326 lines
9.8 KiB
Python
326 lines
9.8 KiB
Python
# quality.py - functions for measuring partitions of a graph
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#
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# Copyright 2015-2019 NetworkX developers.
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#
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# This file is part of NetworkX.
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#
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# NetworkX is distributed under a BSD license; see LICENSE.txt for more
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# information.
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"""Functions for measuring the quality of a partition (into
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communities).
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"""
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from functools import wraps
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from itertools import product
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import networkx as nx
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from networkx import NetworkXError
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from networkx.utils import not_implemented_for
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from networkx.algorithms.community.community_utils import is_partition
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__all__ = ['coverage', 'modularity', 'performance']
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class NotAPartition(NetworkXError):
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"""Raised if a given collection is not a partition.
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"""
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def __init__(self, G, collection):
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msg = '{} is not a valid partition of the graph {}'
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msg = msg.format(G, collection)
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super(NotAPartition, self).__init__(msg)
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def require_partition(func):
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"""Decorator to check that a valid partition is input to a function
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Raises :exc:`networkx.NetworkXError` if the partition is not valid.
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This decorator should be used on functions whose first two arguments
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are a graph and a partition of the nodes of that graph (in that
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order)::
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>>> @require_partition
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... def foo(G, partition):
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... print('partition is valid!')
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...
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>>> G = nx.complete_graph(5)
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>>> partition = [{0, 1}, {2, 3}, {4}]
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>>> foo(G, partition)
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partition is valid!
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>>> partition = [{0}, {2, 3}, {4}]
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>>> foo(G, partition) # doctest: +IGNORE_EXCEPTION_DETAIL
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Traceback (most recent call last):
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...
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NetworkXError: `partition` is not a valid partition of the nodes of G
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>>> partition = [{0, 1}, {1, 2, 3}, {4}]
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>>> foo(G, partition) # doctest: +IGNORE_EXCEPTION_DETAIL
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Traceback (most recent call last):
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...
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NetworkXError: `partition` is not a valid partition of the nodes of G
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"""
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@wraps(func)
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def new_func(*args, **kw):
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# Here we assume that the first two arguments are (G, partition).
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if not is_partition(*args[:2]):
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raise nx.NetworkXError('`partition` is not a valid partition of'
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' the nodes of G')
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return func(*args, **kw)
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return new_func
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def intra_community_edges(G, partition):
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"""Returns the number of intra-community edges for a partition of `G`.
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Parameters
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----------
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G : NetworkX graph.
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partition : iterable of sets of nodes
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This must be a partition of the nodes of `G`.
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The "intra-community edges" are those edges joining a pair of nodes
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in the same block of the partition.
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"""
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return sum(G.subgraph(block).size() for block in partition)
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def inter_community_edges(G, partition):
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"""Returns the number of inter-community edges for a prtition of `G`.
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according to the given
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partition of the nodes of `G`.
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Parameters
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----------
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G : NetworkX graph.
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partition : iterable of sets of nodes
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This must be a partition of the nodes of `G`.
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The *inter-community edges* are those edges joining a pair of nodes
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in different blocks of the partition.
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Implementation note: this function creates an intermediate graph
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that may require the same amount of memory as that of `G`.
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"""
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# Alternate implementation that does not require constructing a new
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# graph object (but does require constructing an affiliation
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# dictionary):
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#
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# aff = dict(chain.from_iterable(((v, block) for v in block)
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# for block in partition))
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# return sum(1 for u, v in G.edges() if aff[u] != aff[v])
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#
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MG = nx.MultiDiGraph if G.is_directed() else nx.MultiGraph
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return nx.quotient_graph(G, partition, create_using=MG).size()
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def inter_community_non_edges(G, partition):
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"""Returns the number of inter-community non-edges according to the
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given partition of the nodes of `G`.
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`G` must be a NetworkX graph.
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`partition` must be a partition of the nodes of `G`.
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A *non-edge* is a pair of nodes (undirected if `G` is undirected)
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that are not adjacent in `G`. The *inter-community non-edges* are
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those non-edges on a pair of nodes in different blocks of the
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partition.
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Implementation note: this function creates two intermediate graphs,
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which may require up to twice the amount of memory as required to
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store `G`.
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"""
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# Alternate implementation that does not require constructing two
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# new graph objects (but does require constructing an affiliation
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# dictionary):
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#
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# aff = dict(chain.from_iterable(((v, block) for v in block)
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# for block in partition))
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# return sum(1 for u, v in nx.non_edges(G) if aff[u] != aff[v])
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#
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return inter_community_edges(nx.complement(G), partition)
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@not_implemented_for('multigraph')
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@require_partition
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def performance(G, partition):
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"""Returns the performance of a partition.
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The *performance* of a partition is the ratio of the number of
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intra-community edges plus inter-community non-edges with the total
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number of potential edges.
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Parameters
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----------
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G : NetworkX graph
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A simple graph (directed or undirected).
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partition : sequence
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Partition of the nodes of `G`, represented as a sequence of
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sets of nodes. Each block of the partition represents a
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community.
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Returns
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-------
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float
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The performance of the partition, as defined above.
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Raises
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------
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NetworkXError
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If `partition` is not a valid partition of the nodes of `G`.
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References
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----------
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.. [1] Santo Fortunato.
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"Community Detection in Graphs".
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*Physical Reports*, Volume 486, Issue 3--5 pp. 75--174
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<https://arxiv.org/abs/0906.0612>
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"""
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# Compute the number of intra-community edges and inter-community
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# edges.
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intra_edges = intra_community_edges(G, partition)
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inter_edges = inter_community_non_edges(G, partition)
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# Compute the number of edges in the complete graph (directed or
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# undirected, as it depends on `G`) on `n` nodes.
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#
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# (If `G` is an undirected graph, we divide by two since we have
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# double-counted each potential edge. We use integer division since
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# `total_pairs` is guaranteed to be even.)
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n = len(G)
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total_pairs = n * (n - 1)
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if not G.is_directed():
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total_pairs //= 2
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return (intra_edges + inter_edges) / total_pairs
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@require_partition
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def coverage(G, partition):
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"""Returns the coverage of a partition.
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The *coverage* of a partition is the ratio of the number of
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intra-community edges to the total number of edges in the graph.
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Parameters
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----------
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G : NetworkX graph
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partition : sequence
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Partition of the nodes of `G`, represented as a sequence of
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sets of nodes. Each block of the partition represents a
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community.
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Returns
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-------
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float
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The coverage of the partition, as defined above.
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Raises
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------
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NetworkXError
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If `partition` is not a valid partition of the nodes of `G`.
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Notes
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-----
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If `G` is a multigraph, the multiplicity of edges is counted.
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References
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----------
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.. [1] Santo Fortunato.
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"Community Detection in Graphs".
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*Physical Reports*, Volume 486, Issue 3--5 pp. 75--174
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<https://arxiv.org/abs/0906.0612>
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"""
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intra_edges = intra_community_edges(G, partition)
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total_edges = G.number_of_edges()
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return intra_edges / total_edges
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def modularity(G, communities, weight='weight'):
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r"""Returns the modularity of the given partition of the graph.
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Modularity is defined in [1]_ as
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.. math::
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Q = \frac{1}{2m} \sum_{ij} \left( A_{ij} - \frac{k_ik_j}{2m}\right)
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\delta(c_i,c_j)
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where $m$ is the number of edges, $A$ is the adjacency matrix of
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`G`, $k_i$ is the degree of $i$ and $\delta(c_i, c_j)$
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is 1 if $i$ and $j$ are in the same community and 0 otherwise.
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Parameters
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----------
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G : NetworkX Graph
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communities : list or iterable of set of nodes
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These node sets must represent a partition of G's nodes.
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Returns
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-------
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Q : float
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The modularity of the paritition.
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Raises
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------
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NotAPartition
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If `communities` is not a partition of the nodes of `G`.
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Examples
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--------
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>>> import networkx.algorithms.community as nx_comm
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>>> G = nx.barbell_graph(3, 0)
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>>> nx_comm.modularity(G, [{0, 1, 2}, {3, 4, 5}])
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0.35714285714285704
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>>> nx_comm.modularity(G, nx_comm.label_propagation_communities(G))
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0.35714285714285704
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References
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----------
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.. [1] M. E. J. Newman *Networks: An Introduction*, page 224.
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Oxford University Press, 2011.
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"""
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if not isinstance(communities, list):
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communities = list(communities)
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if not is_partition(G, communities):
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raise NotAPartition(G, communities)
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multigraph = G.is_multigraph()
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directed = G.is_directed()
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m = G.size(weight=weight)
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if directed:
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out_degree = dict(G.out_degree(weight=weight))
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in_degree = dict(G.in_degree(weight=weight))
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norm = 1 / m
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else:
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out_degree = dict(G.degree(weight=weight))
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in_degree = out_degree
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norm = 1 / (2 * m)
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def val(u, v):
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try:
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if multigraph:
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w = sum(d.get(weight, 1) for k, d in G[u][v].items())
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else:
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w = G[u][v].get(weight, 1)
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except KeyError:
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w = 0
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# Double count self-loops if the graph is undirected.
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if u == v and not directed:
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w *= 2
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return w - in_degree[u] * out_degree[v] * norm
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Q = sum(val(u, v) for c in communities for u, v in product(c, repeat=2))
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return Q * norm
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