Warning

This documents an unmaintained version of NetworkX. Please upgrade to a maintained version and see the current NetworkX documentation.

# Source code for networkx.algorithms.richclub

# -*- coding: utf-8 -*-
#    Aric Hagberg <hagberg@lanl.gov>
#    Dan Schult <dschult@colgate.edu>
#    Pieter Swart <swart@lanl.gov>
#
# Authors: Ben Edwards (bedwards@cs.unm.edu)
#          Aric Hagberg (hagberg@lanl.gov)
"""Functions for computing rich-club coefficients."""
from __future__ import division

import networkx as nx
from networkx.utils import accumulate
from networkx.utils import not_implemented_for

__all__ = ['rich_club_coefficient']

[docs]@not_implemented_for('directed')
@not_implemented_for('multigraph')
def rich_club_coefficient(G, normalized=True, Q=100, seed=None):
r"""Returns the rich-club coefficient of the graph G.

For each degree *k*, the *rich-club coefficient* is the ratio of the
number of actual to the number of potential edges for nodes with
degree greater than *k*:

.. math::

\phi(k) = \frac{2 E_k}{N_k (N_k - 1)}

where N_k is the number of nodes with degree larger than *k*, and
E_k is the number of edges among those nodes.

Parameters
----------
G : NetworkX graph
Undirected graph with neither parallel edges nor self-loops.
normalized : bool (optional)
Normalize using randomized network as in _
Q : float (optional, default=100)
If normalized is True, perform Q * m double-edge
swaps, where m is the number of edges in G, to use as a
null-model for normalization.
seed : integer, random_state, or None (default)
Indicator of random number generation state.
See :ref:Randomness<randomness>.

Returns
-------
rc : dictionary
A dictionary, keyed by degree, with rich-club coefficient values.

Examples
--------
>>> G = nx.Graph([(0, 1), (0, 2), (1, 2), (1, 3), (1, 4), (4, 5)])
>>> rc = nx.rich_club_coefficient(G, normalized=False)
>>> rc # doctest: +SKIP
0.4

Notes
-----
The rich club definition and algorithm are found in _.  This
algorithm ignores any edge weights and is not defined for directed
graphs or graphs with parallel edges or self loops.

Estimates for appropriate values of Q are found in _.

References
----------
..  Julian J. McAuley, Luciano da Fontoura Costa,
and Tibério S. Caetano,
"The rich-club phenomenon across complex network hierarchies",
Applied Physics Letters Vol 91 Issue 8, August 2007.
https://arxiv.org/abs/physics/0701290
..  R. Milo, N. Kashtan, S. Itzkovitz, M. E. J. Newman, U. Alon,
"Uniform generation of random graphs with arbitrary degree
sequences", 2006. https://arxiv.org/abs/cond-mat/0312028
"""
if nx.number_of_selfloops(G) > 0:
raise Exception('rich_club_coefficient is not implemented for '
'graphs with self loops.')
rc = _compute_rc(G)
if normalized:
# make R a copy of G, randomize with Q*|E| double edge swaps
# and use rich_club coefficient of R to normalize
R = G.copy()
E = R.number_of_edges()
nx.double_edge_swap(R, Q * E, max_tries=Q * E * 10, seed=seed)
rcran = _compute_rc(R)
rc = {k: v / rcran[k] for k, v in rc.items()}
return rc

def _compute_rc(G):
"""Returns the rich-club coefficient for each degree in the graph
G.

G is an undirected graph without multiedges.

Returns a dictionary mapping degree to rich-club coefficient for
that degree.

"""
deghist = nx.degree_histogram(G)
total = sum(deghist)
# Compute the number of nodes with degree greater than k, for each
# degree k (omitting the last entry, which is zero).
nks = (total - cs for cs in accumulate(deghist) if total - cs > 1)
# Create a sorted list of pairs of edge endpoint degrees.
#
# The list is sorted in reverse order so that we can pop from the
# right side of the list later, instead of popping from the left
# side of the list, which would have a linear time cost.
edge_degrees = sorted((sorted(map(G.degree, e)) for e in G.edges()),
reverse=True)
ek = G.number_of_edges()
k1, k2 = edge_degrees.pop()
rc = {}
for d, nk in enumerate(nks):
while k1 <= d:
if len(edge_degrees) == 0:
ek = 0
break
k1, k2 = edge_degrees.pop()
ek -= 1
rc[d] = 2 * ek / (nk * (nk - 1))
return rc