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

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"planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
author guerler
date Fri, 31 Jul 2020 00:32:28 -0400
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1 # -*- coding: utf-8 -*-
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2 # Copyright (C) 2014 ysitu <ysitu@users.noreply.github.com>
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3 # All rights reserved.
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4 # BSD license.
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5 #
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6 # Author: ysitu <ysitu@users.noreply.github.com>
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7 """
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8 Algebraic connectivity and Fiedler vectors of undirected graphs.
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9 """
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10 from functools import partial
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11 import networkx as nx
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12 from networkx.utils import not_implemented_for
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13 from networkx.utils import reverse_cuthill_mckee_ordering
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14 from networkx.utils import random_state
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15
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16 try:
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17 from numpy import array, asmatrix, asarray, dot, ndarray, ones, sqrt, zeros
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18 from numpy.linalg import norm, qr
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19 from numpy.random import normal
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20 from scipy.linalg import eigh, inv
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21 from scipy.sparse import csc_matrix, spdiags
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22 from scipy.sparse.linalg import eigsh, lobpcg
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23 __all__ = ['algebraic_connectivity', 'fiedler_vector', 'spectral_ordering']
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24 except ImportError:
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25 __all__ = []
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26
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27 try:
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28 from scipy.linalg.blas import dasum, daxpy, ddot
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29 except ImportError:
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30 if __all__:
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31 # Make sure the imports succeeded.
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32 # Use minimal replacements if BLAS is unavailable from SciPy.
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33 dasum = partial(norm, ord=1)
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34 ddot = dot
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35
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36 def daxpy(x, y, a):
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37 y += a * x
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38 return y
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39
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40
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41 class _PCGSolver(object):
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42 """Preconditioned conjugate gradient method.
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43
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44 To solve Ax = b:
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45 M = A.diagonal() # or some other preconditioner
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46 solver = _PCGSolver(lambda x: A * x, lambda x: M * x)
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47 x = solver.solve(b)
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48
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49 The inputs A and M are functions which compute
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50 matrix multiplication on the argument.
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51 A - multiply by the matrix A in Ax=b
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52 M - multiply by M, the preconditioner surragate for A
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53
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54 Warning: There is no limit on number of iterations.
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55 """
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56
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57 def __init__(self, A, M):
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58 self._A = A
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59 self._M = M or (lambda x: x.copy())
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60
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61 def solve(self, B, tol):
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62 B = asarray(B)
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63 X = ndarray(B.shape, order='F')
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64 for j in range(B.shape[1]):
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65 X[:, j] = self._solve(B[:, j], tol)
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66 return X
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67
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68 def _solve(self, b, tol):
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69 A = self._A
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70 M = self._M
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71 tol *= dasum(b)
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72 # Initialize.
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73 x = zeros(b.shape)
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74 r = b.copy()
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75 z = M(r)
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76 rz = ddot(r, z)
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77 p = z.copy()
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78 # Iterate.
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79 while True:
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80 Ap = A(p)
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81 alpha = rz / ddot(p, Ap)
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82 x = daxpy(p, x, a=alpha)
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83 r = daxpy(Ap, r, a=-alpha)
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84 if dasum(r) < tol:
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85 return x
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86 z = M(r)
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87 beta = ddot(r, z)
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88 beta, rz = beta / rz, beta
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89 p = daxpy(p, z, a=beta)
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90
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91
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92 class _CholeskySolver(object):
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93 """Cholesky factorization.
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94
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95 To solve Ax = b:
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96 solver = _CholeskySolver(A)
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97 x = solver.solve(b)
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98
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99 optional argument `tol` on solve method is ignored but included
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100 to match _PCGsolver API.
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101 """
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102
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103 def __init__(self, A):
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104 if not self._cholesky:
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105 raise nx.NetworkXError('Cholesky solver unavailable.')
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106 self._chol = self._cholesky(A)
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107
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108 def solve(self, B, tol=None):
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109 return self._chol(B)
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110
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111 try:
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112 from scikits.sparse.cholmod import cholesky
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113 _cholesky = cholesky
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114 except ImportError:
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115 _cholesky = None
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116
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117
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118 class _LUSolver(object):
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119 """LU factorization.
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120
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121 To solve Ax = b:
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122 solver = _LUSolver(A)
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123 x = solver.solve(b)
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124
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125 optional argument `tol` on solve method is ignored but included
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126 to match _PCGsolver API.
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127 """
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128
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129 def __init__(self, A):
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130 if not self._splu:
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131 raise nx.NetworkXError('LU solver unavailable.')
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132 self._LU = self._splu(A)
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133
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134 def solve(self, B, tol=None):
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135 B = asarray(B)
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136 X = ndarray(B.shape, order='F')
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137 for j in range(B.shape[1]):
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138 X[:, j] = self._LU.solve(B[:, j])
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139 return X
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140
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141 try:
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142 from scipy.sparse.linalg import splu
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143 _splu = partial(splu, permc_spec='MMD_AT_PLUS_A', diag_pivot_thresh=0.,
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144 options={'Equil': True, 'SymmetricMode': True})
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145 except ImportError:
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146 _splu = None
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147
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148
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149 def _preprocess_graph(G, weight):
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150 """Compute edge weights and eliminate zero-weight edges.
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151 """
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152 if G.is_directed():
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153 H = nx.MultiGraph()
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154 H.add_nodes_from(G)
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155 H.add_weighted_edges_from(((u, v, e.get(weight, 1.))
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156 for u, v, e in G.edges(data=True)
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157 if u != v), weight=weight)
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158 G = H
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159 if not G.is_multigraph():
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160 edges = ((u, v, abs(e.get(weight, 1.)))
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161 for u, v, e in G.edges(data=True) if u != v)
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162 else:
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163 edges = ((u, v, sum(abs(e.get(weight, 1.)) for e in G[u][v].values()))
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164 for u, v in G.edges() if u != v)
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165 H = nx.Graph()
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166 H.add_nodes_from(G)
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167 H.add_weighted_edges_from((u, v, e) for u, v, e in edges if e != 0)
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168 return H
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169
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170
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171 def _rcm_estimate(G, nodelist):
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172 """Estimate the Fiedler vector using the reverse Cuthill-McKee ordering.
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173 """
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174 G = G.subgraph(nodelist)
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175 order = reverse_cuthill_mckee_ordering(G)
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176 n = len(nodelist)
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177 index = dict(zip(nodelist, range(n)))
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178 x = ndarray(n, dtype=float)
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179 for i, u in enumerate(order):
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180 x[index[u]] = i
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181 x -= (n - 1) / 2.
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182 return x
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183
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184
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185 def _tracemin_fiedler(L, X, normalized, tol, method):
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186 """Compute the Fiedler vector of L using the TraceMIN-Fiedler algorithm.
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187
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188 The Fiedler vector of a connected undirected graph is the eigenvector
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189 corresponding to the second smallest eigenvalue of the Laplacian matrix of
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190 of the graph. This function starts with the Laplacian L, not the Graph.
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191
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192 Parameters
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193 ----------
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194 L : Laplacian of a possibly weighted or normalized, but undirected graph
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195
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196 X : Initial guess for a solution. Usually a matrix of random numbers.
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197 This function allows more than one column in X to identify more than
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198 one eigenvector if desired.
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199
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200 normalized : bool
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201 Whether the normalized Laplacian matrix is used.
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202
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203 tol : float
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204 Tolerance of relative residual in eigenvalue computation.
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205 Warning: There is no limit on number of iterations.
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206
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207 method : string
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208 Should be 'tracemin_pcg', 'tracemin_chol' or 'tracemin_lu'.
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209 Otherwise exception is raised.
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210
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211 Returns
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212 -------
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213 sigma, X : Two NumPy arrays of floats.
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214 The lowest eigenvalues and corresponding eigenvectors of L.
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215 The size of input X determines the size of these outputs.
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216 As this is for Fiedler vectors, the zero eigenvalue (and
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217 constant eigenvector) are avoided.
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218 """
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219 n = X.shape[0]
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220
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221 if normalized:
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222 # Form the normalized Laplacian matrix and determine the eigenvector of
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223 # its nullspace.
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224 e = sqrt(L.diagonal())
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225 D = spdiags(1. / e, [0], n, n, format='csr')
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226 L = D * L * D
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227 e *= 1. / norm(e, 2)
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228
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229 if normalized:
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230 def project(X):
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231 """Make X orthogonal to the nullspace of L.
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232 """
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233 X = asarray(X)
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234 for j in range(X.shape[1]):
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235 X[:, j] -= dot(X[:, j], e) * e
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236 else:
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237 def project(X):
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238 """Make X orthogonal to the nullspace of L.
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239 """
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240 X = asarray(X)
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241 for j in range(X.shape[1]):
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242 X[:, j] -= X[:, j].sum() / n
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243
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244 if method == 'tracemin_pcg':
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245 D = L.diagonal().astype(float)
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246 solver = _PCGSolver(lambda x: L * x, lambda x: D * x)
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247 elif method == 'tracemin_chol' or method == 'tracemin_lu':
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248 # Convert A to CSC to suppress SparseEfficiencyWarning.
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249 A = csc_matrix(L, dtype=float, copy=True)
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250 # Force A to be nonsingular. Since A is the Laplacian matrix of a
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251 # connected graph, its rank deficiency is one, and thus one diagonal
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252 # element needs to modified. Changing to infinity forces a zero in the
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253 # corresponding element in the solution.
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254 i = (A.indptr[1:] - A.indptr[:-1]).argmax()
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255 A[i, i] = float('inf')
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256 if method == 'tracemin_chol':
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257 solver = _CholeskySolver(A)
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258 else:
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259 solver = _LUSolver(A)
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260 else:
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261 raise nx.NetworkXError('Unknown linear system solver: ' + method)
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262
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263 # Initialize.
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264 Lnorm = abs(L).sum(axis=1).flatten().max()
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265 project(X)
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266 W = asmatrix(ndarray(X.shape, order='F'))
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267
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268 while True:
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269 # Orthonormalize X.
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270 X = qr(X)[0]
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271 # Compute iteration matrix H.
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272 W[:, :] = L * X
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273 H = X.T * W
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274 sigma, Y = eigh(H, overwrite_a=True)
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275 # Compute the Ritz vectors.
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276 X *= Y
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277 # Test for convergence exploiting the fact that L * X == W * Y.
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278 res = dasum(W * asmatrix(Y)[:, 0] - sigma[0] * X[:, 0]) / Lnorm
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279 if res < tol:
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280 break
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281 # Compute X = L \ X / (X' * (L \ X)).
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282 # L \ X can have an arbitrary projection on the nullspace of L,
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283 # which will be eliminated.
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284 W[:, :] = solver.solve(X, tol)
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285 X = (inv(W.T * X) * W.T).T # Preserves Fortran storage order.
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286 project(X)
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287
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288 return sigma, asarray(X)
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289
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290
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291 def _get_fiedler_func(method):
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292 """Returns a function that solves the Fiedler eigenvalue problem.
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293 """
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294 if method == "tracemin": # old style keyword <v2.1
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295 method = "tracemin_pcg"
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296 if method in ("tracemin_pcg", "tracemin_chol", "tracemin_lu"):
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297 def find_fiedler(L, x, normalized, tol, seed):
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298 q = 1 if method == 'tracemin_pcg' else min(4, L.shape[0] - 1)
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299 X = asmatrix(seed.normal(size=(q, L.shape[0]))).T
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300 sigma, X = _tracemin_fiedler(L, X, normalized, tol, method)
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301 return sigma[0], X[:, 0]
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302 elif method == 'lanczos' or method == 'lobpcg':
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303 def find_fiedler(L, x, normalized, tol, seed):
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304 L = csc_matrix(L, dtype=float)
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305 n = L.shape[0]
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306 if normalized:
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307 D = spdiags(1. / sqrt(L.diagonal()), [0], n, n, format='csc')
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308 L = D * L * D
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309 if method == 'lanczos' or n < 10:
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310 # Avoid LOBPCG when n < 10 due to
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311 # https://github.com/scipy/scipy/issues/3592
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312 # https://github.com/scipy/scipy/pull/3594
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313 sigma, X = eigsh(L, 2, which='SM', tol=tol,
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314 return_eigenvectors=True)
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315 return sigma[1], X[:, 1]
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316 else:
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317 X = asarray(asmatrix(x).T)
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318 M = spdiags(1. / L.diagonal(), [0], n, n)
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319 Y = ones(n)
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320 if normalized:
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321 Y /= D.diagonal()
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322 sigma, X = lobpcg(L, X, M=M, Y=asmatrix(Y).T, tol=tol,
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323 maxiter=n, largest=False)
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324 return sigma[0], X[:, 0]
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325 else:
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326 raise nx.NetworkXError("unknown method '%s'." % method)
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327
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328 return find_fiedler
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329
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330
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331 @random_state(5)
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332 @not_implemented_for('directed')
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333 def algebraic_connectivity(G, weight='weight', normalized=False, tol=1e-8,
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334 method='tracemin_pcg', seed=None):
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335 """Returns the algebraic connectivity of an undirected graph.
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336
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337 The algebraic connectivity of a connected undirected graph is the second
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338 smallest eigenvalue of its Laplacian matrix.
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339
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340 Parameters
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341 ----------
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342 G : NetworkX graph
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343 An undirected graph.
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344
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345 weight : object, optional (default: None)
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346 The data key used to determine the weight of each edge. If None, then
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347 each edge has unit weight.
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348
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349 normalized : bool, optional (default: False)
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350 Whether the normalized Laplacian matrix is used.
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351
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352 tol : float, optional (default: 1e-8)
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353 Tolerance of relative residual in eigenvalue computation.
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354
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355 method : string, optional (default: 'tracemin_pcg')
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356 Method of eigenvalue computation. It must be one of the tracemin
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357 options shown below (TraceMIN), 'lanczos' (Lanczos iteration)
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358 or 'lobpcg' (LOBPCG).
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359
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360 The TraceMIN algorithm uses a linear system solver. The following
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361 values allow specifying the solver to be used.
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362
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363 =============== ========================================
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364 Value Solver
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365 =============== ========================================
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366 'tracemin_pcg' Preconditioned conjugate gradient method
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367 'tracemin_chol' Cholesky factorization
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368 'tracemin_lu' LU factorization
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369 =============== ========================================
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370
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371 seed : integer, random_state, or None (default)
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372 Indicator of random number generation state.
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373 See :ref:`Randomness<randomness>`.
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374
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375 Returns
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376 -------
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377 algebraic_connectivity : float
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378 Algebraic connectivity.
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379
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380 Raises
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381 ------
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382 NetworkXNotImplemented
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383 If G is directed.
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384
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385 NetworkXError
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386 If G has less than two nodes.
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387
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388 Notes
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389 -----
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390 Edge weights are interpreted by their absolute values. For MultiGraph's,
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391 weights of parallel edges are summed. Zero-weighted edges are ignored.
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392
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393 To use Cholesky factorization in the TraceMIN algorithm, the
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394 :samp:`scikits.sparse` package must be installed.
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395
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396 See Also
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397 --------
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398 laplacian_matrix
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399 """
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400 if len(G) < 2:
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401 raise nx.NetworkXError('graph has less than two nodes.')
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402 G = _preprocess_graph(G, weight)
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403 if not nx.is_connected(G):
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404 return 0.
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405
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406 L = nx.laplacian_matrix(G)
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407 if L.shape[0] == 2:
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408 return 2. * L[0, 0] if not normalized else 2.
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409
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410 find_fiedler = _get_fiedler_func(method)
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411 x = None if method != 'lobpcg' else _rcm_estimate(G, G)
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412 sigma, fiedler = find_fiedler(L, x, normalized, tol, seed)
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413 return sigma
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414
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415
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416 @random_state(5)
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417 @not_implemented_for('directed')
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418 def fiedler_vector(G, weight='weight', normalized=False, tol=1e-8,
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419 method='tracemin_pcg', seed=None):
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420 """Returns the Fiedler vector of a connected undirected graph.
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421
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422 The Fiedler vector of a connected undirected graph is the eigenvector
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423 corresponding to the second smallest eigenvalue of the Laplacian matrix of
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424 of the graph.
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425
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426 Parameters
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427 ----------
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428 G : NetworkX graph
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429 An undirected graph.
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430
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431 weight : object, optional (default: None)
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432 The data key used to determine the weight of each edge. If None, then
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433 each edge has unit weight.
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434
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435 normalized : bool, optional (default: False)
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436 Whether the normalized Laplacian matrix is used.
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437
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438 tol : float, optional (default: 1e-8)
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439 Tolerance of relative residual in eigenvalue computation.
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440
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441 method : string, optional (default: 'tracemin_pcg')
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442 Method of eigenvalue computation. It must be one of the tracemin
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443 options shown below (TraceMIN), 'lanczos' (Lanczos iteration)
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444 or 'lobpcg' (LOBPCG).
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445
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446 The TraceMIN algorithm uses a linear system solver. The following
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447 values allow specifying the solver to be used.
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448
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449 =============== ========================================
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450 Value Solver
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451 =============== ========================================
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452 'tracemin_pcg' Preconditioned conjugate gradient method
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453 'tracemin_chol' Cholesky factorization
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454 'tracemin_lu' LU factorization
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455 =============== ========================================
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456
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457 seed : integer, random_state, or None (default)
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458 Indicator of random number generation state.
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459 See :ref:`Randomness<randomness>`.
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460
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461 Returns
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462 -------
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463 fiedler_vector : NumPy array of floats.
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464 Fiedler vector.
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465
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466 Raises
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467 ------
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468 NetworkXNotImplemented
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469 If G is directed.
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470
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471 NetworkXError
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472 If G has less than two nodes or is not connected.
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473
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474 Notes
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475 -----
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476 Edge weights are interpreted by their absolute values. For MultiGraph's,
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477 weights of parallel edges are summed. Zero-weighted edges are ignored.
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478
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479 To use Cholesky factorization in the TraceMIN algorithm, the
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480 :samp:`scikits.sparse` package must be installed.
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481
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482 See Also
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483 --------
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484 laplacian_matrix
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485 """
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486 if len(G) < 2:
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487 raise nx.NetworkXError('graph has less than two nodes.')
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488 G = _preprocess_graph(G, weight)
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489 if not nx.is_connected(G):
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490 raise nx.NetworkXError('graph is not connected.')
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491
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492 if len(G) == 2:
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493 return array([1., -1.])
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494
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495 find_fiedler = _get_fiedler_func(method)
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496 L = nx.laplacian_matrix(G)
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497 x = None if method != 'lobpcg' else _rcm_estimate(G, G)
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498 sigma, fiedler = find_fiedler(L, x, normalized, tol, seed)
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499 return fiedler
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500
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501
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502 @random_state(5)
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503 def spectral_ordering(G, weight='weight', normalized=False, tol=1e-8,
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504 method='tracemin_pcg', seed=None):
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505 """Compute the spectral_ordering of a graph.
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506
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507 The spectral ordering of a graph is an ordering of its nodes where nodes
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508 in the same weakly connected components appear contiguous and ordered by
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509 their corresponding elements in the Fiedler vector of the component.
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510
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511 Parameters
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512 ----------
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513 G : NetworkX graph
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514 A graph.
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515
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516 weight : object, optional (default: None)
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517 The data key used to determine the weight of each edge. If None, then
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518 each edge has unit weight.
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519
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520 normalized : bool, optional (default: False)
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521 Whether the normalized Laplacian matrix is used.
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522
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523 tol : float, optional (default: 1e-8)
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524 Tolerance of relative residual in eigenvalue computation.
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525
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526 method : string, optional (default: 'tracemin_pcg')
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527 Method of eigenvalue computation. It must be one of the tracemin
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528 options shown below (TraceMIN), 'lanczos' (Lanczos iteration)
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529 or 'lobpcg' (LOBPCG).
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530
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531 The TraceMIN algorithm uses a linear system solver. The following
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532 values allow specifying the solver to be used.
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533
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534 =============== ========================================
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535 Value Solver
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536 =============== ========================================
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537 'tracemin_pcg' Preconditioned conjugate gradient method
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538 'tracemin_chol' Cholesky factorization
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539 'tracemin_lu' LU factorization
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540 =============== ========================================
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541
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542 seed : integer, random_state, or None (default)
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543 Indicator of random number generation state.
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544 See :ref:`Randomness<randomness>`.
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545
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546 Returns
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547 -------
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548 spectral_ordering : NumPy array of floats.
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549 Spectral ordering of nodes.
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550
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551 Raises
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552 ------
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553 NetworkXError
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554 If G is empty.
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555
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556 Notes
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557 -----
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558 Edge weights are interpreted by their absolute values. For MultiGraph's,
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559 weights of parallel edges are summed. Zero-weighted edges are ignored.
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560
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561 To use Cholesky factorization in the TraceMIN algorithm, the
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562 :samp:`scikits.sparse` package must be installed.
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563
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564 See Also
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565 --------
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566 laplacian_matrix
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567 """
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568 if len(G) == 0:
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569 raise nx.NetworkXError('graph is empty.')
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570 G = _preprocess_graph(G, weight)
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571
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572 find_fiedler = _get_fiedler_func(method)
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573 order = []
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574 for component in nx.connected_components(G):
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575 size = len(component)
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576 if size > 2:
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577 L = nx.laplacian_matrix(G, component)
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578 x = None if method != 'lobpcg' else _rcm_estimate(G, component)
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579 sigma, fiedler = find_fiedler(L, x, normalized, tol, seed)
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580 sort_info = zip(fiedler, range(size), component)
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581 order.extend(u for x, c, u in sorted(sort_info))
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582 else:
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583 order.extend(component)
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584
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585 return order
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586
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587
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588 # fixture for pytest
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589 def setup_module(module):
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590 import pytest
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591 numpy = pytest.importorskip('numpy')
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592 scipy.sparse = pytest.importorskip('scipy.sparse')