"""Functions for detecting communities based on modularity."""
from collections import defaultdict
import networkx as nx
from networkx.algorithms.community.quality import modularity
from networkx.utils.mapped_queue import MappedQueue
from networkx.utils import not_implemented_for
__all__ = [
"greedy_modularity_communities",
"naive_greedy_modularity_communities",
"_naive_greedy_modularity_communities",
]
[docs]def greedy_modularity_communities(G, weight=None, resolution=1, n_communities=1):
r"""Find communities in G using greedy modularity maximization.
This function uses Clauset-Newman-Moore greedy modularity maximization [2]_.
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 or until number of communities `n_communities` is reached.
This function maximizes the generalized modularity, where `resolution`
is the resolution parameter, often expressed as $\gamma$.
See :func:`~networkx.algorithms.community.quality.modularity`.
Parameters
----------
G : NetworkX graph
weight : string or None, optional (default=None)
The name of an edge attribute that holds the numerical value used
as a weight. If None, then each edge has weight 1.
The degree is the sum of the edge weights adjacent to the node.
resolution : float (default=1)
If resolution is less than 1, modularity favors larger communities.
Greater than 1 favors smaller communities.
n_communities: int
Desired number of communities: the community merging process is
terminated once this number of communities is reached, or until
modularity can not be further increased. Must be between 1 and the
total number of nodes in `G`. Default is ``1``, meaning the community
merging process continues until all nodes are in the same community
or until the best community structure is found.
Returns
-------
partition: list
A list of frozensets of nodes, one for each community.
Sorted by length with largest communities first.
Examples
--------
>>> from networkx.algorithms.community import greedy_modularity_communities
>>> G = nx.karate_club_graph()
>>> c = greedy_modularity_communities(G)
>>> sorted(c[0])
[8, 14, 15, 18, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33]
See Also
--------
modularity
References
----------
.. [1] Newman, M. E. J. "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.
.. [3] Reichardt and Bornholdt "Statistical Mechanics of Community
Detection" Phys. Rev. E74, 2006.
.. [4] Newman, M. E. J."Analysis of weighted networks"
Physical Review E 70(5 Pt 2):056131, 2004.
"""
directed = G.is_directed()
N = G.number_of_nodes()
if (n_communities < 1) or (n_communities > N):
raise ValueError(
f"n_communities must be between 1 and {N}. Got {n_communities}"
)
# Count edges (or the sum of edge-weights for weighted graphs)
m = G.size(weight)
q0 = 1 / m
# Calculate degrees (notation from the papers)
# a : the fraction of (weighted) out-degree for each node
# b : the fraction of (weighted) in-degree for each node
if directed:
a = {node: deg_out * q0 for node, deg_out in G.out_degree(weight=weight)}
b = {node: deg_in * q0 for node, deg_in in G.in_degree(weight=weight)}
else:
a = b = {node: deg * q0 * 0.5 for node, deg in G.degree(weight=weight)}
# this preliminary step collects the edge weights for each node pair
# It handles multigraph and digraph and works fine for graph.
dq_dict = defaultdict(lambda: defaultdict(float))
for u, v, wt in G.edges(data=weight, default=1):
if u == v:
continue
dq_dict[u][v] += wt
dq_dict[v][u] += wt
# now scale and subtract the expected edge-weights term
for u, nbrdict in dq_dict.items():
for v, wt in nbrdict.items():
dq_dict[u][v] = q0 * wt - resolution * (a[u] * b[v] + b[u] * a[v])
# Use -dq to get a max_heap instead of a min_heap
# dq_heap holds a heap for each node's neighbors
dq_heap = {u: MappedQueue({(u, v): -dq for v, dq in dq_dict[u].items()}) for u in G}
# H -> all_dq_heap holds a heap with the best items for each node
H = MappedQueue([dq_heap[n].heap[0] for n in G if len(dq_heap[n]) > 0])
# Initialize single-node communities
communities = {n: frozenset([n]) for n in G}
# Merge communities until we can't improve modularity or until desired number of
# communities (n_communities) is reached.
while len(H) > n_communities:
# Find best merge
# Remove from heap of row maxes
# Ties will be broken by choosing the pair with lowest min community id
try:
negdq, u, v = H.pop()
except IndexError:
break
dq = -negdq
# Remove best merge from row u heap
dq_heap[u].pop()
# Push new row max onto H
if len(dq_heap[u]) > 0:
H.push(dq_heap[u].heap[0])
# If this element was also at the root of row v, we need to remove the
# duplicate entry from H
if dq_heap[v].heap[0] == (v, u):
H.remove((v, u))
# Remove best merge from row v heap
dq_heap[v].remove((v, u))
# Push new row max onto H
if len(dq_heap[v]) > 0:
H.push(dq_heap[v].heap[0])
else:
# Duplicate wasn't in H, just remove from row v heap
dq_heap[v].remove((v, u))
# Stop when change is non-positive (no improvement possible)
if dq <= 0:
break
# Perform merge
communities[v] = frozenset(communities[u] | communities[v])
del communities[u]
# Get neighbor communities connected to the merged communities
u_nbrs = set(dq_dict[u])
v_nbrs = set(dq_dict[v])
all_nbrs = (u_nbrs | v_nbrs) - {u, v}
both_nbrs = u_nbrs & v_nbrs
# Update dq for merge of u into v
for w in all_nbrs:
# Calculate new dq value
if w in both_nbrs:
dq_vw = dq_dict[v][w] + dq_dict[u][w]
elif w in v_nbrs:
dq_vw = dq_dict[v][w] - resolution * (a[u] * b[w] + a[w] * b[u])
else: # w in u_nbrs
dq_vw = dq_dict[u][w] - resolution * (a[v] * b[w] + a[w] * b[v])
# Update rows v and w
for row, col in [(v, w), (w, v)]:
dq_heap_row = dq_heap[row]
# Update dict for v,w only (u is removed below)
dq_dict[row][col] = dq_vw
# Save old max of per-row heap
if len(dq_heap_row) > 0:
d_oldmax = dq_heap_row.heap[0]
else:
d_oldmax = None
# Add/update heaps
d = (row, col)
d_negdq = -dq_vw
# Save old value for finding heap index
if w in v_nbrs:
# Update existing element in per-row heap
dq_heap_row.update(d, d, priority=d_negdq)
else:
# We're creating a new nonzero element, add to heap
dq_heap_row.push(d, priority=d_negdq)
# Update heap of row maxes if necessary
if d_oldmax is None:
# No entries previously in this row, push new max
H.push(d, priority=d_negdq)
else:
# We've updated an entry in this row, has the max changed?
row_max = dq_heap_row.heap[0]
if d_oldmax != row_max or d_oldmax.priority != row_max.priority:
H.update(d_oldmax, row_max)
# Remove row/col u from dq_dict matrix
for w in dq_dict[u]:
# Remove from dict
dq_old = dq_dict[w][u]
del dq_dict[w][u]
# Remove from heaps if we haven't already
if w != v:
# Remove both row and column
for row, col in [(w, u), (u, w)]:
dq_heap_row = dq_heap[row]
# Check if replaced dq is row max
d_old = (row, col)
if dq_heap_row.heap[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.heap[0])
else:
# Only update per-row heap
dq_heap_row.remove(d_old)
del dq_dict[u]
# Mark row u as deleted, but keep placeholder
dq_heap[u] = MappedQueue()
# Merge u into v and update a
a[v] += a[u]
a[u] = 0
if directed:
b[v] += b[u]
b[u] = 0
return sorted(communities.values(), key=len, reverse=True)
[docs]@not_implemented_for("directed")
@not_implemented_for("multigraph")
def naive_greedy_modularity_communities(G, resolution=1, weight=None):
r"""Find communities in G using greedy modularity maximization.
This implementation is O(n^4), much slower than alternatives, but it is
provided as an easy-to-understand reference implementation.
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.
This function maximizes the generalized modularity, where `resolution`
is the resolution parameter, often expressed as $\gamma$.
See :func:`~networkx.algorithms.community.quality.modularity`.
Parameters
----------
G : NetworkX graph
resolution : float (default=1)
If resolution is less than 1, modularity favors larger communities.
Greater than 1 favors smaller communities.
weight : string or None, optional (default=None)
The name of an edge attribute that holds the numerical value used
as a weight. If None, then each edge has weight 1.
The degree is the sum of the edge weights adjacent to the node.
Returns
-------
list
A list of sets of nodes, one for each community.
Sorted by length with largest communities first.
Examples
--------
>>> from networkx.algorithms.community import \
... naive_greedy_modularity_communities
>>> G = nx.karate_club_graph()
>>> c = naive_greedy_modularity_communities(G)
>>> sorted(c[0])
[8, 14, 15, 18, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33]
See Also
--------
greedy_modularity_communities
modularity
"""
# 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, resolution=resolution, weight=weight)
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, resolution=resolution, weight=weight
)
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
return sorted((c for c in communities if len(c) > 0), key=len, reverse=True)
# old name
_naive_greedy_modularity_communities = naive_greedy_modularity_communities