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annotate planemo/lib/python3.7/site-packages/networkx/generators/expanders.py @ 1:56ad4e20f292 draft

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"planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
author guerler
date Fri, 31 Jul 2020 00:32:28 -0400
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1 # -*- coding: utf-8 -*-
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2 # Copyright 2014 "cheebee7i".
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3 # Copyright 2014 "alexbrc".
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4 # Copyright 2014 Jeffrey Finkelstein <jeffrey.finkelstein@gmail.com>.
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5 """Provides explicit constructions of expander graphs.
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6
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7 """
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8 import itertools
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9 import networkx as nx
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10
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11 __all__ = ['margulis_gabber_galil_graph', 'chordal_cycle_graph']
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12
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13
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14 # Other discrete torus expanders can be constructed by using the following edge
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15 # sets. For more information, see Chapter 4, "Expander Graphs", in
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16 # "Pseudorandomness", by Salil Vadhan.
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17 #
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18 # For a directed expander, add edges from (x, y) to:
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19 #
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20 # (x, y),
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21 # ((x + 1) % n, y),
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22 # (x, (y + 1) % n),
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23 # (x, (x + y) % n),
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24 # (-y % n, x)
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25 #
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26 # For an undirected expander, add the reverse edges.
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27 #
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28 # Also appearing in the paper of Gabber and Galil:
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29 #
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30 # (x, y),
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31 # (x, (x + y) % n),
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32 # (x, (x + y + 1) % n),
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33 # ((x + y) % n, y),
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34 # ((x + y + 1) % n, y)
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35 #
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36 # and:
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37 #
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38 # (x, y),
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39 # ((x + 2*y) % n, y),
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40 # ((x + (2*y + 1)) % n, y),
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41 # ((x + (2*y + 2)) % n, y),
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42 # (x, (y + 2*x) % n),
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43 # (x, (y + (2*x + 1)) % n),
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44 # (x, (y + (2*x + 2)) % n),
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45 #
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46 def margulis_gabber_galil_graph(n, create_using=None):
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47 r"""Returns the Margulis-Gabber-Galil undirected MultiGraph on `n^2` nodes.
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48
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49 The undirected MultiGraph is regular with degree `8`. Nodes are integer
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50 pairs. The second-largest eigenvalue of the adjacency matrix of the graph
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51 is at most `5 \sqrt{2}`, regardless of `n`.
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52
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53 Parameters
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54 ----------
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55 n : int
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56 Determines the number of nodes in the graph: `n^2`.
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57 create_using : NetworkX graph constructor, optional (default MultiGraph)
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58 Graph type to create. If graph instance, then cleared before populated.
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59
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60 Returns
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61 -------
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62 G : graph
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63 The constructed undirected multigraph.
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64
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65 Raises
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66 ------
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67 NetworkXError
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68 If the graph is directed or not a multigraph.
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69
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70 """
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71 G = nx.empty_graph(0, create_using, default=nx.MultiGraph)
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72 if G.is_directed() or not G.is_multigraph():
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73 msg = "`create_using` must be an undirected multigraph."
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74 raise nx.NetworkXError(msg)
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75
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76 for (x, y) in itertools.product(range(n), repeat=2):
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77 for (u, v) in (((x + 2 * y) % n, y), ((x + (2 * y + 1)) % n, y),
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78 (x, (y + 2 * x) % n), (x, (y + (2 * x + 1)) % n)):
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79 G.add_edge((x, y), (u, v))
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80 G.graph['name'] = "margulis_gabber_galil_graph({0})".format(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'] = "chordal_cycle_graph({0})".format(p)
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144 return G