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# louvain_communities

louvain_communities(G, weight='weight', resolution=1, threshold=1e-07, max_level=None, 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`, or until `max_levels` is reached).

  Be careful with self-loops in the input graph. These are treated as previously reduced communities – as if the process had been started in the middle of the algorithm. Large self-loop edge weights thus represent strong communities and in practice may be hard to add other nodes to. If your input graph edge weights for self-loops do not represent already reduced communities you may want to remove the self-loops before inputting that graph.

  Parameters:
    G NetworkX graph
    weight string 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.

    resolution float, optional (default=1)

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

    threshold float, 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.

    max_level int or None, optional (default=None)

      The maximum number of levels (steps of the algorithm) to compute. Must be a positive integer or None. If None, then there is no max level and the threshold parameter determines the stopping condition.

    seed integer, 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}]