"""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 import not_implemented_for
from networkx.utils.mapped_queue import MappedQueue
__all__ = [
"greedy_modularity_communities",
"naive_greedy_modularity_communities",
]
def _greedy_modularity_communities_generator(G, weight=None, resolution=1):
r"""Yield community partitions of G and the modularity change at each step.
This function performs Clauset-Newman-Moore greedy modularity maximization [2]_
At each step of the process it yields the change in modularity that will occur in
the next step followed by yielding the new community partition after that step.
Greedy modularity maximization begins with each node in its own community
and repeatedly joins the pair of communities that lead to the largest
modularity until one community contains all nodes (the partition has one set).
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.
Yields
------
Alternating yield statements produce the following two objects:
communities: dict_values
A dict_values of frozensets of nodes, one for each community.
This represents a partition of the nodes of the graph into communities.
The first yield is the partition with each node in its own community.
dq: float
The change in modularity when merging the next two communities
that leads to the largest modularity.
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()
# 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}
yield communities.values()
# Merge the two communities that lead to the largest 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:
negdq, u, v = H.pop()
except IndexError:
break
dq = -negdq
yield dq
# 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))
# 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
yield communities.values()
[docs]
@nx._dispatchable(edge_attrs="weight")
def greedy_modularity_communities(
G,
weight=None,
resolution=1,
cutoff=1,
best_n=None,
):
r"""Find communities in G using greedy modularity maximization.
This function uses Clauset-Newman-Moore greedy modularity maximization [2]_
to find the community partition with the largest modularity.
Greedy modularity maximization begins with each node in its own community
and repeatedly joins the pair of communities that lead to the largest
modularity until no further increase in modularity is possible (a maximum).
Two keyword arguments adjust the stopping condition. `cutoff` is a lower
limit on the number of communities so you can stop the process before
reaching a maximum (used to save computation time). `best_n` is an upper
limit on the number of communities so you can make the process continue
until at most n communities remain even if the maximum modularity occurs
for more. To obtain exactly n communities, set both `cutoff` and `best_n` to n.
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, optional (default=1)
If resolution is less than 1, modularity favors larger communities.
Greater than 1 favors smaller communities.
cutoff : int, optional (default=1)
A minimum number of communities below which the merging process stops.
The process stops at this number of communities even if modularity
is not maximized. The goal is to let the user stop the process early.
The process stops before the cutoff if it finds a maximum of modularity.
best_n : int or None, optional (default=None)
A maximum number of communities above which the merging process will
not stop. This forces community merging to continue after modularity
starts to decrease until `best_n` communities remain.
If ``None``, don't force it to continue beyond a maximum.
Raises
------
ValueError : If the `cutoff` or `best_n` value is not in the range
``[1, G.number_of_nodes()]``, or if `best_n` < `cutoff`.
Returns
-------
communities: list
A list of frozensets of nodes, one for each community.
Sorted by length with largest communities first.
Examples
--------
>>> G = nx.karate_club_graph()
>>> c = nx.community.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.
"""
if not G.size():
return [{n} for n in G]
if (cutoff < 1) or (cutoff > G.number_of_nodes()):
raise ValueError(f"cutoff must be between 1 and {len(G)}. Got {cutoff}.")
if best_n is not None:
if (best_n < 1) or (best_n > G.number_of_nodes()):
raise ValueError(f"best_n must be between 1 and {len(G)}. Got {best_n}.")
if best_n < cutoff:
raise ValueError(f"Must have best_n >= cutoff. Got {best_n} < {cutoff}")
if best_n == 1:
return [set(G)]
else:
best_n = G.number_of_nodes()
# retrieve generator object to construct output
community_gen = _greedy_modularity_communities_generator(
G, weight=weight, resolution=resolution
)
# construct the first best community
communities = next(community_gen)
# continue merging communities until one of the breaking criteria is satisfied
while len(communities) > cutoff:
try:
dq = next(community_gen)
# StopIteration occurs when communities are the connected components
except StopIteration:
communities = sorted(communities, key=len, reverse=True)
# if best_n requires more merging, merge big sets for highest modularity
while len(communities) > best_n:
comm1, comm2, *rest = communities
communities = [comm1 ^ comm2]
communities.extend(rest)
return communities
# keep going unless max_mod is reached or best_n says to merge more
if dq < 0 and len(communities) <= best_n:
break
communities = next(community_gen)
return sorted(communities, key=len, reverse=True)
[docs]
@not_implemented_for("directed")
@not_implemented_for("multigraph")
@nx._dispatchable(edge_attrs="weight")
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
Graph must be simple and undirected.
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
--------
>>> G = nx.karate_club_graph()
>>> c = nx.community.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 = [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)
