Note

This documents the development version of NetworkX. Documentation for the current release can be found here.

# 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=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. 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  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

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}
```