# 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>
#
# TODO:
# - 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__ = [
'greedy_modularity_communities',
'_naive_greedy_modularity_communities']
[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