Mercurial > repos > shellac > sam_consensus_v3
annotate env/lib/python3.9/sitepackages/networkx/linalg/algebraicconnectivity.py @ 0:4f3585e2f14b draft default tip
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author  shellac 

date  Mon, 22 Mar 2021 18:12:50 +0000 
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1 """ 
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2 Algebraic connectivity and Fiedler vectors of undirected graphs. 
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3 """ 
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4 from functools import partial 
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5 import networkx as nx 
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6 from networkx.utils import not_implemented_for 
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7 from networkx.utils import reverse_cuthill_mckee_ordering 
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8 from networkx.utils import random_state 
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9 
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10 try: 
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11 from numpy import array, asarray, dot, ndarray, ones, sqrt, zeros, atleast_2d 
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12 from numpy.linalg import norm, qr 
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13 from scipy.linalg import eigh, inv 
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14 from scipy.sparse import csc_matrix, spdiags 
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15 from scipy.sparse.linalg import eigsh, lobpcg 
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16 
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17 __all__ = ["algebraic_connectivity", "fiedler_vector", "spectral_ordering"] 
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18 except ImportError: 
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19 __all__ = [] 
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20 
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21 try: 
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22 from scipy.linalg.blas import dasum, daxpy, ddot 
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23 except ImportError: 
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24 if __all__: 
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25 # Make sure the imports succeeded. 
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26 # Use minimal replacements if BLAS is unavailable from SciPy. 
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27 dasum = partial(norm, ord=1) 
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28 ddot = dot 
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29 
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30 def daxpy(x, y, a): 
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31 y += a * x 
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32 return y 
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33 
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34 
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35 class _PCGSolver: 
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36 """Preconditioned conjugate gradient method. 
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37 
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38 To solve Ax = b: 
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39 M = A.diagonal() # or some other preconditioner 
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40 solver = _PCGSolver(lambda x: A * x, lambda x: M * x) 
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41 x = solver.solve(b) 
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42 
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43 The inputs A and M are functions which compute 
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44 matrix multiplication on the argument. 
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45 A  multiply by the matrix A in Ax=b 
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46 M  multiply by M, the preconditioner surragate for A 
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47 
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48 Warning: There is no limit on number of iterations. 
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49 """ 
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50 
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51 def __init__(self, A, M): 
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52 self._A = A 
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53 self._M = M or (lambda x: x.copy()) 
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54 
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55 def solve(self, B, tol): 
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56 B = asarray(B) 
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57 X = ndarray(B.shape, order="F") 
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58 for j in range(B.shape[1]): 
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59 X[:, j] = self._solve(B[:, j], tol) 
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60 return X 
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61 
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62 def _solve(self, b, tol): 
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63 A = self._A 
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64 M = self._M 
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65 tol *= dasum(b) 
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66 # Initialize. 
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67 x = zeros(b.shape) 
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68 r = b.copy() 
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69 z = M(r) 
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70 rz = ddot(r, z) 
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71 p = z.copy() 
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72 # Iterate. 
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73 while True: 
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74 Ap = A(p) 
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75 alpha = rz / ddot(p, Ap) 
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76 x = daxpy(p, x, a=alpha) 
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77 r = daxpy(Ap, r, a=alpha) 
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78 if dasum(r) < tol: 
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79 return x 
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80 z = M(r) 
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81 beta = ddot(r, z) 
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82 beta, rz = beta / rz, beta 
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83 p = daxpy(p, z, a=beta) 
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84 
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85 
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86 class _CholeskySolver: 
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87 """Cholesky factorization. 
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88 
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89 To solve Ax = b: 
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90 solver = _CholeskySolver(A) 
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91 x = solver.solve(b) 
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92 
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93 optional argument `tol` on solve method is ignored but included 
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94 to match _PCGsolver API. 
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95 """ 
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96 
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97 def __init__(self, A): 
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98 if not self._cholesky: 
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99 raise nx.NetworkXError("Cholesky solver unavailable.") 
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100 self._chol = self._cholesky(A) 
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101 
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102 def solve(self, B, tol=None): 
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103 return self._chol(B) 
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104 
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105 try: 
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106 from scikits.sparse.cholmod import cholesky 
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107 
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108 _cholesky = cholesky 
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109 except ImportError: 
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110 _cholesky = None 
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111 
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112 
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113 class _LUSolver: 
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114 """LU factorization. 
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115 
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116 To solve Ax = b: 
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117 solver = _LUSolver(A) 
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118 x = solver.solve(b) 
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119 
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120 optional argument `tol` on solve method is ignored but included 
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121 to match _PCGsolver API. 
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122 """ 
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123 
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124 def __init__(self, A): 
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125 if not self._splu: 
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126 raise nx.NetworkXError("LU solver unavailable.") 
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127 self._LU = self._splu(A) 
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128 
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129 def solve(self, B, tol=None): 
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130 B = asarray(B) 
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131 X = ndarray(B.shape, order="F") 
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132 for j in range(B.shape[1]): 
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133 X[:, j] = self._LU.solve(B[:, j]) 
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134 return X 
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135 
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136 try: 
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137 from scipy.sparse.linalg import splu 
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138 
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139 _splu = partial( 
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140 splu, 
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141 permc_spec="MMD_AT_PLUS_A", 
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142 diag_pivot_thresh=0.0, 
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143 options={"Equil": True, "SymmetricMode": True}, 
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144 ) 
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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 zeroweight 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( 
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156 ((u, v, e.get(weight, 1.0)) for u, v, e in G.edges(data=True) if u != v), 
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157 weight=weight, 
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158 ) 
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159 G = H 
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160 if not G.is_multigraph(): 
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161 edges = ( 
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162 (u, v, abs(e.get(weight, 1.0))) for u, v, e in G.edges(data=True) if u != v 
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163 ) 
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164 else: 
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165 edges = ( 
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166 (u, v, sum(abs(e.get(weight, 1.0)) for e in G[u][v].values())) 
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167 for u, v in G.edges() 
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168 if u != v 
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169 ) 
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170 H = nx.Graph() 
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171 H.add_nodes_from(G) 
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172 H.add_weighted_edges_from((u, v, e) for u, v, e in edges if e != 0) 
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173 return H 
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174 
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175 
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176 def _rcm_estimate(G, nodelist): 
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177 """Estimate the Fiedler vector using the reverse CuthillMcKee ordering. 
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178 """ 
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179 G = G.subgraph(nodelist) 
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180 order = reverse_cuthill_mckee_ordering(G) 
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181 n = len(nodelist) 
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182 index = dict(zip(nodelist, range(n))) 
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183 x = ndarray(n, dtype=float) 
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184 for i, u in enumerate(order): 
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185 x[index[u]] = i 
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186 x = (n  1) / 2.0 
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187 return x 
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188 
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189 
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190 def _tracemin_fiedler(L, X, normalized, tol, method): 
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191 """Compute the Fiedler vector of L using the TraceMINFiedler algorithm. 
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192 
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193 The Fiedler vector of a connected undirected graph is the eigenvector 
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194 corresponding to the second smallest eigenvalue of the Laplacian matrix of 
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195 of the graph. This function starts with the Laplacian L, not the Graph. 
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196 
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197 Parameters 
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198  
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199 L : Laplacian of a possibly weighted or normalized, but undirected graph 
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200 
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201 X : Initial guess for a solution. Usually a matrix of random numbers. 
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202 This function allows more than one column in X to identify more than 
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203 one eigenvector if desired. 
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204 
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205 normalized : bool 
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206 Whether the normalized Laplacian matrix is used. 
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207 
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208 tol : float 
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209 Tolerance of relative residual in eigenvalue computation. 
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210 Warning: There is no limit on number of iterations. 
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211 
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212 method : string 
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213 Should be 'tracemin_pcg', 'tracemin_chol' or 'tracemin_lu'. 
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214 Otherwise exception is raised. 
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215 
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216 Returns 
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217  
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218 sigma, X : Two NumPy arrays of floats. 
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219 The lowest eigenvalues and corresponding eigenvectors of L. 
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220 The size of input X determines the size of these outputs. 
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221 As this is for Fiedler vectors, the zero eigenvalue (and 
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222 constant eigenvector) are avoided. 
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223 """ 
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224 n = X.shape[0] 
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225 
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226 if normalized: 
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227 # Form the normalized Laplacian matrix and determine the eigenvector of 
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228 # its nullspace. 
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229 e = sqrt(L.diagonal()) 
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230 D = spdiags(1.0 / e, [0], n, n, format="csr") 
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231 L = D * L * D 
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232 e *= 1.0 / norm(e, 2) 
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233 
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234 if normalized: 
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235 
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236 def project(X): 
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237 """Make X orthogonal to the nullspace of L. 
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238 """ 
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239 X = asarray(X) 
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240 for j in range(X.shape[1]): 
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241 X[:, j] = dot(X[:, j], e) * e 
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242 
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243 else: 
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244 
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245 def project(X): 
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246 """Make X orthogonal to the nullspace of L. 
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247 """ 
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248 X = asarray(X) 
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249 for j in range(X.shape[1]): 
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250 X[:, j] = X[:, j].sum() / n 
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251 
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252 if method == "tracemin_pcg": 
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253 D = L.diagonal().astype(float) 
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254 solver = _PCGSolver(lambda x: L * x, lambda x: D * x) 
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255 elif method == "tracemin_chol" or method == "tracemin_lu": 
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256 # Convert A to CSC to suppress SparseEfficiencyWarning. 
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257 A = csc_matrix(L, dtype=float, copy=True) 
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258 # Force A to be nonsingular. Since A is the Laplacian matrix of a 
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259 # connected graph, its rank deficiency is one, and thus one diagonal 
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260 # element needs to modified. Changing to infinity forces a zero in the 
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261 # corresponding element in the solution. 
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262 i = (A.indptr[1:]  A.indptr[:1]).argmax() 
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263 A[i, i] = float("inf") 
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264 if method == "tracemin_chol": 
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265 solver = _CholeskySolver(A) 
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266 else: 
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267 solver = _LUSolver(A) 
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268 else: 
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269 raise nx.NetworkXError("Unknown linear system solver: " + method) 
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270 
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271 # Initialize. 
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272 Lnorm = abs(L).sum(axis=1).flatten().max() 
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273 project(X) 
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274 W = ndarray(X.shape, order="F") 
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275 
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276 while True: 
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277 # Orthonormalize X. 
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278 X = qr(X)[0] 
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279 # Compute iteration matrix H. 
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280 W[:, :] = L @ X 
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281 H = X.T @ W 
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282 sigma, Y = eigh(H, overwrite_a=True) 
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283 # Compute the Ritz vectors. 
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284 X = X @ Y 
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285 # Test for convergence exploiting the fact that L * X == W * Y. 
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286 res = dasum(W @ Y[:, 0]  sigma[0] * X[:, 0]) / Lnorm 
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287 if res < tol: 
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288 break 
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289 # Compute X = L \ X / (X' * (L \ X)). 
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290 # L \ X can have an arbitrary projection on the nullspace of L, 
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291 # which will be eliminated. 
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292 W[:, :] = solver.solve(X, tol) 
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293 X = (inv(W.T @ X) @ W.T).T # Preserves Fortran storage order. 
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294 project(X) 
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295 
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296 return sigma, asarray(X) 
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297 
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298 
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299 def _get_fiedler_func(method): 
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300 """Returns a function that solves the Fiedler eigenvalue problem. 
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301 """ 
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302 if method == "tracemin": # old style keyword <v2.1 
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303 method = "tracemin_pcg" 
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304 if method in ("tracemin_pcg", "tracemin_chol", "tracemin_lu"): 
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305 
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306 def find_fiedler(L, x, normalized, tol, seed): 
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307 q = 1 if method == "tracemin_pcg" else min(4, L.shape[0]  1) 
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308 X = asarray(seed.normal(size=(q, L.shape[0]))).T 
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309 sigma, X = _tracemin_fiedler(L, X, normalized, tol, method) 
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310 return sigma[0], X[:, 0] 
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311 
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312 elif method == "lanczos" or method == "lobpcg": 
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313 
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314 def find_fiedler(L, x, normalized, tol, seed): 
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315 L = csc_matrix(L, dtype=float) 
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316 n = L.shape[0] 
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317 if normalized: 
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318 D = spdiags(1.0 / sqrt(L.diagonal()), [0], n, n, format="csc") 
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319 L = D * L * D 
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320 if method == "lanczos" or n < 10: 
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321 # Avoid LOBPCG when n < 10 due to 
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322 # https://github.com/scipy/scipy/issues/3592 
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323 # https://github.com/scipy/scipy/pull/3594 
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324 sigma, X = eigsh(L, 2, which="SM", tol=tol, return_eigenvectors=True) 
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325 return sigma[1], X[:, 1] 
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326 else: 
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327 X = asarray(atleast_2d(x).T) 
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328 M = spdiags(1.0 / L.diagonal(), [0], n, n) 
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329 Y = ones(n) 
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330 if normalized: 
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331 Y /= D.diagonal() 
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332 sigma, X = lobpcg( 
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333 L, X, M=M, Y=atleast_2d(Y).T, tol=tol, maxiter=n, largest=False 
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334 ) 
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335 return sigma[0], X[:, 0] 
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336 
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337 else: 
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338 raise nx.NetworkXError(f"unknown method '{method}'.") 
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339 
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340 return find_fiedler 
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341 
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342 
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343 @random_state(5) 
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344 @not_implemented_for("directed") 
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345 def algebraic_connectivity( 
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346 G, weight="weight", normalized=False, tol=1e8, method="tracemin_pcg", seed=None 
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347 ): 
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348 """Returns the algebraic connectivity of an undirected graph. 
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349 
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350 The algebraic connectivity of a connected undirected graph is the second 
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351 smallest eigenvalue of its Laplacian matrix. 
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352 
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353 Parameters 
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354  
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355 G : NetworkX graph 
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356 An undirected graph. 
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357 
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358 weight : object, optional (default: None) 
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359 The data key used to determine the weight of each edge. If None, then 
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360 each edge has unit weight. 
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361 
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362 normalized : bool, optional (default: False) 
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363 Whether the normalized Laplacian matrix is used. 
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364 
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365 tol : float, optional (default: 1e8) 
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366 Tolerance of relative residual in eigenvalue computation. 
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367 
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368 method : string, optional (default: 'tracemin_pcg') 
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369 Method of eigenvalue computation. It must be one of the tracemin 
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370 options shown below (TraceMIN), 'lanczos' (Lanczos iteration) 
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371 or 'lobpcg' (LOBPCG). 
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372 
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373 The TraceMIN algorithm uses a linear system solver. The following 
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374 values allow specifying the solver to be used. 
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375 
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376 =============== ======================================== 
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377 Value Solver 
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378 =============== ======================================== 
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379 'tracemin_pcg' Preconditioned conjugate gradient method 
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380 'tracemin_chol' Cholesky factorization 
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381 'tracemin_lu' LU factorization 
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382 =============== ======================================== 
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383 
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384 seed : integer, random_state, or None (default) 
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385 Indicator of random number generation state. 
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386 See :ref:`Randomness<randomness>`. 
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387 
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388 Returns 
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389  
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390 algebraic_connectivity : float 
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391 Algebraic connectivity. 
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392 
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393 Raises 
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394  
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395 NetworkXNotImplemented 
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396 If G is directed. 
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397 
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398 NetworkXError 
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399 If G has less than two nodes. 
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400 
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401 Notes 
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402  
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403 Edge weights are interpreted by their absolute values. For MultiGraph's, 
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404 weights of parallel edges are summed. Zeroweighted edges are ignored. 
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405 
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406 To use Cholesky factorization in the TraceMIN algorithm, the 
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407 :samp:`scikits.sparse` package must be installed. 
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408 
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409 See Also 
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410  
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411 laplacian_matrix 
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412 """ 
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413 if len(G) < 2: 
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414 raise nx.NetworkXError("graph has less than two nodes.") 
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415 G = _preprocess_graph(G, weight) 
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416 if not nx.is_connected(G): 
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417 return 0.0 
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418 
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419 L = nx.laplacian_matrix(G) 
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420 if L.shape[0] == 2: 
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421 return 2.0 * L[0, 0] if not normalized else 2.0 
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422 
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423 find_fiedler = _get_fiedler_func(method) 
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424 x = None if method != "lobpcg" else _rcm_estimate(G, G) 
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425 sigma, fiedler = find_fiedler(L, x, normalized, tol, seed) 
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426 return sigma 
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427 
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428 
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429 @random_state(5) 
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430 @not_implemented_for("directed") 
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431 def fiedler_vector( 
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432 G, weight="weight", normalized=False, tol=1e8, method="tracemin_pcg", seed=None 
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433 ): 
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434 """Returns the Fiedler vector of a connected undirected graph. 
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435 
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436 The Fiedler vector of a connected undirected graph is the eigenvector 
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437 corresponding to the second smallest eigenvalue of the Laplacian matrix of 
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438 of the graph. 
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439 
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440 Parameters 
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441  
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442 G : NetworkX graph 
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443 An undirected graph. 
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444 
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445 weight : object, optional (default: None) 
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446 The data key used to determine the weight of each edge. If None, then 
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447 each edge has unit weight. 
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448 
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449 normalized : bool, optional (default: False) 
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450 Whether the normalized Laplacian matrix is used. 
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451 
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452 tol : float, optional (default: 1e8) 
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453 Tolerance of relative residual in eigenvalue computation. 
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454 
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455 method : string, optional (default: 'tracemin_pcg') 
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456 Method of eigenvalue computation. It must be one of the tracemin 
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457 options shown below (TraceMIN), 'lanczos' (Lanczos iteration) 
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458 or 'lobpcg' (LOBPCG). 
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459 
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460 The TraceMIN algorithm uses a linear system solver. The following 
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461 values allow specifying the solver to be used. 
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462 
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463 =============== ======================================== 
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464 Value Solver 
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465 =============== ======================================== 
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466 'tracemin_pcg' Preconditioned conjugate gradient method 
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467 'tracemin_chol' Cholesky factorization 
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468 'tracemin_lu' LU factorization 
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469 =============== ======================================== 
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470 
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471 seed : integer, random_state, or None (default) 
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472 Indicator of random number generation state. 
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473 See :ref:`Randomness<randomness>`. 
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474 
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475 Returns 
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476  
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477 fiedler_vector : NumPy array of floats. 
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478 Fiedler vector. 
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479 
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480 Raises 
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481  
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482 NetworkXNotImplemented 
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483 If G is directed. 
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484 
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485 NetworkXError 
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486 If G has less than two nodes or is not connected. 
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487 
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488 Notes 
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489  
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490 Edge weights are interpreted by their absolute values. For MultiGraph's, 
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491 weights of parallel edges are summed. Zeroweighted edges are ignored. 
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492 
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493 To use Cholesky factorization in the TraceMIN algorithm, the 
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494 :samp:`scikits.sparse` package must be installed. 
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495 
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496 See Also 
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497  
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498 laplacian_matrix 
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499 """ 
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500 if len(G) < 2: 
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501 raise nx.NetworkXError("graph has less than two nodes.") 
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502 G = _preprocess_graph(G, weight) 
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503 if not nx.is_connected(G): 
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504 raise nx.NetworkXError("graph is not connected.") 
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505 
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506 if len(G) == 2: 
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507 return array([1.0, 1.0]) 
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508 
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509 find_fiedler = _get_fiedler_func(method) 
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510 L = nx.laplacian_matrix(G) 
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511 x = None if method != "lobpcg" else _rcm_estimate(G, G) 
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512 sigma, fiedler = find_fiedler(L, x, normalized, tol, seed) 
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513 return fiedler 
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514 
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515 
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516 @random_state(5) 
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517 def spectral_ordering( 
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518 G, weight="weight", normalized=False, tol=1e8, method="tracemin_pcg", seed=None 
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519 ): 
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520 """Compute the spectral_ordering of a graph. 
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521 
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522 The spectral ordering of a graph is an ordering of its nodes where nodes 
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523 in the same weakly connected components appear contiguous and ordered by 
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524 their corresponding elements in the Fiedler vector of the component. 
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525 
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526 Parameters 
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527  
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528 G : NetworkX graph 
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529 A graph. 
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530 
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531 weight : object, optional (default: None) 
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532 The data key used to determine the weight of each edge. If None, then 
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533 each edge has unit weight. 
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534 
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535 normalized : bool, optional (default: False) 
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536 Whether the normalized Laplacian matrix is used. 
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537 
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538 tol : float, optional (default: 1e8) 
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539 Tolerance of relative residual in eigenvalue computation. 
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540 
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541 method : string, optional (default: 'tracemin_pcg') 
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542 Method of eigenvalue computation. It must be one of the tracemin 
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543 options shown below (TraceMIN), 'lanczos' (Lanczos iteration) 
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544 or 'lobpcg' (LOBPCG). 
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545 
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546 The TraceMIN algorithm uses a linear system solver. The following 
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547 values allow specifying the solver to be used. 
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548 
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549 =============== ======================================== 
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550 Value Solver 
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551 =============== ======================================== 
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552 'tracemin_pcg' Preconditioned conjugate gradient method 
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553 'tracemin_chol' Cholesky factorization 
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554 'tracemin_lu' LU factorization 
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555 =============== ======================================== 
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556 
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557 seed : integer, random_state, or None (default) 
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558 Indicator of random number generation state. 
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559 See :ref:`Randomness<randomness>`. 
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560 
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561 Returns 
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562  
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563 spectral_ordering : NumPy array of floats. 
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564 Spectral ordering of nodes. 
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565 
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566 Raises 
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567  
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568 NetworkXError 
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569 If G is empty. 
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570 
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571 Notes 
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572  
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573 Edge weights are interpreted by their absolute values. For MultiGraph's, 
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574 weights of parallel edges are summed. Zeroweighted edges are ignored. 
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575 
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576 To use Cholesky factorization in the TraceMIN algorithm, the 
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577 :samp:`scikits.sparse` package must be installed. 
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578 
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579 See Also 
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580  
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581 laplacian_matrix 
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582 """ 
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583 if len(G) == 0: 
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584 raise nx.NetworkXError("graph is empty.") 
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585 G = _preprocess_graph(G, weight) 
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586 
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587 find_fiedler = _get_fiedler_func(method) 
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588 order = [] 
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589 for component in nx.connected_components(G): 
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590 size = len(component) 
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591 if size > 2: 
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592 L = nx.laplacian_matrix(G, component) 
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593 x = None if method != "lobpcg" else _rcm_estimate(G, component) 
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594 sigma, fiedler = find_fiedler(L, x, normalized, tol, seed) 
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595 sort_info = zip(fiedler, range(size), component) 
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596 order.extend(u for x, c, u in sorted(sort_info)) 
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597 else: 
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598 order.extend(component) 
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599 
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600 return order 