louvain_communities

louvain_communities(G, weight='weight', resolution=1, threshold=1e-07, seed=None)

Find the best partition of a graph using the Louvain Community Detection Algorithm.

Louvain Community Detection Algorithm is a simple method to extract the community structure of a network. This is a heuristic method based on modularity optimization. [1]

The algorithm works in 2 steps. On the first step it assigns every node to be in its own community and then for each node it tries to find the maximum positive modularity gain by moving each node to all of its neighbor communities. If no positive gain is achieved the node remains in its original community.

The modularity gain obtained by moving an isolated node \(i\) into a community \(C\) can easily be calculated by the following formula (combining [1] [2] and some algebra):

\[\Delta Q = \frac{k_{i,in}}{2m} - \gamma\frac{ \Sigma_{tot} \cdot k_i}{2m^2}\]

where \(m\) is the size of the graph, \(k_{i,in}\) is the sum of the weights of the links from \(i\) to nodes in \(C\), \(k_i\) is the sum of the weights of the links incident to node \(i\), \(\Sigma_{tot}\) is the sum of the weights of the links incident to nodes in \(C\) and \(\gamma\) is the resolution parameter.

For the directed case the modularity gain can be computed using this formula according to [3]

\[\Delta Q = \frac{k_{i,in}}{m} - \gamma\frac{k_i^{out} \cdot\Sigma_{tot}^{in} + k_i^{in} \cdot \Sigma_{tot}^{out}}{m^2}\]

where \(k_i^{out}\), \(k_i^{in}\) are the outer and inner weighted degrees of node \(i\) and \(\Sigma_{tot}^{in}\), \(\Sigma_{tot}^{out}\) are the sum of in-going and out-going links incident to nodes in \(C\).

The first phase continues until no individual move can improve the modularity.

The second phase consists in building a new network whose nodes are now the communities found in the first phase. To do so, the weights of the links between the new nodes are given by the sum of the weight of the links between nodes in the corresponding two communities. Once this phase is complete it is possible to reapply the first phase creating bigger communities with increased modularity.

The above two phases are executed until no modularity gain is achieved (or is less than the threshold).

Parameters:

GNetworkX graph

weightstring or None, optional (default=”weight”)

The name of an edge attribute that holds the numerical value used as a weight. If None then each edge has weight 1.

resolutionfloat, optional (default=1)

If resolution is less than 1, the algorithm favors larger communities. Greater than 1 favors smaller communities

thresholdfloat, optional (default=0.0000001)

Modularity gain threshold for each level. If the gain of modularity between 2 levels of the algorithm is less than the given threshold then the algorithm stops and returns the resulting communities.

seedinteger, random_state, or None (default)

Indicator of random number generation state. See Randomness.

Returns:

list

A list of sets (partition of G). Each set represents one community and contains all the nodes that constitute it.

See also

louvain_partitions

Notes

The order in which the nodes are considered can affect the final output. In the algorithm the ordering happens using a random shuffle.

References

[1] (1,2)

Blondel, V.D. et al. Fast unfolding of communities in large networks. J. Stat. Mech 10008, 1-12(2008). https://doi.org/10.1088/1742-5468/2008/10/P10008

[2]

Traag, V.A., Waltman, L. & van Eck, N.J. From Louvain to Leiden: guaranteeing well-connected communities. Sci Rep 9, 5233 (2019). https://doi.org/10.1038/s41598-019-41695-z

[3]

Nicolas Dugué, Anthony Perez. Directed Louvain : maximizing modularity in directed networks. [Research Report] Université d’Orléans. 2015. hal-01231784. https://hal.archives-ouvertes.fr/hal-01231784

Examples

>>> import networkx as nx
>>> G = nx.petersen_graph()
>>> nx.community.louvain_communities(G, seed=123)
[{0, 4, 5, 7, 9}, {1, 2, 3, 6, 8}]