networkx.generators.community.LFR_benchmark_graph¶

LFR_benchmark_graph
(n, tau1, tau2, mu, average_degree=None, min_degree=None, max_degree=None, min_community=None, max_community=None, tol=1e07, max_iters=500, seed=None)[source]¶ Returns the LFR benchmark graph.
This algorithm proceeds as follows:
Find a degree sequence with a power law distribution, and minimum value
min_degree
, which has approximate average degreeaverage_degree
. This is accomplished by eitherspecifying
min_degree
and notaverage_degree
,specifying
average_degree
and notmin_degree
, in which case a suitable minimum degree will be found.
max_degree
can also be specified, otherwise it will be set ton
. Each node u will havemu 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.Generate community sizes according to a power law distribution with exponent
tau2
. Ifmin_community
andmax_community
are not specified they will be selected to bemin_degree
andmax_degree
, respectively. Community sizes are generated until the sum of their sizes equalsn
.Each node will be randomly assigned a community with the condition that the community is large enough for the node’s intracommunity 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.Each node u then adds
(1  mu) mathrm{deg}(u)
intracommunity edges andmu mathrm{deg}(u)
intercommunity 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 intracommunity 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 aNetworkXError
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 aNetworkXError
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 (integer, random_state, or None (default)) – Indicator of random number generation state. See Randomness.
 Returns
G – 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. Return type
NetworkX graph
 Raises
If any of the parameters do not meet their upper and lower bounds:
tau1
andtau2
must be strictly greater than 1.mu
must be in [0, 1].max_degree
must be in {1, …, n}.min_community
andmax_community
must be in {0, …, n}.
If not exactly one of
average_degree
andmin_degree
is specified.If
min_degree
is not specified and a suitablemin_degree
cannot be found.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.generators.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].
Rather than connecting the graph via a configuration model then rewiring to match the intracommunity and intercommunity degrees, we do this wiring explicitly at the end, which should be equivalent.
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