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annotate planemo/lib/python3.7/site-packages/networkx/algorithms/threshold.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 # Copyright (C) 2004-2019 by
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2 # Aric Hagberg <hagberg@lanl.gov>
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3 # Dan Schult <dschult@colgate.edu>
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4 # Pieter Swart <swart@lanl.gov>
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5 # All rights reserved.
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6 # BSD license.
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7 #
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8 # Authors: Aric Hagberg (hagberg@lanl.gov)
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9 # Pieter Swart (swart@lanl.gov)
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10 # Dan Schult (dschult@colgate.edu)
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11 """
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12 Threshold Graphs - Creation, manipulation and identification.
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13 """
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14 from math import sqrt
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15 import networkx as nx
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16 from networkx.utils import py_random_state
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17
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18 __all__ = ['is_threshold_graph', 'find_threshold_graph']
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19
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20
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21 def is_threshold_graph(G):
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22 """
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23 Returns True if G is a threshold graph.
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24 """
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25 return is_threshold_sequence(list(d for n, d in G.degree()))
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26
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27
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28 def is_threshold_sequence(degree_sequence):
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29 """
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30 Returns True if the sequence is a threshold degree seqeunce.
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31
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32 Uses the property that a threshold graph must be constructed by
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33 adding either dominating or isolated nodes. Thus, it can be
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34 deconstructed iteratively by removing a node of degree zero or a
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35 node that connects to the remaining nodes. If this deconstruction
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36 failes then the sequence is not a threshold sequence.
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37 """
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38 ds = degree_sequence[:] # get a copy so we don't destroy original
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39 ds.sort()
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40 while ds:
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41 if ds[0] == 0: # if isolated node
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42 ds.pop(0) # remove it
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43 continue
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44 if ds[-1] != len(ds) - 1: # is the largest degree node dominating?
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45 return False # no, not a threshold degree sequence
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46 ds.pop() # yes, largest is the dominating node
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47 ds = [d - 1 for d in ds] # remove it and decrement all degrees
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48 return True
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49
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50
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51 def creation_sequence(degree_sequence, with_labels=False, compact=False):
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52 """
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53 Determines the creation sequence for the given threshold degree sequence.
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54
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55 The creation sequence is a list of single characters 'd'
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56 or 'i': 'd' for dominating or 'i' for isolated vertices.
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57 Dominating vertices are connected to all vertices present when it
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58 is added. The first node added is by convention 'd'.
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59 This list can be converted to a string if desired using "".join(cs)
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60
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61 If with_labels==True:
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62 Returns a list of 2-tuples containing the vertex number
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63 and a character 'd' or 'i' which describes the type of vertex.
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64
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65 If compact==True:
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66 Returns the creation sequence in a compact form that is the number
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67 of 'i's and 'd's alternating.
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68 Examples:
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69 [1,2,2,3] represents d,i,i,d,d,i,i,i
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70 [3,1,2] represents d,d,d,i,d,d
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71
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72 Notice that the first number is the first vertex to be used for
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73 construction and so is always 'd'.
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74
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75 with_labels and compact cannot both be True.
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76
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77 Returns None if the sequence is not a threshold sequence
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78 """
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79 if with_labels and compact:
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80 raise ValueError("compact sequences cannot be labeled")
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81
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82 # make an indexed copy
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83 if isinstance(degree_sequence, dict): # labeled degree seqeunce
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84 ds = [[degree, label] for (label, degree) in degree_sequence.items()]
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85 else:
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86 ds = [[d, i] for i, d in enumerate(degree_sequence)]
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87 ds.sort()
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88 cs = [] # creation sequence
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89 while ds:
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90 if ds[0][0] == 0: # isolated node
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91 (d, v) = ds.pop(0)
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92 if len(ds) > 0: # make sure we start with a d
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93 cs.insert(0, (v, 'i'))
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94 else:
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95 cs.insert(0, (v, 'd'))
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96 continue
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97 if ds[-1][0] != len(ds) - 1: # Not dominating node
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98 return None # not a threshold degree sequence
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99 (d, v) = ds.pop()
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100 cs.insert(0, (v, 'd'))
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101 ds = [[d[0] - 1, d[1]] for d in ds] # decrement due to removing node
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102
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103 if with_labels:
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104 return cs
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105 if compact:
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106 return make_compact(cs)
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107 return [v[1] for v in cs] # not labeled
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108
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109
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110 def make_compact(creation_sequence):
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111 """
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112 Returns the creation sequence in a compact form
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113 that is the number of 'i's and 'd's alternating.
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114
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115 Examples
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116 --------
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117 >>> from networkx.algorithms.threshold import make_compact
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118 >>> make_compact(['d', 'i', 'i', 'd', 'd', 'i', 'i', 'i'])
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119 [1, 2, 2, 3]
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120 >>> make_compact(['d', 'd', 'd', 'i', 'd', 'd'])
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121 [3, 1, 2]
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122
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123 Notice that the first number is the first vertex
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124 to be used for construction and so is always 'd'.
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125
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126 Labeled creation sequences lose their labels in the
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127 compact representation.
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128
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129 >>> make_compact([3, 1, 2])
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130 [3, 1, 2]
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131 """
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132 first = creation_sequence[0]
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133 if isinstance(first, str): # creation sequence
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134 cs = creation_sequence[:]
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135 elif isinstance(first, tuple): # labeled creation sequence
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136 cs = [s[1] for s in creation_sequence]
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137 elif isinstance(first, int): # compact creation sequence
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138 return creation_sequence
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139 else:
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140 raise TypeError("Not a valid creation sequence type")
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141
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142 ccs = []
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143 count = 1 # count the run lengths of d's or i's.
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144 for i in range(1, len(cs)):
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145 if cs[i] == cs[i - 1]:
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146 count += 1
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147 else:
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148 ccs.append(count)
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149 count = 1
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150 ccs.append(count) # don't forget the last one
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151 return ccs
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152
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153
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154 def uncompact(creation_sequence):
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155 """
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156 Converts a compact creation sequence for a threshold
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157 graph to a standard creation sequence (unlabeled).
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158 If the creation_sequence is already standard, return it.
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159 See creation_sequence.
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160 """
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161 first = creation_sequence[0]
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162 if isinstance(first, str): # creation sequence
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163 return creation_sequence
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164 elif isinstance(first, tuple): # labeled creation sequence
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165 return creation_sequence
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166 elif isinstance(first, int): # compact creation sequence
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167 ccscopy = creation_sequence[:]
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168 else:
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169 raise TypeError("Not a valid creation sequence type")
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170 cs = []
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171 while ccscopy:
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172 cs.extend(ccscopy.pop(0) * ['d'])
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173 if ccscopy:
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174 cs.extend(ccscopy.pop(0) * ['i'])
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175 return cs
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176
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177
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178 def creation_sequence_to_weights(creation_sequence):
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179 """
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180 Returns a list of node weights which create the threshold
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181 graph designated by the creation sequence. The weights
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182 are scaled so that the threshold is 1.0. The order of the
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183 nodes is the same as that in the creation sequence.
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184 """
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185 # Turn input sequence into a labeled creation sequence
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186 first = creation_sequence[0]
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187 if isinstance(first, str): # creation sequence
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188 if isinstance(creation_sequence, list):
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189 wseq = creation_sequence[:]
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190 else:
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191 wseq = list(creation_sequence) # string like 'ddidid'
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192 elif isinstance(first, tuple): # labeled creation sequence
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193 wseq = [v[1] for v in creation_sequence]
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194 elif isinstance(first, int): # compact creation sequence
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195 wseq = uncompact(creation_sequence)
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196 else:
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197 raise TypeError("Not a valid creation sequence type")
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198 # pass through twice--first backwards
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199 wseq.reverse()
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200 w = 0
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201 prev = 'i'
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202 for j, s in enumerate(wseq):
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203 if s == 'i':
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204 wseq[j] = w
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205 prev = s
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206 elif prev == 'i':
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207 prev = s
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208 w += 1
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209 wseq.reverse() # now pass through forwards
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210 for j, s in enumerate(wseq):
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211 if s == 'd':
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212 wseq[j] = w
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213 prev = s
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214 elif prev == 'd':
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215 prev = s
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216 w += 1
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217 # Now scale weights
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218 if prev == 'd':
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219 w += 1
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220 wscale = 1. / float(w)
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221 return [ww * wscale for ww in wseq]
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222 # return wseq
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223
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224
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225 def weights_to_creation_sequence(weights, threshold=1, with_labels=False, compact=False):
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226 """
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227 Returns a creation sequence for a threshold graph
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228 determined by the weights and threshold given as input.
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229 If the sum of two node weights is greater than the
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230 threshold value, an edge is created between these nodes.
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231
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232 The creation sequence is a list of single characters 'd'
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233 or 'i': 'd' for dominating or 'i' for isolated vertices.
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234 Dominating vertices are connected to all vertices present
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235 when it is added. The first node added is by convention 'd'.
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236
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237 If with_labels==True:
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238 Returns a list of 2-tuples containing the vertex number
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239 and a character 'd' or 'i' which describes the type of vertex.
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240
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241 If compact==True:
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242 Returns the creation sequence in a compact form that is the number
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243 of 'i's and 'd's alternating.
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244 Examples:
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245 [1,2,2,3] represents d,i,i,d,d,i,i,i
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246 [3,1,2] represents d,d,d,i,d,d
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247
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248 Notice that the first number is the first vertex to be used for
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249 construction and so is always 'd'.
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250
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251 with_labels and compact cannot both be True.
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252 """
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253 if with_labels and compact:
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254 raise ValueError("compact sequences cannot be labeled")
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255
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256 # make an indexed copy
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257 if isinstance(weights, dict): # labeled weights
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258 wseq = [[w, label] for (label, w) in weights.items()]
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259 else:
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260 wseq = [[w, i] for i, w in enumerate(weights)]
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261 wseq.sort()
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262 cs = [] # creation sequence
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263 cutoff = threshold - wseq[-1][0]
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264 while wseq:
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265 if wseq[0][0] < cutoff: # isolated node
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266 (w, label) = wseq.pop(0)
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267 cs.append((label, 'i'))
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268 else:
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269 (w, label) = wseq.pop()
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270 cs.append((label, 'd'))
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271 cutoff = threshold - wseq[-1][0]
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272 if len(wseq) == 1: # make sure we start with a d
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273 (w, label) = wseq.pop()
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274 cs.append((label, 'd'))
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275 # put in correct order
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276 cs.reverse()
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277
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278 if with_labels:
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279 return cs
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280 if compact:
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281 return make_compact(cs)
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282 return [v[1] for v in cs] # not labeled
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283
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284
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285 # Manipulating NetworkX.Graphs in context of threshold graphs
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286 def threshold_graph(creation_sequence, create_using=None):
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287 """
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288 Create a threshold graph from the creation sequence or compact
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289 creation_sequence.
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290
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291 The input sequence can be a
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292
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293 creation sequence (e.g. ['d','i','d','d','d','i'])
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294 labeled creation sequence (e.g. [(0,'d'),(2,'d'),(1,'i')])
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295 compact creation sequence (e.g. [2,1,1,2,0])
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296
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297 Use cs=creation_sequence(degree_sequence,labeled=True)
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298 to convert a degree sequence to a creation sequence.
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299
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300 Returns None if the sequence is not valid
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301 """
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302 # Turn input sequence into a labeled creation sequence
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303 first = creation_sequence[0]
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304 if isinstance(first, str): # creation sequence
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305 ci = list(enumerate(creation_sequence))
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306 elif isinstance(first, tuple): # labeled creation sequence
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307 ci = creation_sequence[:]
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308 elif isinstance(first, int): # compact creation sequence
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309 cs = uncompact(creation_sequence)
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310 ci = list(enumerate(cs))
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311 else:
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312 print("not a valid creation sequence type")
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313 return None
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314
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315 G = nx.empty_graph(0, create_using)
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316 if G.is_directed():
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317 raise nx.NetworkXError("Directed Graph not supported")
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318
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319 G.name = "Threshold Graph"
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320
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321 # add nodes and edges
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322 # if type is 'i' just add nodea
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323 # if type is a d connect to everything previous
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324 while ci:
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325 (v, node_type) = ci.pop(0)
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326 if node_type == 'd': # dominating type, connect to all existing nodes
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327 # We use `for u in list(G):` instead of
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328 # `for u in G:` because we edit the graph `G` in
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329 # the loop. Hence using an iterator will result in
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330 # `RuntimeError: dictionary changed size during iteration`
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331 for u in list(G):
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332 G.add_edge(v, u)
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333 G.add_node(v)
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334 return G
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335
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336
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337 def find_alternating_4_cycle(G):
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338 """
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339 Returns False if there aren't any alternating 4 cycles.
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340 Otherwise returns the cycle as [a,b,c,d] where (a,b)
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341 and (c,d) are edges and (a,c) and (b,d) are not.
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342 """
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343 for (u, v) in G.edges():
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344 for w in G.nodes():
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345 if not G.has_edge(u, w) and u != w:
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346 for x in G.neighbors(w):
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347 if not G.has_edge(v, x) and v != x:
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348 return [u, v, w, x]
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349 return False
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350
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351
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352 def find_threshold_graph(G, create_using=None):
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353 """
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354 Return a threshold subgraph that is close to largest in G.
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355 The threshold graph will contain the largest degree node in G.
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356
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357 """
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358 return threshold_graph(find_creation_sequence(G), create_using)
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359
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360
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361 def find_creation_sequence(G):
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362 """
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363 Find a threshold subgraph that is close to largest in G.
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364 Returns the labeled creation sequence of that threshold graph.
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365 """
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366 cs = []
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367 # get a local pointer to the working part of the graph
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368 H = G
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369 while H.order() > 0:
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370 # get new degree sequence on subgraph
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371 dsdict = dict(H.degree())
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372 ds = [(d, v) for v, d in dsdict.items()]
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373 ds.sort()
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374 # Update threshold graph nodes
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375 if ds[-1][0] == 0: # all are isolated
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376 cs.extend(zip(dsdict, ['i'] * (len(ds) - 1) + ['d']))
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377 break # Done!
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378 # pull off isolated nodes
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379 while ds[0][0] == 0:
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380 (d, iso) = ds.pop(0)
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381 cs.append((iso, 'i'))
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382 # find new biggest node
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383 (d, bigv) = ds.pop()
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384 # add edges of star to t_g
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385 cs.append((bigv, 'd'))
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386 # form subgraph of neighbors of big node
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guerler
parents:
diff changeset
387 H = H.subgraph(H.neighbors(bigv))
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guerler
parents:
diff changeset
388 cs.reverse()
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guerler
parents:
diff changeset
389 return cs
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
390
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guerler
parents:
diff changeset
391
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
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parents:
diff changeset
392 # Properties of Threshold Graphs
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parents:
diff changeset
393 def triangles(creation_sequence):
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guerler
parents:
diff changeset
394 """
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
395 Compute number of triangles in the threshold graph with the
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
396 given creation sequence.
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
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parents:
diff changeset
397 """
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
398 # shortcut algorithm that doesn't require computing number
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
399 # of triangles at each node.
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guerler
parents:
diff changeset
400 cs = creation_sequence # alias
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
401 dr = cs.count("d") # number of d's in sequence
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
402 ntri = dr * (dr - 1) * (dr - 2) / 6 # number of triangles in clique of nd d's
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
403 # now add dr choose 2 triangles for every 'i' in sequence where
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
404 # dr is the number of d's to the right of the current i
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guerler
parents:
diff changeset
405 for i, typ in enumerate(cs):
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
406 if typ == "i":
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
407 ntri += dr * (dr - 1) / 2
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
408 else:
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parents:
diff changeset
409 dr -= 1
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
410 return ntri
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guerler
parents:
diff changeset
411
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guerler
parents:
diff changeset
412
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
413 def triangle_sequence(creation_sequence):
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guerler
parents:
diff changeset
414 """
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
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parents:
diff changeset
415 Return triangle sequence for the given threshold graph creation sequence.
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
416
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
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diff changeset
417 """
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guerler
parents:
diff changeset
418 cs = creation_sequence
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
419 seq = []
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
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diff changeset
420 dr = cs.count("d") # number of d's to the right of the current pos
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
421 dcur = (dr - 1) * (dr - 2) // 2 # number of triangles through a node of clique dr
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
422 irun = 0 # number of i's in the last run
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
423 drun = 0 # number of d's in the last run
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parents:
diff changeset
424 for i, sym in enumerate(cs):
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
425 if sym == "d":
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
426 drun += 1
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
427 tri = dcur + (dr - 1) * irun # new triangles at this d
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
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428 else: # cs[i]="i":
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
429 if prevsym == "d": # new string of i's
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
430 dcur += (dr - 1) * irun # accumulate shared shortest paths
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
431 irun = 0 # reset i run counter
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
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parents:
diff changeset
432 dr -= drun # reduce number of d's to right
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
433 drun = 0 # reset d run counter
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
434 irun += 1
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
435 tri = dr * (dr - 1) // 2 # new triangles at this i
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
436 seq.append(tri)
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
437 prevsym = sym
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
438 return seq
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
439
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parents:
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440
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parents:
diff changeset
441 def cluster_sequence(creation_sequence):
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diff changeset
442 """
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
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parents:
diff changeset
443 Return cluster sequence for the given threshold graph creation sequence.
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
444 """
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
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diff changeset
445 triseq = triangle_sequence(creation_sequence)
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
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diff changeset
446 degseq = degree_sequence(creation_sequence)
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
447 cseq = []
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
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448 for i, deg in enumerate(degseq):
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
449 tri = triseq[i]
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
450 if deg <= 1: # isolated vertex or single pair gets cc 0
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guerler
parents:
diff changeset
451 cseq.append(0)
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
452 continue
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
453 max_size = (deg * (deg - 1)) // 2
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
454 cseq.append(float(tri) / float(max_size))
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
455 return cseq
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
456
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
457
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
458 def degree_sequence(creation_sequence):
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
459 """
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
460 Return degree sequence for the threshold graph with the given
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
461 creation sequence
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
462 """
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
463 cs = creation_sequence # alias
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
464 seq = []
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
465 rd = cs.count("d") # number of d to the right
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
466 for i, sym in enumerate(cs):
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
467 if sym == "d":
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
468 rd -= 1
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
469 seq.append(rd + i)
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
470 else:
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
471 seq.append(rd)
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
472 return seq
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
473
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
474
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
475 def density(creation_sequence):
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
476 """
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
477 Return the density of the graph with this creation_sequence.
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
478 The density is the fraction of possible edges present.
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
479 """
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
480 N = len(creation_sequence)
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
481 two_size = sum(degree_sequence(creation_sequence))
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
482 two_possible = N * (N - 1)
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
483 den = two_size / float(two_possible)
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
484 return den
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
485
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
486
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
487 def degree_correlation(creation_sequence):
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
488 """
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
489 Return the degree-degree correlation over all edges.
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
490 """
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
491 cs = creation_sequence
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
492 s1 = 0 # deg_i*deg_j
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
493 s2 = 0 # deg_i^2+deg_j^2
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
494 s3 = 0 # deg_i+deg_j
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
495 m = 0 # number of edges
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
496 rd = cs.count("d") # number of d nodes to the right
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
497 rdi = [i for i, sym in enumerate(cs) if sym == "d"] # index of "d"s
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
498 ds = degree_sequence(cs)
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
499 for i, sym in enumerate(cs):
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
500 if sym == "d":
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
501 if i != rdi[0]:
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
502 print("Logic error in degree_correlation", i, rdi)
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
503 raise ValueError
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
504 rdi.pop(0)
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
505 degi = ds[i]
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
506 for dj in rdi:
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
507 degj = ds[dj]
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
508 s1 += degj * degi
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
509 s2 += degi**2 + degj**2
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
510 s3 += degi + degj
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
511 m += 1
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
512 denom = (2 * m * s2 - s3 * s3)
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
513 numer = (4 * m * s1 - s3 * s3)
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
514 if denom == 0:
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
515 if numer == 0:
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
516 return 1
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
517 raise ValueError("Zero Denominator but Numerator is %s" % numer)
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guerler
parents:
diff changeset
518 return numer / float(denom)
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guerler
parents:
diff changeset
519
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
520
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
521 def shortest_path(creation_sequence, u, v):
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
522 """
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
523 Find the shortest path between u and v in a
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
524 threshold graph G with the given creation_sequence.
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guerler
parents:
diff changeset
525
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
526 For an unlabeled creation_sequence, the vertices
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
527 u and v must be integers in (0,len(sequence)) referring
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
528 to the position of the desired vertices in the sequence.
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guerler
parents:
diff changeset
529
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
530 For a labeled creation_sequence, u and v are labels of veritices.
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
531
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guerler
parents:
diff changeset
532 Use cs=creation_sequence(degree_sequence,with_labels=True)
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
533 to convert a degree sequence to a creation sequence.
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guerler
parents:
diff changeset
534
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
535 Returns a list of vertices from u to v.
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
536 Example: if they are neighbors, it returns [u,v]
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
537 """
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
538 # Turn input sequence into a labeled creation sequence
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
539 first = creation_sequence[0]
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
540 if isinstance(first, str): # creation sequence
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
541 cs = [(i, creation_sequence[i]) for i in range(len(creation_sequence))]
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
542 elif isinstance(first, tuple): # labeled creation sequence
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guerler
parents:
diff changeset
543 cs = creation_sequence[:]
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
544 elif isinstance(first, int): # compact creation sequence
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
545 ci = uncompact(creation_sequence)
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
546 cs = [(i, ci[i]) for i in range(len(ci))]
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guerler
parents:
diff changeset
547 else:
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
548 raise TypeError("Not a valid creation sequence type")
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guerler
parents:
diff changeset
549
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guerler
parents:
diff changeset
550 verts = [s[0] for s in cs]
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guerler
parents:
diff changeset
551 if v not in verts:
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guerler
parents:
diff changeset
552 raise ValueError("Vertex %s not in graph from creation_sequence" % v)
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guerler
parents:
diff changeset
553 if u not in verts:
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
554 raise ValueError("Vertex %s not in graph from creation_sequence" % u)
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
555 # Done checking
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
556 if u == v:
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
557 return [u]
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
558
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guerler
parents:
diff changeset
559 uindex = verts.index(u)
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
560 vindex = verts.index(v)
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
561 bigind = max(uindex, vindex)
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
562 if cs[bigind][1] == 'd':
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
563 return [u, v]
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
564 # must be that cs[bigind][1]=='i'
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guerler
parents:
diff changeset
565 cs = cs[bigind:]
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
566 while cs:
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
567 vert = cs.pop()
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
568 if vert[1] == 'd':
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
569 return [u, vert[0], v]
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
570 # All after u are type 'i' so no connection
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guerler
parents:
diff changeset
571 return -1
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
572
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guerler
parents:
diff changeset
573
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parents:
diff changeset
574 def shortest_path_length(creation_sequence, i):
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
575 """
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
576 Return the shortest path length from indicated node to
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
577 every other node for the threshold graph with the given
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
578 creation sequence.
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guerler
parents:
diff changeset
579 Node is indicated by index i in creation_sequence unless
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
580 creation_sequence is labeled in which case, i is taken to
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
581 be the label of the node.
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guerler
parents:
diff changeset
582
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guerler
parents:
diff changeset
583 Paths lengths in threshold graphs are at most 2.
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
584 Length to unreachable nodes is set to -1.
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
585 """
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
586 # Turn input sequence into a labeled creation sequence
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guerler
parents:
diff changeset
587 first = creation_sequence[0]
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guerler
parents:
diff changeset
588 if isinstance(first, str): # creation sequence
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guerler
parents:
diff changeset
589 if isinstance(creation_sequence, list):
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guerler
parents:
diff changeset
590 cs = creation_sequence[:]
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guerler
parents:
diff changeset
591 else:
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guerler
parents:
diff changeset
592 cs = list(creation_sequence)
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
593 elif isinstance(first, tuple): # labeled creation sequence
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
594 cs = [v[1] for v in creation_sequence]
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guerler
parents:
diff changeset
595 i = [v[0] for v in creation_sequence].index(i)
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guerler
parents:
diff changeset
596 elif isinstance(first, int): # compact creation sequence
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
597 cs = uncompact(creation_sequence)
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parents:
diff changeset
598 else:
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
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parents:
diff changeset
599 raise TypeError("Not a valid creation sequence type")
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guerler
parents:
diff changeset
600
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guerler
parents:
diff changeset
601 # Compute
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
602 N = len(cs)
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
603 spl = [2] * N # length 2 to every node
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
604 spl[i] = 0 # except self which is 0
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
605 # 1 for all d's to the right
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
606 for j in range(i + 1, N):
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
607 if cs[j] == "d":
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
608 spl[j] = 1
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
609 if cs[i] == 'd': # 1 for all nodes to the left
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
610 for j in range(i):
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
611 spl[j] = 1
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
612 # and -1 for any trailing i to indicate unreachable
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
613 for j in range(N - 1, 0, -1):
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
614 if cs[j] == "d":
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
615 break
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
616 spl[j] = -1
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
617 return spl
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
618
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
619
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
620 def betweenness_sequence(creation_sequence, normalized=True):
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
621 """
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
622 Return betweenness for the threshold graph with the given creation
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
623 sequence. The result is unscaled. To scale the values
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
624 to the iterval [0,1] divide by (n-1)*(n-2).
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
625 """
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
626 cs = creation_sequence
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
627 seq = [] # betweenness
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
628 lastchar = 'd' # first node is always a 'd'
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
629 dr = float(cs.count("d")) # number of d's to the right of curren pos
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
630 irun = 0 # number of i's in the last run
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
631 drun = 0 # number of d's in the last run
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
632 dlast = 0.0 # betweenness of last d
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
633 for i, c in enumerate(cs):
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
634 if c == 'd': # cs[i]=="d":
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
635 # betweennees = amt shared with eariler d's and i's
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
636 # + new isolated nodes covered
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
637 # + new paths to all previous nodes
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
638 b = dlast + (irun - 1) * irun / dr + 2 * irun * (i - drun - irun) / dr
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
639 drun += 1 # update counter
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
640 else: # cs[i]="i":
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
641 if lastchar == 'd': # if this is a new run of i's
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
642 dlast = b # accumulate betweenness
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
643 dr -= drun # update number of d's to the right
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
644 drun = 0 # reset d counter
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
645 irun = 0 # reset i counter
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
646 b = 0 # isolated nodes have zero betweenness
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
647 irun += 1 # add another i to the run
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
648 seq.append(float(b))
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
649 lastchar = c
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
650
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
651 # normalize by the number of possible shortest paths
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
652 if normalized:
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
653 order = len(cs)
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
654 scale = 1.0 / ((order - 1) * (order - 2))
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
655 seq = [s * scale for s in seq]
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
656
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
657 return seq
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
658
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
659
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
660 def eigenvectors(creation_sequence):
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
661 """
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
662 Return a 2-tuple of Laplacian eigenvalues and eigenvectors
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
663 for the threshold network with creation_sequence.
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
664 The first value is a list of eigenvalues.
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
665 The second value is a list of eigenvectors.
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
666 The lists are in the same order so corresponding eigenvectors
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
667 and eigenvalues are in the same position in the two lists.
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
668
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
669 Notice that the order of the eigenvalues returned by eigenvalues(cs)
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
670 may not correspond to the order of these eigenvectors.
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
671 """
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
672 ccs = make_compact(creation_sequence)
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
673 N = sum(ccs)
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
674 vec = [0] * N
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
675 val = vec[:]
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
676 # get number of type d nodes to the right (all for first node)
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
677 dr = sum(ccs[::2])
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
678
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
679 nn = ccs[0]
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
680 vec[0] = [1. / sqrt(N)] * N
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
681 val[0] = 0
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
682 e = dr
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
683 dr -= nn
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
684 type_d = True
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
685 i = 1
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
686 dd = 1
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
687 while dd < nn:
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
688 scale = 1. / sqrt(dd * dd + i)
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
689 vec[i] = i * [-scale] + [dd * scale] + [0] * (N - i - 1)
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
690 val[i] = e
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
691 i += 1
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
692 dd += 1
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
693 if len(ccs) == 1:
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
694 return (val, vec)
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
695 for nn in ccs[1:]:
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
696 scale = 1. / sqrt(nn * i * (i + nn))
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
697 vec[i] = i * [-nn * scale] + nn * [i * scale] + [0] * (N - i - nn)
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
698 # find eigenvalue
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
699 type_d = not type_d
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
700 if type_d:
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
701 e = i + dr
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
702 dr -= nn
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
703 else:
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
704 e = dr
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
705 val[i] = e
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
706 st = i
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
707 i += 1
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
708 dd = 1
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
709 while dd < nn:
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
710 scale = 1. / sqrt(i - st + dd * dd)
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
711 vec[i] = [0] * st + (i - st) * [-scale] + [dd * scale] + [0] * (N - i - 1)
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
712 val[i] = e
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
713 i += 1
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
714 dd += 1
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
715 return (val, vec)
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
716
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
717
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
718 def spectral_projection(u, eigenpairs):
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
719 """
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
720 Returns the coefficients of each eigenvector
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
721 in a projection of the vector u onto the normalized
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
722 eigenvectors which are contained in eigenpairs.
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
723
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
724 eigenpairs should be a list of two objects. The
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
725 first is a list of eigenvalues and the second a list
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
726 of eigenvectors. The eigenvectors should be lists.
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
727
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
728 There's not a lot of error checking on lengths of
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
729 arrays, etc. so be careful.
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
730 """
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
731 coeff = []
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
732 evect = eigenpairs[1]
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
733 for ev in evect:
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
734 c = sum([evv * uv for (evv, uv) in zip(ev, u)])
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
735 coeff.append(c)
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
736 return coeff
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
737
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
738
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
739 def eigenvalues(creation_sequence):
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
740 """
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
741 Return sequence of eigenvalues of the Laplacian of the threshold
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
742 graph for the given creation_sequence.
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
743
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
744 Based on the Ferrer's diagram method. The spectrum is integral
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
745 and is the conjugate of the degree sequence.
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
746
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
747 See::
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
748
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
749 @Article{degree-merris-1994,
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
750 author = {Russel Merris},
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
751 title = {Degree maximal graphs are Laplacian integral},
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
752 journal = {Linear Algebra Appl.},
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
753 year = {1994},
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
754 volume = {199},
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
755 pages = {381--389},
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
756 }
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
757
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
758 """
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
759 degseq = degree_sequence(creation_sequence)
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
760 degseq.sort()
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
761 eiglist = [] # zero is always one eigenvalue
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
762 eig = 0
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
763 row = len(degseq)
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
764 bigdeg = degseq.pop()
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
765 while row:
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
766 if bigdeg < row:
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
767 eiglist.append(eig)
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
768 row -= 1
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
769 else:
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
770 eig += 1
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
771 if degseq:
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
772 bigdeg = degseq.pop()
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
773 else:
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
774 bigdeg = 0
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
775 return eiglist
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
776
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
777
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
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parents:
diff changeset
778 # Threshold graph creation routines
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
779
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
780 @py_random_state(2)
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
781 def random_threshold_sequence(n, p, seed=None):
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
782 """
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
783 Create a random threshold sequence of size n.
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
784 A creation sequence is built by randomly choosing d's with
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
785 probabiliy p and i's with probability 1-p.
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
786
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
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parents:
diff changeset
787 s=nx.random_threshold_sequence(10,0.5)
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
788
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
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parents:
diff changeset
789 returns a threshold sequence of length 10 with equal
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
790 probably of an i or a d at each position.
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
791
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
792 A "random" threshold graph can be built with
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
793
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
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parents:
diff changeset
794 G=nx.threshold_graph(s)
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
795
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parents:
diff changeset
796 seed : integer, random_state, or None (default)
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
797 Indicator of random number generation state.
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
798 See :ref:`Randomness<randomness>`.
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parents:
diff changeset
799 """
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
800 if not (0 <= p <= 1):
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
801 raise ValueError("p must be in [0,1]")
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guerler
parents:
diff changeset
802
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
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parents:
diff changeset
803 cs = ['d'] # threshold sequences always start with a d
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guerler
parents:
diff changeset
804 for i in range(1, n):
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guerler
parents:
diff changeset
805 if seed.random() < p:
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parents:
diff changeset
806 cs.append('d')
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
807 else:
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parents:
diff changeset
808 cs.append('i')
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
809 return cs
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
810
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
811
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
812 # maybe *_d_threshold_sequence routines should
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
813 # be (or be called from) a single routine with a more descriptive name
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
814 # and a keyword parameter?
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
815 def right_d_threshold_sequence(n, m):
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
816 """
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
817 Create a skewed threshold graph with a given number
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
818 of vertices (n) and a given number of edges (m).
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
819
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
820 The routine returns an unlabeled creation sequence
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
821 for the threshold graph.
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
822
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
823 FIXME: describe algorithm
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
824
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diff changeset
825 """
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
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diff changeset
826 cs = ['d'] + ['i'] * (n - 1) # create sequence with n insolated nodes
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guerler
parents:
diff changeset
827
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
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diff changeset
828 # m <n : not enough edges, make disconnected
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
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parents:
diff changeset
829 if m < n:
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
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parents:
diff changeset
830 cs[m] = 'd'
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guerler
parents:
diff changeset
831 return cs
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
832
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
833 # too many edges
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
834 if m > n * (n - 1) / 2:
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
835 raise ValueError("Too many edges for this many nodes.")
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guerler
parents:
diff changeset
836
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parents:
diff changeset
837 # connected case m >n-1
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parents:
diff changeset
838 ind = n - 1
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guerler
parents:
diff changeset
839 sum = n - 1
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parents:
diff changeset
840 while sum < m:
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diff changeset
841 cs[ind] = 'd'
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guerler
parents:
diff changeset
842 ind -= 1
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guerler
parents:
diff changeset
843 sum += ind
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guerler
parents:
diff changeset
844 ind = m - (sum - ind)
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parents:
diff changeset
845 cs[ind] = 'd'
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guerler
parents:
diff changeset
846 return cs
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guerler
parents:
diff changeset
847
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guerler
parents:
diff changeset
848
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guerler
parents:
diff changeset
849 def left_d_threshold_sequence(n, m):
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guerler
parents:
diff changeset
850 """
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
851 Create a skewed threshold graph with a given number
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
852 of vertices (n) and a given number of edges (m).
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
853
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
854 The routine returns an unlabeled creation sequence
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
855 for the threshold graph.
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
856
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
857 FIXME: describe algorithm
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
858
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
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diff changeset
859 """
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
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diff changeset
860 cs = ['d'] + ['i'] * (n - 1) # create sequence with n insolated nodes
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
861
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
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parents:
diff changeset
862 # m <n : not enough edges, make disconnected
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
863 if m < n:
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
864 cs[m] = 'd'
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
865 return cs
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
866
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
867 # too many edges
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
868 if m > n * (n - 1) / 2:
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
869 raise ValueError("Too many edges for this many nodes.")
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
870
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
871 # Connected case when M>N-1
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
872 cs[n - 1] = 'd'
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guerler
parents:
diff changeset
873 sum = n - 1
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guerler
parents:
diff changeset
874 ind = 1
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
875 while sum < m:
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guerler
parents:
diff changeset
876 cs[ind] = 'd'
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guerler
parents:
diff changeset
877 sum += ind
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
878 ind += 1
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guerler
parents:
diff changeset
879 if sum > m: # be sure not to change the first vertex
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
880 cs[sum - m] = 'i'
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guerler
parents:
diff changeset
881 return cs
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guerler
parents:
diff changeset
882
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
883
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guerler
parents:
diff changeset
884 @py_random_state(3)
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guerler
parents:
diff changeset
885 def swap_d(cs, p_split=1.0, p_combine=1.0, seed=None):
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
886 """
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
887 Perform a "swap" operation on a threshold sequence.
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
888
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
889 The swap preserves the number of nodes and edges
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
890 in the graph for the given sequence.
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
891 The resulting sequence is still a threshold sequence.
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
892
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guerler
parents:
diff changeset
893 Perform one split and one combine operation on the
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
894 'd's of a creation sequence for a threshold graph.
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
895 This operation maintains the number of nodes and edges
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
896 in the graph, but shifts the edges from node to node
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
897 maintaining the threshold quality of the graph.
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
898
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
899 seed : integer, random_state, or None (default)
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
900 Indicator of random number generation state.
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
901 See :ref:`Randomness<randomness>`.
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
902 """
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
903 # preprocess the creation sequence
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
904 dlist = [i for (i, node_type) in enumerate(cs[1:-1]) if node_type == 'd']
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
905 # split
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guerler
parents:
diff changeset
906 if seed.random() < p_split:
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guerler
parents:
diff changeset
907 choice = seed.choice(dlist)
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
908 split_to = seed.choice(range(choice))
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
909 flip_side = choice - split_to
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
910 if split_to != flip_side and cs[split_to] == 'i' and cs[flip_side] == 'i':
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
911 cs[choice] = 'i'
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
912 cs[split_to] = 'd'
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
913 cs[flip_side] = 'd'
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
914 dlist.remove(choice)
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
915 # don't add or combine may reverse this action
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
916 # dlist.extend([split_to,flip_side])
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
917 # print >>sys.stderr,"split at %s to %s and %s"%(choice,split_to,flip_side)
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
918 # combine
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
919 if seed.random() < p_combine and dlist:
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
920 first_choice = seed.choice(dlist)
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
921 second_choice = seed.choice(dlist)
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
922 target = first_choice + second_choice
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
923 if target >= len(cs) or cs[target] == 'd' or first_choice == second_choice:
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
924 return cs
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
925 # OK to combine
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
926 cs[first_choice] = 'i'
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
927 cs[second_choice] = 'i'
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
928 cs[target] = 'd'
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
929 # print >>sys.stderr,"combine %s and %s to make %s."%(first_choice,second_choice,target)
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
930
56ad4e20f292 "planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
guerler
parents:
diff changeset
931 return cs