# quality.py - functions for measuring partitions of a graph
#
# Copyright 2015-2019 NetworkX developers.
#
# This file is part of NetworkX.
#
# NetworkX is distributed under a BSD license; see LICENSE.txt for more
# information.
"""Functions for measuring the quality of a partition (into
communities).
"""
from __future__ import division
from functools import wraps
from itertools import product
import networkx as nx
from networkx import NetworkXError
from networkx.utils import not_implemented_for
from networkx.algorithms.community.community_utils import is_partition
__all__ = ['coverage', 'modularity', 'performance']
class NotAPartition(NetworkXError):
"""Raised if a given collection is not a partition.
"""
def __init__(self, G, collection):
msg = '{} is not a valid partition of the graph {}'
msg = msg.format(G, collection)
super(NotAPartition, self).__init__(msg)
def require_partition(func):
"""Decorator that raises an exception if a partition is not a valid
partition of the nodes of a graph.
Raises :exc:`networkx.NetworkXError` if the partition is not valid.
This decorator should be used on functions whose first two arguments
are a graph and a partition of the nodes of that graph (in that
order)::
>>> @require_partition
... def foo(G, partition):
... print('partition is valid!')
...
>>> G = nx.complete_graph(5)
>>> partition = [{0, 1}, {2, 3}, {4}]
>>> foo(G, partition)
partition is valid!
>>> partition = [{0}, {2, 3}, {4}]
>>> foo(G, partition) # doctest: +IGNORE_EXCEPTION_DETAIL
Traceback (most recent call last):
...
NetworkXError: `partition` is not a valid partition of the nodes of G
>>> partition = [{0, 1}, {1, 2, 3}, {4}]
>>> foo(G, partition) # doctest: +IGNORE_EXCEPTION_DETAIL
Traceback (most recent call last):
...
NetworkXError: `partition` is not a valid partition of the nodes of G
"""
@wraps(func)
def new_func(*args, **kw):
# Here we assume that the first two arguments are (G, partition).
if not is_partition(*args[:2]):
raise nx.NetworkXError('`partition` is not a valid partition of'
' the nodes of G')
return func(*args, **kw)
return new_func
def intra_community_edges(G, partition):
"""Returns the number of intra-community edges according to the given
partition of the nodes of `G`.
`G` must be a NetworkX graph.
`partition` must be a partition of the nodes of `G`.
The "intra-community edges" are those edges joining a pair of nodes
in the same block of the partition.
"""
return sum(G.subgraph(block).size() for block in partition)
def inter_community_edges(G, partition):
"""Returns the number of inter-community edges according to the given
partition of the nodes of `G`.
`G` must be a NetworkX graph.
`partition` must be a partition of the nodes of `G`.
The *inter-community edges* are those edges joining a pair of nodes
in different blocks of the partition.
Implementation note: this function creates an intermediate graph
that may require the same amount of memory as required to store
`G`.
"""
# Alternate implementation that does not require constructing a new
# graph object (but does require constructing an affiliation
# dictionary):
#
# aff = dict(chain.from_iterable(((v, block) for v in block)
# for block in partition))
# return sum(1 for u, v in G.edges() if aff[u] != aff[v])
#
if G.is_directed():
return nx.quotient_graph(G, partition, create_using=nx.MultiDiGraph()).size()
else:
return nx.quotient_graph(G, partition, create_using=nx.MultiGraph()).size()
def inter_community_non_edges(G, partition):
"""Returns the number of inter-community non-edges according to the
given partition of the nodes of `G`.
`G` must be a NetworkX graph.
`partition` must be a partition of the nodes of `G`.
A *non-edge* is a pair of nodes (undirected if `G` is undirected)
that are not adjacent in `G`. The *inter-community non-edges* are
those non-edges on a pair of nodes in different blocks of the
partition.
Implementation note: this function creates two intermediate graphs,
which may require up to twice the amount of memory as required to
store `G`.
"""
# Alternate implementation that does not require constructing two
# new graph objects (but does require constructing an affiliation
# dictionary):
#
# aff = dict(chain.from_iterable(((v, block) for v in block)
# for block in partition))
# return sum(1 for u, v in nx.non_edges(G) if aff[u] != aff[v])
#
return inter_community_edges(nx.complement(G), partition)
@not_implemented_for('multigraph')
@require_partition
def performance(G, partition):
"""Returns the performance of a partition.
The *performance* of a partition is the ratio of the number of
intra-community edges plus inter-community non-edges with the total
number of potential edges.
Parameters
----------
G : NetworkX graph
A simple graph (directed or undirected).
partition : sequence
Partition of the nodes of `G`, represented as a sequence of
sets of nodes. Each block of the partition represents a
community.
Returns
-------
float
The performance of the partition, as defined above.
Raises
------
NetworkXError
If `partition` is not a valid partition of the nodes of `G`.
References
----------
.. [1] Santo Fortunato.
"Community Detection in Graphs".
*Physical Reports*, Volume 486, Issue 3--5 pp. 75--174
"""
# Compute the number of intra-community edges and inter-community
# edges.
intra_edges = intra_community_edges(G, partition)
inter_edges = inter_community_non_edges(G, partition)
# Compute the number of edges in the complete graph (directed or
# undirected, as it depends on `G`) on `n` nodes.
#
# (If `G` is an undirected graph, we divide by two since we have
# double-counted each potential edge. We use integer division since
# `total_pairs` is guaranteed to be even.)
n = len(G)
total_pairs = n * (n - 1)
if not G.is_directed():
total_pairs //= 2
return (intra_edges + inter_edges) / total_pairs
@require_partition
def coverage(G, partition):
"""Returns the coverage of a partition.
The *coverage* of a partition is the ratio of the number of
intra-community edges to the total number of edges in the graph.
Parameters
----------
G : NetworkX graph
partition : sequence
Partition of the nodes of `G`, represented as a sequence of
sets of nodes. Each block of the partition represents a
community.
Returns
-------
float
The coverage of the partition, as defined above.
Raises
------
NetworkXError
If `partition` is not a valid partition of the nodes of `G`.
Notes
-----
If `G` is a multigraph, the multiplicity of edges is counted.
References
----------
.. [1] Santo Fortunato.
"Community Detection in Graphs".
*Physical Reports*, Volume 486, Issue 3--5 pp. 75--174
"""
intra_edges = intra_community_edges(G, partition)
total_edges = G.number_of_edges()
return intra_edges / total_edges
def modularity(G, communities, weight='weight'):
r"""Returns the modularity of the given partition of the graph.
Modularity is defined in [1]_ as
.. math::
Q = \frac{1}{2m} \sum_{ij} \left( A_{ij} - \frac{k_ik_j}{2m}\right)
\delta(c_i,c_j)
where $m$ is the number of edges, $A$ is the adjacency matrix of
`G`, $k_i$ is the degree of $i$ and $\delta(c_i, c_j)$
is 1 if $i$ and $j$ are in the same community and 0 otherwise.
Parameters
----------
G : NetworkX Graph
communities : list
List of sets of nodes of `G` representing a partition of the
nodes.
Returns
-------
Q : float
The modularity of the paritition.
Raises
------
NotAPartition
If `communities` is not a partition of the nodes of `G`.
Examples
--------
>>> G = nx.barbell_graph(3, 0)
>>> nx.algorithms.community.modularity(G, [{0, 1, 2}, {3, 4, 5}])
0.35714285714285704
References
----------
.. [1] M. E. J. Newman *Networks: An Introduction*, page 224.
Oxford University Press, 2011.
"""
if not is_partition(G, communities):
raise NotAPartition(G, communities)
multigraph = G.is_multigraph()
directed = G.is_directed()
m = G.size(weight=weight)
if directed:
out_degree = dict(G.out_degree(weight=weight))
in_degree = dict(G.in_degree(weight=weight))
norm = 1 / m
else:
out_degree = dict(G.degree(weight=weight))
in_degree = out_degree
norm = 1 / (2 * m)
def val(u, v):
try:
if multigraph:
w = sum(d.get(weight, 1) for k, d in G[u][v].items())
else:
w = G[u][v].get(weight, 1)
except KeyError:
w = 0
# Double count self-loops if the graph is undirected.
if u == v and not directed:
w *= 2
return w - in_degree[u] * out_degree[v] * norm
Q = sum(val(u, v) for c in communities for u, v in product(c, repeat=2))
return Q * norm