Mercurial > repos > shellac > sam_consensus_v3
annotate env/lib/python3.9/site-packages/networkx/generators/expanders.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 """Provides explicit constructions of expander graphs. |
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2 |
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3 """ |
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4 import itertools |
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5 import networkx as nx |
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6 |
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7 __all__ = ["margulis_gabber_galil_graph", "chordal_cycle_graph", "paley_graph"] |
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8 |
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9 |
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10 # Other discrete torus expanders can be constructed by using the following edge |
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11 # sets. For more information, see Chapter 4, "Expander Graphs", in |
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12 # "Pseudorandomness", by Salil Vadhan. |
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13 # |
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14 # For a directed expander, add edges from (x, y) to: |
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15 # |
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16 # (x, y), |
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17 # ((x + 1) % n, y), |
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18 # (x, (y + 1) % n), |
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19 # (x, (x + y) % n), |
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20 # (-y % n, x) |
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21 # |
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22 # For an undirected expander, add the reverse edges. |
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23 # |
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24 # Also appearing in the paper of Gabber and Galil: |
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25 # |
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26 # (x, y), |
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27 # (x, (x + y) % n), |
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28 # (x, (x + y + 1) % n), |
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29 # ((x + y) % n, y), |
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30 # ((x + y + 1) % n, y) |
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31 # |
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32 # and: |
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33 # |
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34 # (x, y), |
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35 # ((x + 2*y) % n, y), |
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36 # ((x + (2*y + 1)) % n, y), |
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37 # ((x + (2*y + 2)) % n, y), |
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38 # (x, (y + 2*x) % n), |
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39 # (x, (y + (2*x + 1)) % n), |
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40 # (x, (y + (2*x + 2)) % n), |
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41 # |
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42 def margulis_gabber_galil_graph(n, create_using=None): |
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43 r"""Returns the Margulis-Gabber-Galil undirected MultiGraph on `n^2` nodes. |
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44 |
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45 The undirected MultiGraph is regular with degree `8`. Nodes are integer |
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46 pairs. The second-largest eigenvalue of the adjacency matrix of the graph |
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47 is at most `5 \sqrt{2}`, regardless of `n`. |
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48 |
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49 Parameters |
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50 ---------- |
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51 n : int |
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52 Determines the number of nodes in the graph: `n^2`. |
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53 create_using : NetworkX graph constructor, optional (default MultiGraph) |
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54 Graph type to create. If graph instance, then cleared before populated. |
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55 |
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56 Returns |
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57 ------- |
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58 G : graph |
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59 The constructed undirected multigraph. |
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60 |
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61 Raises |
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62 ------ |
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63 NetworkXError |
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64 If the graph is directed or not a multigraph. |
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65 |
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66 """ |
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67 G = nx.empty_graph(0, create_using, default=nx.MultiGraph) |
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68 if G.is_directed() or not G.is_multigraph(): |
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69 msg = "`create_using` must be an undirected multigraph." |
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70 raise nx.NetworkXError(msg) |
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71 |
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72 for (x, y) in itertools.product(range(n), repeat=2): |
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73 for (u, v) in ( |
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74 ((x + 2 * y) % n, y), |
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75 ((x + (2 * y + 1)) % n, y), |
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76 (x, (y + 2 * x) % n), |
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77 (x, (y + (2 * x + 1)) % n), |
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78 ): |
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79 G.add_edge((x, y), (u, v)) |
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80 G.graph["name"] = f"margulis_gabber_galil_graph({n})" |
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81 return G |
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82 |
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83 |
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84 def chordal_cycle_graph(p, create_using=None): |
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85 """Returns the chordal cycle graph on `p` nodes. |
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86 |
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87 The returned graph is a cycle graph on `p` nodes with chords joining each |
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88 vertex `x` to its inverse modulo `p`. This graph is a (mildly explicit) |
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89 3-regular expander [1]_. |
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90 |
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91 `p` *must* be a prime number. |
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92 |
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93 Parameters |
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94 ---------- |
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95 p : a prime number |
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96 |
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97 The number of vertices in the graph. This also indicates where the |
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98 chordal edges in the cycle will be created. |
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99 |
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100 create_using : NetworkX graph constructor, optional (default=nx.Graph) |
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101 Graph type to create. If graph instance, then cleared before populated. |
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102 |
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103 Returns |
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104 ------- |
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105 G : graph |
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106 The constructed undirected multigraph. |
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107 |
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108 Raises |
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109 ------ |
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110 NetworkXError |
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111 |
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112 If `create_using` indicates directed or not a multigraph. |
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113 |
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114 References |
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115 ---------- |
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116 |
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117 .. [1] Theorem 4.4.2 in A. Lubotzky. "Discrete groups, expanding graphs and |
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118 invariant measures", volume 125 of Progress in Mathematics. |
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119 Birkhäuser Verlag, Basel, 1994. |
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120 |
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121 """ |
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122 G = nx.empty_graph(0, create_using, default=nx.MultiGraph) |
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123 if G.is_directed() or not G.is_multigraph(): |
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124 msg = "`create_using` must be an undirected multigraph." |
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125 raise nx.NetworkXError(msg) |
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126 |
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127 for x in range(p): |
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128 left = (x - 1) % p |
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129 right = (x + 1) % p |
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130 # Here we apply Fermat's Little Theorem to compute the multiplicative |
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131 # inverse of x in Z/pZ. By Fermat's Little Theorem, |
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132 # |
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133 # x^p = x (mod p) |
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134 # |
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135 # Therefore, |
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136 # |
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137 # x * x^(p - 2) = 1 (mod p) |
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138 # |
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139 # The number 0 is a special case: we just let its inverse be itself. |
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140 chord = pow(x, p - 2, p) if x > 0 else 0 |
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141 for y in (left, right, chord): |
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142 G.add_edge(x, y) |
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143 G.graph["name"] = f"chordal_cycle_graph({p})" |
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144 return G |
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145 |
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146 |
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147 def paley_graph(p, create_using=None): |
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148 """Returns the Paley (p-1)/2-regular graph on p nodes. |
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149 |
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150 The returned graph is a graph on Z/pZ with edges between x and y |
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151 if and only if x-y is a nonzero square in Z/pZ. |
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152 |
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153 If p = 1 mod 4, -1 is a square in Z/pZ and therefore x-y is a square if and |
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154 only if y-x is also a square, i.e the edges in the Paley graph are symmetric. |
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155 |
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156 If p = 3 mod 4, -1 is not a square in Z/pZ and therefore either x-y or y-x |
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157 is a square in Z/pZ but not both. |
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158 |
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159 Note that a more general definition of Paley graphs extends this construction |
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160 to graphs over q=p^n vertices, by using the finite field F_q instead of Z/pZ. |
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161 This construction requires to compute squares in general finite fields and is |
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162 not what is implemented here (i.e paley_graph(25) does not return the true |
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163 Paley graph associated with 5^2). |
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164 |
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165 Parameters |
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166 ---------- |
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167 p : int, an odd prime number. |
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168 |
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169 create_using : NetworkX graph constructor, optional (default=nx.Graph) |
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170 Graph type to create. If graph instance, then cleared before populated. |
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171 |
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172 Returns |
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173 ------- |
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174 G : graph |
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175 The constructed directed graph. |
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176 |
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177 Raises |
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178 ------ |
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179 NetworkXError |
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180 If the graph is a multigraph. |
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181 |
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182 References |
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183 ---------- |
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184 Chapter 13 in B. Bollobas, Random Graphs. Second edition. |
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185 Cambridge Studies in Advanced Mathematics, 73. |
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186 Cambridge University Press, Cambridge (2001). |
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187 """ |
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188 G = nx.empty_graph(0, create_using, default=nx.DiGraph) |
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189 if G.is_multigraph(): |
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190 msg = "`create_using` cannot be a multigraph." |
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191 raise nx.NetworkXError(msg) |
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192 |
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193 # Compute the squares in Z/pZ. |
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194 # Make it a set to uniquify (there are exactly (p-1)/2 squares in Z/pZ |
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195 # when is prime). |
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196 square_set = {(x ** 2) % p for x in range(1, p) if (x ** 2) % p != 0} |
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197 |
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198 for x in range(p): |
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199 for x2 in square_set: |
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200 G.add_edge(x, (x + x2) % p) |
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201 G.graph["name"] = f"paley({p})" |
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202 return G |