-
Notifications
You must be signed in to change notification settings - Fork 0
/
louvain.py
268 lines (233 loc) · 10.7 KB
/
louvain.py
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
# Tested on NetworkX 1.11
import networkx as nx
from collections import defaultdict
import random
from itertools import product
def modularity(G, partition):
"""Returns the modularity of the partition of an undirected graph G.
Definition as given in:
M. E. J. Newman. Networks: An Introduction, page 224.
Oxford University Press, 2011.
"""
m = G.size(weight="weight")
degrees = dict(G.degree(weight="weight"))
Q = 0
for community in partition:
for u, v in product(community, repeat=2):
try:
w = G[u][v].get("weight", 1)
except KeyError:
w = 0
if u == v:
# Double count self-loop weight.
w *= 2
Q += w - degrees[u] * degrees[v] / (2 * m)
return Q / (2 * m)
class Louvain:
def __init__(self, G, verbose=False, randomized=False):
# SETTINGS
self.verbose = verbose
self.randomized = randomized
# Create helper to track network statistics.
# We use the coarse_grain_graph in the iterations.
self.tracker = CommunityTracker()
self.original_graph = G
self.coarse_grain_graph = G
# self.community_history keeps track of the community maps
# from each iteration.
self.community_history = []
self.iteration_count = 0
self.finished = False
# Final community map and list of communities created at end.
self.community_map = None
self.communities = None
def run(self):
"""Runs the iterations of the Louvain method until finished then
generates the final community map.
"""
while not self.finished:
self.iterate()
if self.verbose:
print("Finished in {} iterations".format(self.iteration_count))
self.community_map = self.generate_community_map(
self.community_history)
self.communities = self.invert_community_map(self.community_map)
def iterate(self):
"""Performs one iteration of the Louvain method on the current graph G.
For each node we move it to a neighbouring community which causes the
greatest increase in modularity (if there is no such positive change we
leave it where it is). We continue this until no more moves can be done
so we have reached a local modularity optimum.
We then create a new coarse grained graph where each node represents a
community for the next iteration.
"""
self.iteration_count += 1
if self.verbose:
print("Iteration: ", self.iteration_count)
# modified if we have made at least one change overall.
# improved if we have made at least one change in the current pass.
modified = False
improved = True
G = self.coarse_grain_graph
self.tracker.initialize_network_statistics(G)
community_map = self.tracker.node_to_community_map
while improved:
improved = False
nodes = G.nodes()
if self.randomized:
nodes = list(G.nodes())
random.seed()
random.shuffle(nodes)
for node in nodes:
best_delta_Q = 0.0
old_community = community_map[node]
new_community = old_community
neighbour_communities = self.get_neighbour_communities(
G, node, community_map)
# Isolate the current node and find the best neighbouring
# community (including checking the original).
old_incident_weight = neighbour_communities.get(
old_community, 0)
self.tracker.remove(node, old_community, old_incident_weight)
for community, incident_wt in neighbour_communities.items():
delta_Q = self.calculate_delta_Q(
G, node, community, incident_wt)
if delta_Q > best_delta_Q:
best_delta_Q = delta_Q
new_community = community
# Move to the best community and check if we actually improved.
new_incident_weight = neighbour_communities[new_community]
self.tracker.insert(node, new_community, new_incident_weight)
if self.verbose:
message = "Moved node {} from community {} to community {}"
print(message.format(node, old_community, new_community))
if new_community != old_community:
improved = True
modified = True
if modified:
self.relabel_community_map(community_map)
self.community_history.append(community_map)
self.coarse_grain_graph = self.generate_coarse_grain_graph(
G, community_map)
else:
# We didn't modify any nodes so we are finished.
self.finished = True
def get_neighbour_communities(self, G, node, community_map):
"""Returns a dictionary with the neighbouring communities as keys and
incident edge weights between node and the community as values.
"""
neighbour_communities = defaultdict(int)
for neighbour in G[node]:
if neighbour != node:
neighbour_community = community_map[neighbour]
w = G[node][neighbour].get("weight", 1)
neighbour_communities[neighbour_community] += w
return neighbour_communities
def calculate_delta_Q(self, G, node, community, incident_weight):
"""Calculate change in modularity from adding isolated node to
community."""
# Sum of the weights of the links incident to nodes in C.
sigma_tot = self.tracker.community_degrees[community]
# Sum of the weights of the links incident to node i.
k_i = self.tracker.degrees[node]
# Sum of the weights of the links from i to nodes in C.
k_i_in = incident_weight
# Sum of the weights of all the links in the network.
m = self.tracker.m
delta_Q = 2 * k_i_in - sigma_tot * k_i / m
return delta_Q
def generate_coarse_grain_graph(self, G, community_map):
"""Generates new coarse grain graph with each community as a single
node.
Weights between nodes are the sum of all weights between respective
communities and self loops are added for the weights of he internal
edges.
"""
new_graph = nx.Graph()
# Create nodes for each community.
for community in set(community_map.values()):
new_graph.add_node(community)
# Create the combined edges from the individual old edges.
for u, v, w in G.edges_iter(data="weight", default=1):
c1 = community_map[u]
c2 = community_map[v]
new_weight = w
if new_graph.has_edge(c1, c2):
new_weight += new_graph[c1][c2].get("weight", 1)
new_graph.add_edge(c1, c2, weight=new_weight)
return new_graph
def relabel_community_map(self, community_map):
"""Relabels communities to be from 0 to n."""
community_labels = set(community_map.values())
relabelled_communities = {j: i for i, j in enumerate(community_labels)}
for node in community_map:
community_map[node] = relabelled_communities[community_map[node]]
def invert_community_map(self, community_map):
"""Inverts a community map from nodes to communities to a list of
lists of nodes where each list of nodes represents one community.
"""
inverted_community_map = defaultdict(list)
for node in community_map:
inverted_community_map[community_map[node]].append(node)
return list(inverted_community_map.values())
def generate_community_map(self, community_history):
"""Builds the final community map using the history of iterations."""
community_map = {node: node for node in self.original_graph}
for node in community_map:
for iteration in community_history:
# Follow iterations to find final community of node.
community_map[node] = iteration[community_map[node]]
return community_map
def detect_communities(G, verbose=False, randomized=False):
"""Returns the detected communities as a list of lists of nodes
representing each community.
Uses the Louvain heuristic from:
Blondel, V.D. et al. Fast unfolding of communities in
large networks. J. Stat. Mech 10008, 1 - 12(2008).
"""
louvain = Louvain(G, verbose=verbose, randomized=randomized)
louvain.run()
return louvain.communities
class CommunityTracker:
"""Class to keep track of network statistics of the network as the
algorithm progresses and nodes move communities.
"""
def __init__(self):
self.node_to_community_map = None
self.m = 0.0
self.degrees = None
self.self_loops = None
self.community_degrees = None
self.community_self_loops = None
def initialize_network_statistics(self, G):
self.node_to_community_map = {}
self.m = G.size(weight="weight")
self.degrees = {}
self.self_loops = {}
# Sum of the weights of the edges incident to nodes in C.
self.community_degrees = {}
# Sum of the weights of the internal edges in C.
self.community_self_loops = {}
# Initialize all nodes in separate communities.
for community, node in enumerate(G):
self.node_to_community_map[node] = community
degree = G.degree(node, weight="weight")
self.degrees[node] = self.community_degrees[community] = degree
self_loop = 0
if G.has_edge(node, node):
self_loop = G[node][node].get("weight", 1)
self.community_self_loops[community] = self.self_loops[node] = self_loop
def remove(self, node, community, incident_weight):
"""Removes node from community and updates statistics given the
incident weight of edges from node to other nodes in community.
"""
self.community_degrees[community] -= self.degrees[node]
self.community_self_loops[community] -= incident_weight + self.self_loops[node]
self.node_to_community_map[node] = None
def insert(self, node, community, incident_weight):
"""Inserts isolated node into community and updates statistics given
the incident weight of edges from node to other nodes in community.
"""
self.community_degrees[community] += self.degrees[node]
self.community_self_loops[community] += incident_weight + self.self_loops[node]
self.node_to_community_map[node] = community