Mercurial > repos > shellac > sam_consensus_v3
annotate env/lib/python3.9/site-packages/networkx/algorithms/community/modularity_max.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 # TODO: |
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2 # - Alter equations for weighted case |
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3 # - Write tests for weighted case |
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4 """Functions for detecting communities based on modularity. |
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5 """ |
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6 |
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7 from networkx.algorithms.community.quality import modularity |
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8 |
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9 from networkx.utils.mapped_queue import MappedQueue |
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10 |
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11 __all__ = [ |
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12 "greedy_modularity_communities", |
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13 "naive_greedy_modularity_communities", |
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14 "_naive_greedy_modularity_communities", |
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15 ] |
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16 |
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17 |
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18 def greedy_modularity_communities(G, weight=None): |
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19 """Find communities in graph using Clauset-Newman-Moore greedy modularity |
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20 maximization. This method currently supports the Graph class and does not |
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21 consider edge weights. |
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22 |
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23 Greedy modularity maximization begins with each node in its own community |
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24 and joins the pair of communities that most increases modularity until no |
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25 such pair exists. |
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26 |
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27 Parameters |
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28 ---------- |
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29 G : NetworkX graph |
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30 |
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31 Returns |
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32 ------- |
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33 Yields sets of nodes, one for each community. |
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34 |
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35 Examples |
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36 -------- |
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37 >>> from networkx.algorithms.community import greedy_modularity_communities |
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38 >>> G = nx.karate_club_graph() |
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39 >>> c = list(greedy_modularity_communities(G)) |
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40 >>> sorted(c[0]) |
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41 [8, 14, 15, 18, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33] |
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42 |
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43 References |
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44 ---------- |
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45 .. [1] M. E. J Newman 'Networks: An Introduction', page 224 |
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46 Oxford University Press 2011. |
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47 .. [2] Clauset, A., Newman, M. E., & Moore, C. |
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48 "Finding community structure in very large networks." |
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49 Physical Review E 70(6), 2004. |
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50 """ |
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51 |
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52 # Count nodes and edges |
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53 N = len(G.nodes()) |
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54 m = sum([d.get("weight", 1) for u, v, d in G.edges(data=True)]) |
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55 q0 = 1.0 / (2.0 * m) |
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56 |
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57 # Map node labels to contiguous integers |
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58 label_for_node = {i: v for i, v in enumerate(G.nodes())} |
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59 node_for_label = {label_for_node[i]: i for i in range(N)} |
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60 |
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61 # Calculate degrees |
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62 k_for_label = G.degree(G.nodes(), weight=weight) |
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63 k = [k_for_label[label_for_node[i]] for i in range(N)] |
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64 |
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65 # Initialize community and merge lists |
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66 communities = {i: frozenset([i]) for i in range(N)} |
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67 merges = [] |
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68 |
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69 # Initial modularity |
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70 partition = [[label_for_node[x] for x in c] for c in communities.values()] |
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71 q_cnm = modularity(G, partition) |
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72 |
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73 # Initialize data structures |
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74 # CNM Eq 8-9 (Eq 8 was missing a factor of 2 (from A_ij + A_ji) |
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75 # a[i]: fraction of edges within community i |
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76 # dq_dict[i][j]: dQ for merging community i, j |
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77 # dq_heap[i][n] : (-dq, i, j) for communitiy i nth largest dQ |
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78 # H[n]: (-dq, i, j) for community with nth largest max_j(dQ_ij) |
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79 a = [k[i] * q0 for i in range(N)] |
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80 dq_dict = { |
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81 i: { |
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82 j: 2 * q0 - 2 * k[i] * k[j] * q0 * q0 |
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83 for j in [node_for_label[u] for u in G.neighbors(label_for_node[i])] |
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84 if j != i |
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85 } |
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86 for i in range(N) |
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87 } |
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88 dq_heap = [ |
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89 MappedQueue([(-dq, i, j) for j, dq in dq_dict[i].items()]) for i in range(N) |
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90 ] |
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91 H = MappedQueue([dq_heap[i].h[0] for i in range(N) if len(dq_heap[i]) > 0]) |
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92 |
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93 # Merge communities until we can't improve modularity |
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94 while len(H) > 1: |
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95 # Find best merge |
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96 # Remove from heap of row maxes |
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97 # Ties will be broken by choosing the pair with lowest min community id |
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98 try: |
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99 dq, i, j = H.pop() |
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100 except IndexError: |
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101 break |
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102 dq = -dq |
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103 # Remove best merge from row i heap |
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104 dq_heap[i].pop() |
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105 # Push new row max onto H |
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106 if len(dq_heap[i]) > 0: |
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107 H.push(dq_heap[i].h[0]) |
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108 # If this element was also at the root of row j, we need to remove the |
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109 # duplicate entry from H |
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110 if dq_heap[j].h[0] == (-dq, j, i): |
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111 H.remove((-dq, j, i)) |
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112 # Remove best merge from row j heap |
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113 dq_heap[j].remove((-dq, j, i)) |
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114 # Push new row max onto H |
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115 if len(dq_heap[j]) > 0: |
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116 H.push(dq_heap[j].h[0]) |
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117 else: |
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118 # Duplicate wasn't in H, just remove from row j heap |
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119 dq_heap[j].remove((-dq, j, i)) |
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120 # Stop when change is non-positive |
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121 if dq <= 0: |
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122 break |
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123 |
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124 # Perform merge |
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125 communities[j] = frozenset(communities[i] | communities[j]) |
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126 del communities[i] |
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127 merges.append((i, j, dq)) |
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128 # New modularity |
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129 q_cnm += dq |
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130 # Get list of communities connected to merged communities |
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131 i_set = set(dq_dict[i].keys()) |
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132 j_set = set(dq_dict[j].keys()) |
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133 all_set = (i_set | j_set) - {i, j} |
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134 both_set = i_set & j_set |
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135 # Merge i into j and update dQ |
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136 for k in all_set: |
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137 # Calculate new dq value |
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138 if k in both_set: |
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139 dq_jk = dq_dict[j][k] + dq_dict[i][k] |
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140 elif k in j_set: |
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141 dq_jk = dq_dict[j][k] - 2.0 * a[i] * a[k] |
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142 else: |
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143 # k in i_set |
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144 dq_jk = dq_dict[i][k] - 2.0 * a[j] * a[k] |
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145 # Update rows j and k |
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146 for row, col in [(j, k), (k, j)]: |
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147 # Save old value for finding heap index |
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148 if k in j_set: |
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149 d_old = (-dq_dict[row][col], row, col) |
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150 else: |
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151 d_old = None |
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152 # Update dict for j,k only (i is removed below) |
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153 dq_dict[row][col] = dq_jk |
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154 # Save old max of per-row heap |
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155 if len(dq_heap[row]) > 0: |
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156 d_oldmax = dq_heap[row].h[0] |
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157 else: |
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158 d_oldmax = None |
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159 # Add/update heaps |
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160 d = (-dq_jk, row, col) |
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161 if d_old is None: |
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162 # We're creating a new nonzero element, add to heap |
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163 dq_heap[row].push(d) |
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164 else: |
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165 # Update existing element in per-row heap |
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166 dq_heap[row].update(d_old, d) |
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167 # Update heap of row maxes if necessary |
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168 if d_oldmax is None: |
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169 # No entries previously in this row, push new max |
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170 H.push(d) |
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171 else: |
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172 # We've updated an entry in this row, has the max changed? |
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173 if dq_heap[row].h[0] != d_oldmax: |
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174 H.update(d_oldmax, dq_heap[row].h[0]) |
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175 |
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176 # Remove row/col i from matrix |
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177 i_neighbors = dq_dict[i].keys() |
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178 for k in i_neighbors: |
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179 # Remove from dict |
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180 dq_old = dq_dict[k][i] |
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181 del dq_dict[k][i] |
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182 # Remove from heaps if we haven't already |
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183 if k != j: |
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184 # Remove both row and column |
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185 for row, col in [(k, i), (i, k)]: |
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186 # Check if replaced dq is row max |
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187 d_old = (-dq_old, row, col) |
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188 if dq_heap[row].h[0] == d_old: |
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189 # Update per-row heap and heap of row maxes |
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190 dq_heap[row].remove(d_old) |
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191 H.remove(d_old) |
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192 # Update row max |
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193 if len(dq_heap[row]) > 0: |
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194 H.push(dq_heap[row].h[0]) |
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195 else: |
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196 # Only update per-row heap |
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197 dq_heap[row].remove(d_old) |
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198 |
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199 del dq_dict[i] |
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200 # Mark row i as deleted, but keep placeholder |
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201 dq_heap[i] = MappedQueue() |
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202 # Merge i into j and update a |
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203 a[j] += a[i] |
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204 a[i] = 0 |
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205 |
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206 communities = [ |
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207 frozenset([label_for_node[i] for i in c]) for c in communities.values() |
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208 ] |
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209 return sorted(communities, key=len, reverse=True) |
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210 |
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211 |
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212 def naive_greedy_modularity_communities(G): |
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213 """Find communities in graph using the greedy modularity maximization. |
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214 This implementation is O(n^4), much slower than alternatives, but it is |
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215 provided as an easy-to-understand reference implementation. |
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216 """ |
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217 # First create one community for each node |
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218 communities = list([frozenset([u]) for u in G.nodes()]) |
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219 # Track merges |
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220 merges = [] |
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221 # Greedily merge communities until no improvement is possible |
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222 old_modularity = None |
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223 new_modularity = modularity(G, communities) |
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224 while old_modularity is None or new_modularity > old_modularity: |
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225 # Save modularity for comparison |
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226 old_modularity = new_modularity |
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227 # Find best pair to merge |
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228 trial_communities = list(communities) |
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229 to_merge = None |
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230 for i, u in enumerate(communities): |
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231 for j, v in enumerate(communities): |
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232 # Skip i=j and empty communities |
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233 if j <= i or len(u) == 0 or len(v) == 0: |
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234 continue |
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235 # Merge communities u and v |
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236 trial_communities[j] = u | v |
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237 trial_communities[i] = frozenset([]) |
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238 trial_modularity = modularity(G, trial_communities) |
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239 if trial_modularity >= new_modularity: |
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240 # Check if strictly better or tie |
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241 if trial_modularity > new_modularity: |
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242 # Found new best, save modularity and group indexes |
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243 new_modularity = trial_modularity |
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244 to_merge = (i, j, new_modularity - old_modularity) |
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245 elif to_merge and min(i, j) < min(to_merge[0], to_merge[1]): |
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246 # Break ties by choosing pair with lowest min id |
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247 new_modularity = trial_modularity |
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248 to_merge = (i, j, new_modularity - old_modularity) |
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249 # Un-merge |
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250 trial_communities[i] = u |
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251 trial_communities[j] = v |
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252 if to_merge is not None: |
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253 # If the best merge improves modularity, use it |
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254 merges.append(to_merge) |
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255 i, j, dq = to_merge |
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256 u, v = communities[i], communities[j] |
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257 communities[j] = u | v |
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258 communities[i] = frozenset([]) |
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259 # Remove empty communities and sort |
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260 communities = [c for c in communities if len(c) > 0] |
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261 yield from sorted(communities, key=lambda x: len(x), reverse=True) |
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262 |
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263 |
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264 # old name |
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265 _naive_greedy_modularity_communities = naive_greedy_modularity_communities |