"""Functions for measuring the quality of a partition (into
communities).
"""
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 = f"{G} is not a valid partition of the graph {collection}"
super().__init__(msg)
def require_partition(func):
"""Decorator to check that a valid partition is input to a function
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)
Traceback (most recent call last):
...
networkx.exception.NetworkXError: `partition` is not a valid partition of the nodes of G
>>> partition = [{0, 1}, {1, 2, 3}, {4}]
>>> foo(G, partition)
Traceback (most recent call last):
...
networkx.exception.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 for a partition of `G`.
Parameters
----------
G : NetworkX graph.
partition : iterable of sets of nodes
This 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 for a prtition of `G`.
according to the given
partition of the nodes of `G`.
Parameters
----------
G : NetworkX graph.
partition : iterable of sets of nodes
This 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 that of `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])
#
MG = nx.MultiDiGraph if G.is_directed() else nx.MultiGraph
return nx.quotient_graph(G, partition, create_using=MG).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)
[docs]@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
<https://arxiv.org/abs/0906.0612>
"""
intra_edges = intra_community_edges(G, partition)
total_edges = G.number_of_edges()
return intra_edges / total_edges
[docs]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.
According to [2]_ (and verified by some algebra) this can be reduced to
.. math::
Q = \sum_{c=1}^{n}
\left[ \frac{L_c}{m} - \left( \frac{k_c}{2m} \right) ^2 \right]
where the sum iterates over all communities $c$, $m$ is the number of edges,
$L_c$ is the number of intra-community links for community $c$,
$k_c$ is the sum of degrees of the nodes in community $c$.
The second formula is the one actually used in calculation of the modularity.
Parameters
----------
G : NetworkX Graph
communities : list or iterable of set of nodes
These node sets must represent a partition of G's nodes.
weight : string or None, optional (default="weight")
The edge attribute that holds the numerical value used
as a weight. If None or an edge does not have that attribute,
then that edge has weight 1.
Returns
-------
Q : float
The modularity of the paritition.
Raises
------
NotAPartition
If `communities` is not a partition of the nodes of `G`.
Examples
--------
>>> import networkx.algorithms.community as nx_comm
>>> G = nx.barbell_graph(3, 0)
>>> nx_comm.modularity(G, [{0, 1, 2}, {3, 4, 5}])
0.35714285714285715
>>> nx_comm.modularity(G, nx_comm.label_propagation_communities(G))
0.35714285714285715
References
----------
.. [1] M. E. J. Newman *Networks: An Introduction*, page 224.
Oxford University Press, 2011.
.. [2] Clauset, Aaron, Mark EJ Newman, and Cristopher Moore.
"Finding community structure in very large networks."
Physical review E 70.6 (2004). <https://arxiv.org/abs/cond-mat/0408187>
"""
if not isinstance(communities, list):
communities = list(communities)
if not is_partition(G, communities):
raise NotAPartition(G, communities)
directed = G.is_directed()
if directed:
out_degree = dict(G.out_degree(weight=weight))
in_degree = dict(G.in_degree(weight=weight))
m = sum(out_degree.values())
norm = 1 / m ** 2
else:
out_degree = in_degree = dict(G.degree(weight=weight))
deg_sum = sum(out_degree.values())
m = deg_sum / 2
norm = 1 / deg_sum ** 2
def community_contribution(community):
comm = set(community)
L_c = sum(wt for u, v, wt in G.edges(comm, data=weight, default=1) if v in comm)
out_degree_sum = sum(out_degree[u] for u in comm)
in_degree_sum = sum(in_degree[u] for u in comm) if directed else out_degree_sum
return L_c / m - out_degree_sum * in_degree_sum * norm
return sum(map(community_contribution, communities))