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Source code for networkx.algorithms.community.modularity_max

# modularity_max.py - functions for finding communities based on modularity
# Copyright 2018 Edward L. Platt
# This file is part of NetworkX
# NetworkX is distributed under a BSD license; see LICENSE.txt for more
# information.
# Authors:
#   Edward L. Platt <ed@elplatt.com>
#   - Alter equations for weighted case
#   - Write tests for weighted case
"""Functions for detecting communities based on modularity.
from __future__ import division

import networkx as nx
from networkx.algorithms.community.quality import modularity

from networkx.utils.mapped_queue import MappedQueue

__all__ = [

[docs]def greedy_modularity_communities(G, weight=None): """Find communities in graph using Clauset-Newman-Moore greedy modularity maximization. This method currently supports the Graph class and does not consider edge weights. Greedy modularity maximization begins with each node in its own community and joins the pair of communities that most increases modularity until no such pair exists. Parameters ---------- G : NetworkX graph Returns ------- Yields sets of nodes, one for each community. Examples -------- >>> from networkx.algorithms.community import greedy_modularity_communities >>> G = nx.karate_club_graph() >>> c = list(greedy_modularity_communities(G)) >>> sorted(c[0]) [8, 14, 15, 18, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33] References ---------- .. [1] M. E. J Newman 'Networks: An Introduction', page 224 Oxford University Press 2011. .. [2] Clauset, A., Newman, M. E., & Moore, C. "Finding community structure in very large networks." Physical Review E 70(6), 2004. """ # Count nodes and edges N = len(G.nodes()) m = sum([d.get('weight', 1) for u, v, d in G.edges(data=True)]) q0 = 1.0 / (2.0*m) # Map node labels to contiguous integers label_for_node = dict((i, v) for i, v in enumerate(G.nodes())) node_for_label = dict((label_for_node[i], i) for i in range(N)) # Calculate degrees k_for_label = G.degree(G.nodes(), weight=weight) k = [k_for_label[label_for_node[i]] for i in range(N)] # Initialize community and merge lists communities = dict((i, frozenset([i])) for i in range(N)) merges = [] # Initial modularity partition = [[label_for_node[x] for x in c] for c in communities.values()] q_cnm = modularity(G, partition) # Initialize data structures # CNM Eq 8-9 (Eq 8 was missing a factor of 2 (from A_ij + A_ji) # a[i]: fraction of edges within community i # dq_dict[i][j]: dQ for merging community i, j # dq_heap[i][n] : (-dq, i, j) for communitiy i nth largest dQ # H[n]: (-dq, i, j) for community with nth largest max_j(dQ_ij) a = [k[i]*q0 for i in range(N)] dq_dict = dict( (i, dict( (j, 2*q0 - 2*k[i]*k[j]*q0*q0) for j in [ node_for_label[u] for u in G.neighbors(label_for_node[i])] if j != i)) for i in range(N)) dq_heap = [ MappedQueue([ (-dq, i, j) for j, dq in dq_dict[i].items()]) for i in range(N)] H = MappedQueue([ dq_heap[i].h[0] for i in range(N) if len(dq_heap[i]) > 0]) # Merge communities until we can't improve modularity while len(H) > 1: # Find best merge # Remove from heap of row maxes # Ties will be broken by choosing the pair with lowest min community id try: dq, i, j = H.pop() except IndexError: break dq = -dq # Remove best merge from row i heap dq_heap[i].pop() # Push new row max onto H if len(dq_heap[i]) > 0: H.push(dq_heap[i].h[0]) # If this element was also at the root of row j, we need to remove the # duplicate entry from H if dq_heap[j].h[0] == (-dq, j, i): H.remove((-dq, j, i)) # Remove best merge from row j heap dq_heap[j].remove((-dq, j, i)) # Push new row max onto H if len(dq_heap[j]) > 0: H.push(dq_heap[j].h[0]) else: # Duplicate wasn't in H, just remove from row j heap dq_heap[j].remove((-dq, j, i)) # Stop when change is non-positive if dq <= 0: break # Perform merge communities[j] = frozenset(communities[i] | communities[j]) del communities[i] merges.append((i, j, dq)) # New modularity q_cnm += dq # Get list of communities connected to merged communities i_set = set(dq_dict[i].keys()) j_set = set(dq_dict[j].keys()) all_set = (i_set | j_set) - set([i, j]) both_set = i_set & j_set # Merge i into j and update dQ for k in all_set: # Calculate new dq value if k in both_set: dq_jk = dq_dict[j][k] + dq_dict[i][k] elif k in j_set: dq_jk = dq_dict[j][k] - 2.0*a[i]*a[k] else: # k in i_set dq_jk = dq_dict[i][k] - 2.0*a[j]*a[k] # Update rows j and k for row, col in [(j, k), (k, j)]: # Save old value for finding heap index if k in j_set: d_old = (-dq_dict[row][col], row, col) else: d_old = None # Update dict for j,k only (i is removed below) dq_dict[row][col] = dq_jk # Save old max of per-row heap if len(dq_heap[row]) > 0: d_oldmax = dq_heap[row].h[0] else: d_oldmax = None # Add/update heaps d = (-dq_jk, row, col) if d_old is None: # We're creating a new nonzero element, add to heap dq_heap[row].push(d) else: # Update existing element in per-row heap dq_heap[row].update(d_old, d) # Update heap of row maxes if necessary if d_oldmax is None: # No entries previously in this row, push new max H.push(d) else: # We've updated an entry in this row, has the max changed? if dq_heap[row].h[0] != d_oldmax: H.update(d_oldmax, dq_heap[row].h[0]) # Remove row/col i from matrix i_neighbors = dq_dict[i].keys() for k in i_neighbors: # Remove from dict dq_old = dq_dict[k][i] del dq_dict[k][i] # Remove from heaps if we haven't already if k != j: # Remove both row and column for row, col in [(k, i), (i, k)]: # Check if replaced dq is row max d_old = (-dq_old, row, col) if dq_heap[row].h[0] == d_old: # Update per-row heap and heap of row maxes dq_heap[row].remove(d_old) H.remove(d_old) # Update row max if len(dq_heap[row]) > 0: H.push(dq_heap[row].h[0]) else: # Only update per-row heap dq_heap[row].remove(d_old) del dq_dict[i] # Mark row i as deleted, but keep placeholder dq_heap[i] = MappedQueue() # Merge i into j and update a a[j] += a[i] a[i] = 0 communities = [ frozenset([label_for_node[i] for i in c]) for c in communities.values()] return sorted(communities, key=len, reverse=True)
def _naive_greedy_modularity_communities(G): """Find communities in graph using the greedy modularity maximization. This implementation is O(n^4), much slower than alternatives, but it is provided as an easy-to-understand reference implementation. """ # First create one community for each node communities = list([frozenset([u]) for u in G.nodes()]) # Track merges merges = [] # Greedily merge communities until no improvement is possible old_modularity = None new_modularity = modularity(G, communities) while old_modularity is None or new_modularity > old_modularity: # Save modularity for comparison old_modularity = new_modularity # Find best pair to merge trial_communities = list(communities) to_merge = None for i, u in enumerate(communities): for j, v in enumerate(communities): # Skip i=j and empty communities if j <= i or len(u) == 0 or len(v) == 0: continue # Merge communities u and v trial_communities[j] = u | v trial_communities[i] = frozenset([]) trial_modularity = modularity(G, trial_communities) if trial_modularity >= new_modularity: # Check if strictly better or tie if trial_modularity > new_modularity: # Found new best, save modularity and group indexes new_modularity = trial_modularity to_merge = (i, j, new_modularity - old_modularity) elif ( to_merge and min(i, j) < min(to_merge[0], to_merge[1]) ): # Break ties by choosing pair with lowest min id new_modularity = trial_modularity to_merge = (i, j, new_modularity - old_modularity) # Un-merge trial_communities[i] = u trial_communities[j] = v if to_merge is not None: # If the best merge improves modularity, use it merges.append(to_merge) i, j, dq = to_merge u, v = communities[i], communities[j] communities[j] = u | v communities[i] = frozenset([]) # Remove empty communities and sort communities = [c for c in communities if len(c) > 0] for com in sorted(communities, key=lambda x: len(x), reverse=True): yield com