networkx.LFR_benchmark_graph

networkx.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 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

    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:

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 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 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.

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

[R263]“Benchmark graphs for testing community detection algorithms”, Andrea Lancichinetti, Santo Fortunato, and Filippo Radicchi, Phys. Rev. E 78, 046110 2008
[R264]http://santo.fortunato.googlepages.com/inthepress2

Examples

Basic usage:

>>> n = 250
>>> tau1 = 3
>>> tau2 = 1.5
>>> mu = 0.1
>>> G = nx.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.node[v]['community']) for v in G}