Mercurial > repos > shellac > sam_consensus_v3
annotate env/lib/python3.9/sitepackages/networkx/algorithms/approximation/kcomponents.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 """ Fast approximation for kcomponent structure 
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2 """ 
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3 import itertools 
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4 from collections import defaultdict 
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5 from collections.abc import Mapping 
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6 
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7 import networkx as nx 
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8 from networkx.exception import NetworkXError 
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9 from networkx.utils import not_implemented_for 
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10 
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11 from networkx.algorithms.approximation import local_node_connectivity 
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12 
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13 
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14 __all__ = ["k_components"] 
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15 
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16 
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17 not_implemented_for("directed") 
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18 
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19 
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20 def k_components(G, min_density=0.95): 
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21 r"""Returns the approximate kcomponent structure of a graph G. 
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22 
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23 A `k`component is a maximal subgraph of a graph G that has, at least, 
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24 node connectivity `k`: we need to remove at least `k` nodes to break it 
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25 into more components. `k`components have an inherent hierarchical 
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26 structure because they are nested in terms of connectivity: a connected 
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27 graph can contain several 2components, each of which can contain 
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28 one or more 3components, and so forth. 
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29 
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30 This implementation is based on the fast heuristics to approximate 
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31 the `k`component structure of a graph [1]_. Which, in turn, it is based on 
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32 a fast approximation algorithm for finding good lower bounds of the number 
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33 of node independent paths between two nodes [2]_. 
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34 
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35 Parameters 
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36  
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37 G : NetworkX graph 
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38 Undirected graph 
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39 
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40 min_density : Float 
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41 Density relaxation threshold. Default value 0.95 
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42 
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43 Returns 
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44  
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45 k_components : dict 
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46 Dictionary with connectivity level `k` as key and a list of 
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47 sets of nodes that form a kcomponent of level `k` as values. 
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48 
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49 
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50 Examples 
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51  
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52 >>> # Petersen graph has 10 nodes and it is triconnected, thus all 
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53 >>> # nodes are in a single component on all three connectivity levels 
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54 >>> from networkx.algorithms import approximation as apxa 
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55 >>> G = nx.petersen_graph() 
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56 >>> k_components = apxa.k_components(G) 
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57 
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58 Notes 
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59  
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60 The logic of the approximation algorithm for computing the `k`component 
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61 structure [1]_ is based on repeatedly applying simple and fast algorithms 
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62 for `k`cores and biconnected components in order to narrow down the 
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63 number of pairs of nodes over which we have to compute White and Newman's 
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64 approximation algorithm for finding node independent paths [2]_. More 
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65 formally, this algorithm is based on Whitney's theorem, which states 
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66 an inclusion relation among node connectivity, edge connectivity, and 
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67 minimum degree for any graph G. This theorem implies that every 
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68 `k`component is nested inside a `k`edgecomponent, which in turn, 
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69 is contained in a `k`core. Thus, this algorithm computes node independent 
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70 paths among pairs of nodes in each biconnected part of each `k`core, 
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71 and repeats this procedure for each `k` from 3 to the maximal core number 
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72 of a node in the input graph. 
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73 
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74 Because, in practice, many nodes of the core of level `k` inside a 
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75 bicomponent actually are part of a component of level k, the auxiliary 
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76 graph needed for the algorithm is likely to be very dense. Thus, we use 
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77 a complement graph data structure (see `AntiGraph`) to save memory. 
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78 AntiGraph only stores information of the edges that are *not* present 
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79 in the actual auxiliary graph. When applying algorithms to this 
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80 complement graph data structure, it behaves as if it were the dense 
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81 version. 
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82 
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83 See also 
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84  
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85 k_components 
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86 
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87 References 
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88  
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89 .. [1] Torrents, J. and F. Ferraro (2015) Structural Cohesion: 
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90 Visualization and Heuristics for Fast Computation. 
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91 https://arxiv.org/pdf/1503.04476v1 
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92 
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93 .. [2] White, Douglas R., and Mark Newman (2001) A Fast Algorithm for 
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94 NodeIndependent Paths. Santa Fe Institute Working Paper #0107035 
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95 http://eclectic.ss.uci.edu/~drwhite/working.pdf 
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96 
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97 .. [3] Moody, J. and D. White (2003). Social cohesion and embeddedness: 
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98 A hierarchical conception of social groups. 
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99 American Sociological Review 68(1), 10328. 
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100 http://www2.asanet.org/journals/ASRFeb03MoodyWhite.pdf 
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101 
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102 """ 
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103 # Dictionary with connectivity level (k) as keys and a list of 
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104 # sets of nodes that form a kcomponent as values 
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105 k_components = defaultdict(list) 
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106 # make a few functions local for speed 
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107 node_connectivity = local_node_connectivity 
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108 k_core = nx.k_core 
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109 core_number = nx.core_number 
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110 biconnected_components = nx.biconnected_components 
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111 density = nx.density 
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112 combinations = itertools.combinations 
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113 # Exact solution for k = {1,2} 
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114 # There is a linear time algorithm for triconnectivity, if we had an 
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115 # implementation available we could start from k = 4. 
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116 for component in nx.connected_components(G): 
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117 # isolated nodes have connectivity 0 
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118 comp = set(component) 
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119 if len(comp) > 1: 
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120 k_components[1].append(comp) 
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121 for bicomponent in nx.biconnected_components(G): 
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122 # avoid considering dyads as bicomponents 
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123 bicomp = set(bicomponent) 
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124 if len(bicomp) > 2: 
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125 k_components[2].append(bicomp) 
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126 # There is no kcomponent of k > maximum core number 
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127 # \kappa(G) <= \lambda(G) <= \delta(G) 
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128 g_cnumber = core_number(G) 
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129 max_core = max(g_cnumber.values()) 
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130 for k in range(3, max_core + 1): 
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131 C = k_core(G, k, core_number=g_cnumber) 
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132 for nodes in biconnected_components(C): 
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133 # Build a subgraph SG induced by the nodes that are part of 
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134 # each biconnected component of the kcore subgraph C. 
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135 if len(nodes) < k: 
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136 continue 
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137 SG = G.subgraph(nodes) 
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138 # Build auxiliary graph 
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139 H = _AntiGraph() 
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140 H.add_nodes_from(SG.nodes()) 
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141 for u, v in combinations(SG, 2): 
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142 K = node_connectivity(SG, u, v, cutoff=k) 
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143 if k > K: 
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144 H.add_edge(u, v) 
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145 for h_nodes in biconnected_components(H): 
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146 if len(h_nodes) <= k: 
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147 continue 
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148 SH = H.subgraph(h_nodes) 
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149 for Gc in _cliques_heuristic(SG, SH, k, min_density): 
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150 for k_nodes in biconnected_components(Gc): 
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151 Gk = nx.k_core(SG.subgraph(k_nodes), k) 
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152 if len(Gk) <= k: 
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153 continue 
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154 k_components[k].append(set(Gk)) 
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155 return k_components 
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156 
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157 
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158 def _cliques_heuristic(G, H, k, min_density): 
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159 h_cnumber = nx.core_number(H) 
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160 for i, c_value in enumerate(sorted(set(h_cnumber.values()), reverse=True)): 
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161 cands = {n for n, c in h_cnumber.items() if c == c_value} 
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162 # Skip checking for overlap for the highest core value 
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163 if i == 0: 
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164 overlap = False 
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165 else: 
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166 overlap = set.intersection( 
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167 *[{x for x in H[n] if x not in cands} for n in cands] 
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168 ) 
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169 if overlap and len(overlap) < k: 
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170 SH = H.subgraph(cands  overlap) 
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171 else: 
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172 SH = H.subgraph(cands) 
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173 sh_cnumber = nx.core_number(SH) 
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174 SG = nx.k_core(G.subgraph(SH), k) 
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175 while not (_same(sh_cnumber) and nx.density(SH) >= min_density): 
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176 # This subgraph must be writable => .copy() 
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177 SH = H.subgraph(SG).copy() 
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178 if len(SH) <= k: 
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179 break 
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180 sh_cnumber = nx.core_number(SH) 
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181 sh_deg = dict(SH.degree()) 
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182 min_deg = min(sh_deg.values()) 
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183 SH.remove_nodes_from(n for n, d in sh_deg.items() if d == min_deg) 
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184 SG = nx.k_core(G.subgraph(SH), k) 
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185 else: 
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186 yield SG 
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187 
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188 
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189 def _same(measure, tol=0): 
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190 vals = set(measure.values()) 
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191 if (max(vals)  min(vals)) <= tol: 
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192 return True 
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193 return False 
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194 
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195 
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196 class _AntiGraph(nx.Graph): 
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197 """ 
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198 Class for complement graphs. 
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199 
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200 The main goal is to be able to work with big and dense graphs with 
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201 a low memory foodprint. 
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202 
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203 In this class you add the edges that *do not exist* in the dense graph, 
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204 the report methods of the class return the neighbors, the edges and 
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205 the degree as if it was the dense graph. Thus it's possible to use 
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206 an instance of this class with some of NetworkX functions. In this 
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207 case we only use kcore, connected_components, and biconnected_components. 
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208 """ 
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209 
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210 all_edge_dict = {"weight": 1} 
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211 
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212 def single_edge_dict(self): 
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213 return self.all_edge_dict 
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214 
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215 edge_attr_dict_factory = single_edge_dict 
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216 
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217 def __getitem__(self, n): 
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218 """Returns a dict of neighbors of node n in the dense graph. 
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219 
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220 Parameters 
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221  
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222 n : node 
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223 A node in the graph. 
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224 
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225 Returns 
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226  
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227 adj_dict : dictionary 
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228 The adjacency dictionary for nodes connected to n. 
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229 
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230 """ 
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231 all_edge_dict = self.all_edge_dict 
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232 return { 
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233 node: all_edge_dict for node in set(self._adj)  set(self._adj[n])  {n} 
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234 } 
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235 
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236 def neighbors(self, n): 
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237 """Returns an iterator over all neighbors of node n in the 
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238 dense graph. 
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239 """ 
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240 try: 
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241 return iter(set(self._adj)  set(self._adj[n])  {n}) 
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242 except KeyError as e: 
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243 raise NetworkXError(f"The node {n} is not in the graph.") from e 
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244 
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245 class AntiAtlasView(Mapping): 
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246 """An adjacency inner dict for AntiGraph""" 
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247 
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248 def __init__(self, graph, node): 
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249 self._graph = graph 
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250 self._atlas = graph._adj[node] 
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251 self._node = node 
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252 
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253 def __len__(self): 
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254 return len(self._graph)  len(self._atlas)  1 
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255 
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256 def __iter__(self): 
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257 return (n for n in self._graph if n not in self._atlas and n != self._node) 
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258 
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259 def __getitem__(self, nbr): 
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260 nbrs = set(self._graph._adj)  set(self._atlas)  {self._node} 
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261 if nbr in nbrs: 
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262 return self._graph.all_edge_dict 
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263 raise KeyError(nbr) 
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264 
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265 class AntiAdjacencyView(AntiAtlasView): 
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266 """An adjacency outer dict for AntiGraph""" 
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267 
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268 def __init__(self, graph): 
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269 self._graph = graph 
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270 self._atlas = graph._adj 
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271 
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272 def __len__(self): 
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273 return len(self._atlas) 
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274 
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275 def __iter__(self): 
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276 return iter(self._graph) 
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277 
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278 def __getitem__(self, node): 
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279 if node not in self._graph: 
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280 raise KeyError(node) 
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281 return self._graph.AntiAtlasView(self._graph, node) 
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282 
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283 @property 
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284 def adj(self): 
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285 return self.AntiAdjacencyView(self) 
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286 
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287 def subgraph(self, nodes): 
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288 """This subgraph method returns a full AntiGraph. Not a View""" 
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289 nodes = set(nodes) 
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290 G = _AntiGraph() 
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291 G.add_nodes_from(nodes) 
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292 for n in G: 
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293 Gnbrs = G.adjlist_inner_dict_factory() 
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294 G._adj[n] = Gnbrs 
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295 for nbr, d in self._adj[n].items(): 
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296 if nbr in G._adj: 
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297 Gnbrs[nbr] = d 
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298 G._adj[nbr][n] = d 
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299 G.graph = self.graph 
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300 return G 
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301 
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302 class AntiDegreeView(nx.reportviews.DegreeView): 
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303 def __iter__(self): 
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304 all_nodes = set(self._succ) 
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305 for n in self._nodes: 
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306 nbrs = all_nodes  set(self._succ[n])  {n} 
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307 yield (n, len(nbrs)) 
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308 
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309 def __getitem__(self, n): 
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310 nbrs = set(self._succ)  set(self._succ[n])  {n} 
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311 # AntiGraph is a ThinGraph so all edges have weight 1 
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312 return len(nbrs) + (n in nbrs) 
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313 
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314 @property 
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315 def degree(self): 
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316 """Returns an iterator for (node, degree) and degree for single node. 
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317 
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318 The node degree is the number of edges adjacent to the node. 
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319 
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320 Parameters 
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321  
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322 nbunch : iterable container, optional (default=all nodes) 
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323 A container of nodes. The container will be iterated 
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324 through once. 
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325 
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326 weight : string or None, optional (default=None) 
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327 The edge attribute that holds the numerical value used 
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328 as a weight. If None, then each edge has weight 1. 
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329 The degree is the sum of the edge weights adjacent to the node. 
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330 
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331 Returns 
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332  
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333 deg: 
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334 Degree of the node, if a single node is passed as argument. 
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335 nd_iter : an iterator 
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336 The iterator returns twotuples of (node, degree). 
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337 
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338 See Also 
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339  
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340 degree 
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341 
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342 Examples 
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343  
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344 >>> G = nx.path_graph(4) 
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345 >>> G.degree(0) # node 0 with degree 1 
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346 1 
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347 >>> list(G.degree([0, 1])) 
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348 [(0, 1), (1, 2)] 
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349 
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350 """ 
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351 return self.AntiDegreeView(self) 
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352 
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353 def adjacency(self): 
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354 """Returns an iterator of (node, adjacency set) tuples for all nodes 
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355 in the dense graph. 
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356 
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357 This is the fastest way to look at every edge. 
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358 For directed graphs, only outgoing adjacencies are included. 
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359 
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360 Returns 
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361  
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362 adj_iter : iterator 
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363 An iterator of (node, adjacency set) for all nodes in 
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364 the graph. 
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365 
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366 """ 
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367 for n in self._adj: 
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368 yield (n, set(self._adj)  set(self._adj[n])  {n}) 