# generators.py - functions for generating graphs with community structure
#
# Copyright 2011 Ben Edwards <bedwards@cs.unm.edu>.
# Copyright 2011 Aric Hagberg <hagberg@lanl.gov>
# Copyright 2015 NetworkX developers.
#
# This file is part of NetworkX.
#
# NetworkX is distributed under a BSD license; see LICENSE.txt for more
# information.
"""Functions for generating graphs with community structure."""
from __future__ import division
import random
# HACK In order to accomodate both SciPy and non-SciPy implementations,
# we need to wrap the SciPy implementation of the zeta function with an
# extra parameter, `tolerance`, which will be ignored.
try:
from scipy.special import zeta as _zeta
def zeta(x, q, tolerance):
return _zeta(x, q)
except ImportError:
def zeta(x, q, tolerance):
"""The Hurwitz zeta function, or the Riemann zeta function of two
arguments.
``x`` must be greater than one and ``q`` must be positive.
This function repeatedly computes subsequent partial sums until
convergence, as decided by ``tolerance``.
"""
z = 0
z_prev = -float('inf')
k = 0
while abs(z - z_prev) > tolerance:
z_prev = z
z += 1 / ((k + q) ** x)
k += 1
return z
import networkx as nx
__all__ = ['LFR_benchmark_graph']
def _zipf_rv_below(gamma, xmin, threshold):
"""Returns a random value chosen from the Zipf distribution,
guaranteed to be less than or equal to the value ``threshold``.
Repeatedly draws values from the Zipf distribution until the
threshold is met, then returns that value.
"""
result = nx.utils.zipf_rv(gamma, xmin)
while result > threshold:
result = nx.utils.zipf_rv(gamma, xmin)
return result
def _powerlaw_sequence(gamma, low, high, condition, length, max_iters):
"""Returns a list of numbers obeying a power law distribution, with
some additional restrictions.
``gamma`` and ``low`` are the parameters for the Zipf distribution.
``high`` is the maximum allowed value for values draw from the Zipf
distribution. For more information, see :func:`_zipf_rv_below`.
``condition`` and ``length`` are Boolean-valued functions on
lists. While generating the list, random values are drawn and
appended to the list until ``length`` is satisfied by the created
list. Once ``condition`` is satisfied, the sequence generated in
this way is returned.
``max_iters`` indicates the number of times to generate a list
satisfying ``length``. If the number of iterations exceeds this
value, :exc:`~networkx.exception.ExceededMaxIterations` is raised.
"""
for i in range(max_iters):
seq = []
while not length(seq):
seq.append(_zipf_rv_below(gamma, low, high))
if condition(seq):
return seq
raise nx.ExceededMaxIterations("Could not create power law sequence")
# TODO Needs documentation.
def _generate_min_degree(gamma, average_degree, max_degree, tolerance,
max_iters):
"""Returns a minimum degree from the given average degree."""
min_deg_top = max_degree
min_deg_bot = 1
min_deg_mid = (min_deg_top - min_deg_bot) / 2 + min_deg_bot
itrs = 0
mid_avg_deg = 0
while abs(mid_avg_deg - average_degree) > tolerance:
if itrs > max_iters:
raise nx.ExceededMaxIterations("Could not match average_degree")
mid_avg_deg = 0
for x in range(int(min_deg_mid), max_degree + 1):
mid_avg_deg += (x ** (-gamma + 1)) / zeta(gamma, min_deg_mid,
tolerance)
if mid_avg_deg > average_degree:
min_deg_top = min_deg_mid
min_deg_mid = (min_deg_top - min_deg_bot) / 2 + min_deg_bot
else:
min_deg_bot = min_deg_mid
min_deg_mid = (min_deg_top - min_deg_bot) / 2 + min_deg_bot
itrs += 1
# return int(min_deg_mid + 0.5)
return round(min_deg_mid)
def _generate_communities(degree_sequence, community_sizes, mu, max_iters):
"""Returns a list of sets, each of which represents a community in
the graph.
``degree_sequence`` is the degree sequence that must be met by the
graph.
``community_sizes`` is the community size distribution that must be
met by the generated list of sets.
``mu`` is a float in the interval [0, 1] indicating the fraction of
intra-community edges incident to each node.
``max_iters`` is the number of times to try to add a node to a
community. This must be greater than the length of
``degree_sequence``, otherwise this function will always fail. If
the number of iterations exceeds this value,
:exc:`~networkx.exception.ExceededMaxIterations` is raised.
The communities returned by this are sets of integers in the set {0,
..., *n* - 1}, where *n* is the length of ``degree_sequence``.
"""
# This assumes the nodes in the graph will be natural numbers.
result = [set() for _ in community_sizes]
n = len(degree_sequence)
free = list(range(n))
for i in range(max_iters):
v = free.pop()
c = random.choice(range(len(community_sizes)))
# s = int(degree_sequence[v] * (1 - mu) + 0.5)
s = round(degree_sequence[v] * (1 - mu))
# If the community is large enough, add the node to the chosen
# community. Otherwise, return it to the list of unaffiliated
# nodes.
if s < community_sizes[c]:
result[c].add(v)
else:
free.append(v)
# If the community is too big, remove a node from it.
if len(result[c]) > community_sizes[c]:
free.append(result[c].pop())
if not free:
return result
msg = 'Could not assign communities; try increasing min_community'
raise nx.ExceededMaxIterations(msg)
[docs]def LFR_benchmark_graph(n, tau1, tau2, mu, average_degree=None,
min_degree=None, max_degree=None, min_community=None,
max_community=None, tol=1.0e-7, max_iters=500,
seed=None):
r"""Returns the LFR benchmark graph for testing community-finding
algorithms.
This algorithm proceeds as follows:
1) Find a degree sequence with a power law distribution, and minimum
value ``min_degree``, which has approximate average degree
``average_degree``. This is accomplished by either
a) specifying ``min_degree`` and not ``average_degree``,
b) specifying ``average_degree`` and not ``min_degree``, in which
case a suitable minimum degree will be found.
``max_degree`` can also be specified, otherwise it will be set to
``n``. Each node *u* will have `\mu \mathrm{deg}(u)` edges
joining it to nodes in communities other than its own and `(1 -
\mu) \mathrm{deg}(u)` edges joining it to nodes in its own
community.
2) Generate community sizes according to a power law distribution
with exponent ``tau2``. If ``min_community`` and
``max_community`` are not specified they will be selected to be
``min_degree`` and ``max_degree``, respectively. Community sizes
are generated until the sum of their sizes equals ``n``.
3) Each node will be randomly assigned a community with the
condition that the community is large enough for the node's
intra-community degree, `(1 - \mu) \mathrm{deg}(u)` as
described in step 2. If a community grows too large, a random node
will be selected for reassignment to a new community, until all
nodes have been assigned a community.
4) Each node *u* then adds `(1 - \mu) \mathrm{deg}(u)`
intra-community edges and `\mu \mathrm{deg}(u)` inter-community
edges.
Parameters
----------
n : int
Number of nodes in the created graph.
tau1 : float
Power law exponent for the degree distribution of the created
graph. This value must be strictly greater than one.
tau2 : float
Power law exponent for the community size distribution in the
created graph. This value must be strictly greater than one.
mu : float
Fraction of intra-community edges incident to each node. This
value must be in the interval [0, 1].
average_degree : float
Desired average degree of nodes in the created graph. This value
must be in the interval [0, *n*]. Exactly one of this and
``min_degree`` must be specified, otherwise a
:exc:`NetworkXError` is raised.
min_degree : int
Minimum degree of nodes in the created graph. This value must be
in the interval [0, *n*]. Exactly one of this and
``average_degree`` must be specified, otherwise a
:exc:`NetworkXError` is raised.
max_degree : int
Maximum degree of nodes in the created graph. If not specified,
this is set to ``n``, the total number of nodes in the graph.
min_community : int
Minimum size of communities in the graph. If not specified, this
is set to ``min_degree``.
max_community : int
Maximum size of communities in the graph. If not specified, this
is set to ``n``, the total number of nodes in the graph.
tol : float
Tolerance when comparing floats, specifically when comparing
average degree values.
max_iters : int
Maximum number of iterations to try to create the community sizes,
degree distribution, and community affiliations.
seed : int
A seed for the random number generator.
Returns
-------
G : NetworkX graph
The LFR benchmark graph generated according to the specified
parameters.
Each node in the graph has a node attribute ``'community'`` that
stores the community (that is, the set of nodes) that includes
it.
Raises
------
NetworkXError
If any of the parameters do not meet their upper and lower bounds:
- ``tau1`` and ``tau2`` must be less than or equal to one.
- ``mu`` must be in [0, 1].
- ``max_degree`` must be in {1, ..., *n*}.
- ``min_community`` and ``max_community`` must be in {0, ...,
*n*}.
If not exactly one of ``average_degree`` and ``min_degree`` is
specified.
If ``min_degree`` is not specified and a suitable ``min_degree``
cannot be found.
ExceededMaxIterations
If a valid degree sequence cannot be created within
``max_iters`` number of iterations.
If a valid set of community sizes cannot be created within
``max_iters`` number of iterations.
If a valid community assignment cannot be created within ``10 *
n * max_iters`` number of iterations.
Examples
--------
Basic usage::
>>> from networkx.algorithms.community import LFR_benchmark_graph
>>> n = 250
>>> tau1 = 3
>>> tau2 = 1.5
>>> mu = 0.1
>>> G = LFR_benchmark_graph(n, tau1, tau2, mu, average_degree=5,
... min_community=20, seed=10)
Continuing the example above, you can get the communities from the
node attributes of the graph::
>>> communities = {frozenset(G.nodes[v]['community']) for v in G}
Notes
-----
This algorithm differs slightly from the original way it was
presented in [1].
1) Rather than connecting the graph via a configuration model then
rewiring to match the intra-community and inter-community
degrees, we do this wiring explicitly at the end, which should be
equivalent.
2) The code posted on the author's website [2] calculates the random
power law distributed variables and their average using
continuous approximations, whereas we use the discrete
distributions here as both degree and community size are
discrete.
Though the authors describe the algorithm as quite robust, testing
during development indicates that a somewhat narrower parameter set
is likely to successfully produce a graph. Some suggestions have
been provided in the event of exceptions.
References
----------
.. [1] "Benchmark graphs for testing community detection algorithms",
Andrea Lancichinetti, Santo Fortunato, and Filippo Radicchi,
Phys. Rev. E 78, 046110 2008
.. [2] http://santo.fortunato.googlepages.com/inthepress2
"""
# Perform some basic parameter validation.
if seed is not None:
random.seed(seed)
if not tau1 > 1:
raise nx.NetworkXError("tau1 must be greater than one")
if not tau2 > 1:
raise nx.NetworkXError("tau2 must be greater than one")
if not 0 <= mu <= 1:
raise nx.NetworkXError("mu must be in the interval [0, 1]")
# Validate parameters for generating the degree sequence.
if max_degree is None:
max_degree = n
elif not 0 < max_degree <= n:
raise nx.NetworkXError("max_degree must be in the interval (0, n]")
if not ((min_degree is None) ^ (average_degree is None)):
raise nx.NetworkXError("Must assign exactly one of min_degree and"
" average_degree")
if min_degree is None:
min_degree = _generate_min_degree(tau1, average_degree, max_degree,
tol, max_iters)
# Generate a degree sequence with a power law distribution.
low, high = min_degree, max_degree
condition = lambda seq: sum(seq) % 2 == 0
length = lambda seq: len(seq) >= n
deg_seq = _powerlaw_sequence(tau1, low, high, condition, length, max_iters)
# Validate parameters for generating the community size sequence.
if min_community is None:
min_community = min(deg_seq)
if max_community is None:
max_community = max(deg_seq)
# Generate a community size sequence with a power law distribution.
#
# TODO The original code incremented the number of iterations each
# time a new Zipf random value was drawn from the distribution. This
# differed from the way the number of iterations was incremented in
# `_powerlaw_degree_sequence`, so this code was changed to match
# that one. As a result, this code is allowed many more chances to
# generate a valid community size sequence.
low, high = min_community, max_community
condition = lambda seq: sum(seq) == n
length = lambda seq: sum(seq) >= n
comms = _powerlaw_sequence(tau2, low, high, condition, length, max_iters)
# Generate the communities based on the given degree sequence and
# community sizes.
max_iters *= 10 * n
communities = _generate_communities(deg_seq, comms, mu, max_iters)
# Finally, generate the benchmark graph based on the given
# communities, joining nodes according to the intra- and
# inter-community degrees.
G = nx.Graph()
G.add_nodes_from(range(n))
for c in communities:
for u in c:
while G.degree(u) < round(deg_seq[u] * (1 - mu)):
v = random.choice(list(c))
G.add_edge(u, v)
while G.degree(u) < deg_seq[u]:
v = random.choice(range(n))
if v not in c:
G.add_edge(u, v)
G.nodes[u]['community'] = c
return G