 ---------- |
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81 b : list |
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82 |
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83 c : list |
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84 |
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85 Returns |
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86 ------- |
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87 iterable |
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88 An iterable over three tuples. |
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89 |
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90 Examples |
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91 -------- |
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92 >>> G = nx.dodecahedral_graph() |
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93 >>> b, c = nx.intersection_array(G) |
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94 >>> list(nx.global_parameters(b, c)) |
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95 [(0, 0, 3), (1, 0, 2), (1, 1, 1), (1, 1, 1), (2, 0, 1), (3, 0, 0)] |
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96 |
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97 References |
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98 ---------- |
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99 .. [1] Weisstein, Eric W. "Global Parameters." |
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100 From MathWorld--A Wolfram Web Resource. |
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101 http://mathworld.wolfram.com/GlobalParameters.html |
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102 |
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103 See Also |
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104 -------- |
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105 intersection_array |
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106 """ |
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107 return ((y, b[0] - x - y, x) for x, y in zip(b + [0], [0] + c)) |
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108 |
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109 |
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110 @not_implemented_for("directed", "multigraph") |
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111 def intersection_array(G): |
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112 """Returns the intersection array of a distance-regular graph. |
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113 |
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114 Given a distance-regular graph G with integers b_i, c_i,i = 0,....,d |
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115 such that for any 2 vertices x,y in G at a distance i=d(x,y), there |
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116 are exactly c_i neighbors of y at a distance of i-1 from x and b_i |
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117 neighbors of y at a distance of i+1 from x. |
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118 |
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119 A distance regular graph's intersection array is given by, |
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120 [b_0,b_1,.....b_{d-1};c_1,c_2,.....c_d] |
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121 |
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122 Parameters |
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123 ---------- |
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124 G: Networkx graph (undirected) |
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125 |
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126 Returns |
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127 ------- |
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128 b,c: tuple of lists |
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129 |
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130 Examples |
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131 -------- |
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132 >>> G = nx.icosahedral_graph() |
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133 >>> nx.intersection_array(G) |
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134 ([5, 2, 1], [1, 2, 5]) |
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135 |
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136 References |
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137 ---------- |
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138 .. [1] Weisstein, Eric W. "Intersection Array." |
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139 From MathWorld--A Wolfram Web Resource. |
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140 http://mathworld.wolfram.com/IntersectionArray.html |
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141 |
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142 See Also |
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143 -------- |
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144 global_parameters |
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145 """ |
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146 # test for regular graph (all degrees must be equal) |
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147 degree = iter(G.degree()) |
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148 (_, k) = next(degree) |
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149 for _, knext in degree: |
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150 if knext != k: |
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151 raise nx.NetworkXError("Graph is not distance regular.") |
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152 k = knext |
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153 path_length = dict(nx.all_pairs_shortest_path_length(G)) |
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154 diameter = max([max(path_length[n].values()) for n in path_length]) |
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155 bint = {} # 'b' intersection array |
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156 cint = {} # 'c' intersection array |
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157 for u in G: |
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158 for v in G: |
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159 try: |
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160 i = path_length[u][v] |
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161 except KeyError as e: # graph must be connected |
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162 raise nx.NetworkXError("Graph is not distance regular.") from e |
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163 # number of neighbors of v at a distance of i-1 from u |
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164 c = len([n for n in G[v] if path_length[n][u] == i - 1]) |
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165 # number of neighbors of v at a distance of i+1 from u |
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166 b = len([n for n in G[v] if path_length[n][u] == i + 1]) |
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167 # b,c are independent of u and v |
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168 if cint.get(i, c) != c or bint.get(i, b) != b: |
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169 raise nx.NetworkXError("Graph is not distance regular") |
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170 bint[i] = b |
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171 cint[i] = c |
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172 return ( |
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173 [bint.get(j, 0) for j in range(diameter)], |
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174 [cint.get(j + 1, 0) for j in range(diameter)], |
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175 ) |
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176 |
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177 |
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178 # TODO There is a definition for directed strongly regular graphs. |
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179 @not_implemented_for("directed", "multigraph") |
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180 def is_strongly_regular(G): |
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181 """Returns True if and only if the given graph is strongly |
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182 regular. |
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183 |
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184 An undirected graph is *strongly regular* if |
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185 |
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186 * it is regular, |
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187 * each pair of adjacent vertices has the same number of neighbors in |
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188 common, |
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189 * each pair of nonadjacent vertices has the same number of neighbors |
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190 in common. |
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191 |
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192 Each strongly regular graph is a distance-regular graph. |
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193 Conversely, if a distance-regular graph has diameter two, then it is |
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194 a strongly regular graph. For more information on distance-regular |
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195 graphs, see :func:`is_distance_regular`. |
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196 |
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197 Parameters |
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198 ---------- |
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199 G : NetworkX graph |
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200 An undirected graph. |
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201 |
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202 Returns |
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203 ------- |
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204 bool |
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205 Whether `G` is strongly regular. |
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206 |
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207 Examples |
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208 -------- |
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209 |
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210 The cycle graph on five vertices is strongly regular. It is |
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211 two-regular, each pair of adjacent vertices has no shared neighbors, |
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212 and each pair of nonadjacent vertices has one shared neighbor:: |
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213 |
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214 >>> G = nx.cycle_graph(5) |
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215 >>> nx.is_strongly_regular(G) |
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216 True |
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217 |
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218 """ |
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219 # Here is an alternate implementation based directly on the |
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220 # definition of strongly regular graphs: |
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221 # |
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222 # return (all_equal(G.degree().values()) |
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223 # and all_equal(len(common_neighbors(G, u, v)) |
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224 # for u, v in G.edges()) |
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225 # and all_equal(len(common_neighbors(G, u, v)) |
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226 # for u, v in non_edges(G))) |
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227 # |
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228 # We instead use the fact that a distance-regular graph of diameter |
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"planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
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229 # two is strongly regular. |
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"planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
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230 return is_distance_regular(G) and diameter(G) == 2 |