Mercurial > repos > shellac > sam_consensus_v3
annotate env/lib/python3.9/site-packages/networkx/algorithms/connectivity/kcomponents.py @ 0:4f3585e2f14b draft default tip
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author | shellac |
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date | Mon, 22 Mar 2021 18:12:50 +0000 |
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1 """ |
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2 Moody and White algorithm for k-components |
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3 """ |
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4 from collections import defaultdict |
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5 from itertools import combinations |
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6 from operator import itemgetter |
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7 |
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8 import networkx as nx |
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9 from networkx.utils import not_implemented_for |
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10 |
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11 # Define the default maximum flow function. |
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12 from networkx.algorithms.flow import edmonds_karp |
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13 |
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14 default_flow_func = edmonds_karp |
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15 |
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16 __all__ = ["k_components"] |
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17 |
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18 |
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19 @not_implemented_for("directed") |
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20 def k_components(G, flow_func=None): |
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21 r"""Returns the k-component 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 2-components, each of which can contain |
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28 one or more 3-components, and so forth. |
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29 |
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30 Parameters |
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31 ---------- |
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32 G : NetworkX graph |
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33 |
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34 flow_func : function |
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35 Function to perform the underlying flow computations. Default value |
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36 :meth:`edmonds_karp`. This function performs better in sparse graphs with |
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37 right tailed degree distributions. :meth:`shortest_augmenting_path` will |
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38 perform better in denser graphs. |
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39 |
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40 Returns |
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41 ------- |
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42 k_components : dict |
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43 Dictionary with all connectivity levels `k` in the input Graph as keys |
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44 and a list of sets of nodes that form a k-component of level `k` as |
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45 values. |
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46 |
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47 Raises |
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48 ------ |
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49 NetworkXNotImplemented |
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50 If the input graph is directed. |
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51 |
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52 Examples |
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53 -------- |
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54 >>> # Petersen graph has 10 nodes and it is triconnected, thus all |
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55 >>> # nodes are in a single component on all three connectivity levels |
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56 >>> G = nx.petersen_graph() |
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57 >>> k_components = nx.k_components(G) |
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58 |
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59 Notes |
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60 ----- |
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61 Moody and White [1]_ (appendix A) provide an algorithm for identifying |
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62 k-components in a graph, which is based on Kanevsky's algorithm [2]_ |
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63 for finding all minimum-size node cut-sets of a graph (implemented in |
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64 :meth:`all_node_cuts` function): |
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65 |
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66 1. Compute node connectivity, k, of the input graph G. |
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67 |
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68 2. Identify all k-cutsets at the current level of connectivity using |
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69 Kanevsky's algorithm. |
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70 |
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71 3. Generate new graph components based on the removal of |
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72 these cutsets. Nodes in a cutset belong to both sides |
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73 of the induced cut. |
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74 |
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75 4. If the graph is neither complete nor trivial, return to 1; |
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76 else end. |
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77 |
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78 This implementation also uses some heuristics (see [3]_ for details) |
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79 to speed up the computation. |
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80 |
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81 See also |
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82 -------- |
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83 node_connectivity |
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84 all_node_cuts |
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85 biconnected_components : special case of this function when k=2 |
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86 k_edge_components : similar to this function, but uses edge-connectivity |
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87 instead of node-connectivity |
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88 |
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89 References |
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90 ---------- |
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91 .. [1] Moody, J. and D. White (2003). Social cohesion and embeddedness: |
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92 A hierarchical conception of social groups. |
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93 American Sociological Review 68(1), 103--28. |
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94 http://www2.asanet.org/journals/ASRFeb03MoodyWhite.pdf |
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95 |
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96 .. [2] Kanevsky, A. (1993). Finding all minimum-size separating vertex |
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97 sets in a graph. Networks 23(6), 533--541. |
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98 http://onlinelibrary.wiley.com/doi/10.1002/net.3230230604/abstract |
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99 |
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100 .. [3] Torrents, J. and F. Ferraro (2015). Structural Cohesion: |
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101 Visualization and Heuristics for Fast Computation. |
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102 https://arxiv.org/pdf/1503.04476v1 |
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103 |
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104 """ |
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105 # Dictionary with connectivity level (k) as keys and a list of |
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106 # sets of nodes that form a k-component as values. Note that |
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107 # k-compoents can overlap (but only k - 1 nodes). |
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108 k_components = defaultdict(list) |
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109 # Define default flow function |
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110 if flow_func is None: |
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111 flow_func = default_flow_func |
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112 # Bicomponents as a base to check for higher order k-components |
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113 for component in nx.connected_components(G): |
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114 # isolated nodes have connectivity 0 |
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115 comp = set(component) |
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116 if len(comp) > 1: |
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117 k_components[1].append(comp) |
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118 bicomponents = [G.subgraph(c) for c in nx.biconnected_components(G)] |
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119 for bicomponent in bicomponents: |
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120 bicomp = set(bicomponent) |
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121 # avoid considering dyads as bicomponents |
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122 if len(bicomp) > 2: |
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123 k_components[2].append(bicomp) |
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124 for B in bicomponents: |
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125 if len(B) <= 2: |
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126 continue |
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127 k = nx.node_connectivity(B, flow_func=flow_func) |
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128 if k > 2: |
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129 k_components[k].append(set(B)) |
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130 # Perform cuts in a DFS like order. |
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131 cuts = list(nx.all_node_cuts(B, k=k, flow_func=flow_func)) |
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132 stack = [(k, _generate_partition(B, cuts, k))] |
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133 while stack: |
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134 (parent_k, partition) = stack[-1] |
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135 try: |
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136 nodes = next(partition) |
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137 C = B.subgraph(nodes) |
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138 this_k = nx.node_connectivity(C, flow_func=flow_func) |
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139 if this_k > parent_k and this_k > 2: |
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140 k_components[this_k].append(set(C)) |
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141 cuts = list(nx.all_node_cuts(C, k=this_k, flow_func=flow_func)) |
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142 if cuts: |
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143 stack.append((this_k, _generate_partition(C, cuts, this_k))) |
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144 except StopIteration: |
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145 stack.pop() |
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146 |
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147 # This is necessary because k-components may only be reported at their |
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148 # maximum k level. But we want to return a dictionary in which keys are |
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149 # connectivity levels and values list of sets of components, without |
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150 # skipping any connectivity level. Also, it's possible that subsets of |
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151 # an already detected k-component appear at a level k. Checking for this |
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152 # in the while loop above penalizes the common case. Thus we also have to |
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153 # _consolidate all connectivity levels in _reconstruct_k_components. |
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154 return _reconstruct_k_components(k_components) |
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155 |
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156 |
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157 def _consolidate(sets, k): |
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158 """Merge sets that share k or more elements. |
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159 |
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160 See: http://rosettacode.org/wiki/Set_consolidation |
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161 |
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162 The iterative python implementation posted there is |
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163 faster than this because of the overhead of building a |
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164 Graph and calling nx.connected_components, but it's not |
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165 clear for us if we can use it in NetworkX because there |
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166 is no licence for the code. |
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167 |
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168 """ |
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169 G = nx.Graph() |
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170 nodes = {i: s for i, s in enumerate(sets)} |
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171 G.add_nodes_from(nodes) |
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172 G.add_edges_from( |
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173 (u, v) for u, v in combinations(nodes, 2) if len(nodes[u] & nodes[v]) >= k |
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174 ) |
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175 for component in nx.connected_components(G): |
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176 yield set.union(*[nodes[n] for n in component]) |
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177 |
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178 |
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179 def _generate_partition(G, cuts, k): |
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180 def has_nbrs_in_partition(G, node, partition): |
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181 for n in G[node]: |
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182 if n in partition: |
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183 return True |
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184 return False |
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185 |
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186 components = [] |
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187 nodes = {n for n, d in G.degree() if d > k} - {n for cut in cuts for n in cut} |
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188 H = G.subgraph(nodes) |
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189 for cc in nx.connected_components(H): |
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190 component = set(cc) |
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191 for cut in cuts: |
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192 for node in cut: |
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193 if has_nbrs_in_partition(G, node, cc): |
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194 component.add(node) |
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195 if len(component) < G.order(): |
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196 components.append(component) |
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197 yield from _consolidate(components, k + 1) |
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198 |
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199 |
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200 def _reconstruct_k_components(k_comps): |
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201 result = dict() |
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202 max_k = max(k_comps) |
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203 for k in reversed(range(1, max_k + 1)): |
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204 if k == max_k: |
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205 result[k] = list(_consolidate(k_comps[k], k)) |
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206 elif k not in k_comps: |
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207 result[k] = list(_consolidate(result[k + 1], k)) |
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208 else: |
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209 nodes_at_k = set.union(*k_comps[k]) |
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210 to_add = [c for c in result[k + 1] if any(n not in nodes_at_k for n in c)] |
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211 if to_add: |
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212 result[k] = list(_consolidate(k_comps[k] + to_add, k)) |
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213 else: |
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214 result[k] = list(_consolidate(k_comps[k], k)) |
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215 return result |
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216 |
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217 |
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218 def build_k_number_dict(kcomps): |
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219 result = {} |
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220 for k, comps in sorted(kcomps.items(), key=itemgetter(0)): |
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221 for comp in comps: |
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222 for node in comp: |
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"planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
shellac
parents:
diff
changeset
|
223 result[node] = k |
4f3585e2f14b
"planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
shellac
parents:
diff
changeset
|
224 return result |