"""Functions for computing clustering of pairs
"""
import itertools
import networkx as nx
__all__ = [
"clustering",
"average_clustering",
"latapy_clustering",
"robins_alexander_clustering",
]
def cc_dot(nu, nv):
return float(len(nu & nv)) / len(nu | nv)
def cc_max(nu, nv):
return float(len(nu & nv)) / max(len(nu), len(nv))
def cc_min(nu, nv):
return float(len(nu & nv)) / min(len(nu), len(nv))
modes = {"dot": cc_dot, "min": cc_min, "max": cc_max}
[docs]def latapy_clustering(G, nodes=None, mode="dot"):
r"""Compute a bipartite clustering coefficient for nodes.
The bipartie clustering coefficient is a measure of local density
of connections defined as [1]_:
.. math::
c_u = \frac{\sum_{v \in N(N(u))} c_{uv} }{|N(N(u))|}
where `N(N(u))` are the second order neighbors of `u` in `G` excluding `u`,
and `c_{uv}` is the pairwise clustering coefficient between nodes
`u` and `v`.
The mode selects the function for `c_{uv}` which can be:
`dot`:
.. math::
c_{uv}=\frac{|N(u)\cap N(v)|}{|N(u) \cup N(v)|}
`min`:
.. math::
c_{uv}=\frac{|N(u)\cap N(v)|}{min(|N(u)|,|N(v)|)}
`max`:
.. math::
c_{uv}=\frac{|N(u)\cap N(v)|}{max(|N(u)|,|N(v)|)}
Parameters
----------
G : graph
A bipartite graph
nodes : list or iterable (optional)
Compute bipartite clustering for these nodes. The default
is all nodes in G.
mode : string
The pariwise bipartite clustering method to be used in the computation.
It must be "dot", "max", or "min".
Returns
-------
clustering : dictionary
A dictionary keyed by node with the clustering coefficient value.
Examples
--------
>>> from networkx.algorithms import bipartite
>>> G = nx.path_graph(4) # path graphs are bipartite
>>> c = bipartite.clustering(G)
>>> c[0]
0.5
>>> c = bipartite.clustering(G, mode="min")
>>> c[0]
1.0
See Also
--------
robins_alexander_clustering
square_clustering
average_clustering
References
----------
.. [1] Latapy, Matthieu, Clémence Magnien, and Nathalie Del Vecchio (2008).
Basic notions for the analysis of large two-mode networks.
Social Networks 30(1), 31--48.
"""
if not nx.algorithms.bipartite.is_bipartite(G):
raise nx.NetworkXError("Graph is not bipartite")
try:
cc_func = modes[mode]
except KeyError as e:
raise nx.NetworkXError(
"Mode for bipartite clustering must be: dot, min or max"
) from e
if nodes is None:
nodes = G
ccs = {}
for v in nodes:
cc = 0.0
nbrs2 = {u for nbr in G[v] for u in G[nbr]} - {v}
for u in nbrs2:
cc += cc_func(set(G[u]), set(G[v]))
if cc > 0.0: # len(nbrs2)>0
cc /= len(nbrs2)
ccs[v] = cc
return ccs
clustering = latapy_clustering
[docs]def average_clustering(G, nodes=None, mode="dot"):
r"""Compute the average bipartite clustering coefficient.
A clustering coefficient for the whole graph is the average,
.. math::
C = \frac{1}{n}\sum_{v \in G} c_v,
where `n` is the number of nodes in `G`.
Similar measures for the two bipartite sets can be defined [1]_
.. math::
C_X = \frac{1}{|X|}\sum_{v \in X} c_v,
where `X` is a bipartite set of `G`.
Parameters
----------
G : graph
a bipartite graph
nodes : list or iterable, optional
A container of nodes to use in computing the average.
The nodes should be either the entire graph (the default) or one of the
bipartite sets.
mode : string
The pariwise bipartite clustering method.
It must be "dot", "max", or "min"
Returns
-------
clustering : float
The average bipartite clustering for the given set of nodes or the
entire graph if no nodes are specified.
Examples
--------
>>> from networkx.algorithms import bipartite
>>> G = nx.star_graph(3) # star graphs are bipartite
>>> bipartite.average_clustering(G)
0.75
>>> X, Y = bipartite.sets(G)
>>> bipartite.average_clustering(G, X)
0.0
>>> bipartite.average_clustering(G, Y)
1.0
See Also
--------
clustering
Notes
-----
The container of nodes passed to this function must contain all of the nodes
in one of the bipartite sets ("top" or "bottom") in order to compute
the correct average bipartite clustering coefficients.
See :mod:`bipartite documentation <networkx.algorithms.bipartite>`
for further details on how bipartite graphs are handled in NetworkX.
References
----------
.. [1] Latapy, Matthieu, Clémence Magnien, and Nathalie Del Vecchio (2008).
Basic notions for the analysis of large two-mode networks.
Social Networks 30(1), 31--48.
"""
if nodes is None:
nodes = G
ccs = latapy_clustering(G, nodes=nodes, mode=mode)
return float(sum(ccs[v] for v in nodes)) / len(nodes)
[docs]def robins_alexander_clustering(G):
r"""Compute the bipartite clustering of G.
Robins and Alexander [1]_ defined bipartite clustering coefficient as
four times the number of four cycles `C_4` divided by the number of
three paths `L_3` in a bipartite graph:
.. math::
CC_4 = \frac{4 * C_4}{L_3}
Parameters
----------
G : graph
a bipartite graph
Returns
-------
clustering : float
The Robins and Alexander bipartite clustering for the input graph.
Examples
--------
>>> from networkx.algorithms import bipartite
>>> G = nx.davis_southern_women_graph()
>>> print(round(bipartite.robins_alexander_clustering(G), 3))
0.468
See Also
--------
latapy_clustering
square_clustering
References
----------
.. [1] Robins, G. and M. Alexander (2004). Small worlds among interlocking
directors: Network structure and distance in bipartite graphs.
Computational & Mathematical Organization Theory 10(1), 69–94.
"""
if G.order() < 4 or G.size() < 3:
return 0
L_3 = _threepaths(G)
if L_3 == 0:
return 0
C_4 = _four_cycles(G)
return (4.0 * C_4) / L_3
def _four_cycles(G):
cycles = 0
for v in G:
for u, w in itertools.combinations(G[v], 2):
cycles += len((set(G[u]) & set(G[w])) - {v})
return cycles / 4
def _threepaths(G):
paths = 0
for v in G:
for u in G[v]:
for w in set(G[u]) - {v}:
paths += len(set(G[w]) - {v, u})
# Divide by two because we count each three path twice
# one for each possible starting point
return paths / 2