# 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=1e-07, max_iters=500, seed=None)[source]#

Returns the LFR benchmark graph.

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

1. specifying min_degree and not average_degree,

2. 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:
nint

Number of nodes in the created graph.

tau1float

Power law exponent for the degree distribution of the created graph. This value must be strictly greater than one.

tau2float

Power law exponent for the community size distribution in the created graph. This value must be strictly greater than one.

mufloat

Fraction of inter-community edges incident to each node. This value must be in the interval [0, 1].

average_degreefloat

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 NetworkXError is raised.

min_degreeint

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 NetworkXError is raised.

max_degreeint

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_communityint

Minimum size of communities in the graph. If not specified, this is set to min_degree.

max_communityint

Maximum size of communities in the graph. If not specified, this is set to n, the total number of nodes in the graph.

tolfloat

Tolerance when comparing floats, specifically when comparing average degree values.

max_itersint

Maximum number of iterations to try to create the community sizes, degree distribution, and community affiliations.

seedinteger, random_state, or None (default)

Indicator of random number generation state. See Randomness.

Returns:
GNetworkX 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 strictly greater than 1.

• 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.

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

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}