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# modularity

modularity(G, communities, weight='weight', resolution=1)

  Returns the modularity of the given partition of the graph.

  Modularity is defined in [1] as

  \[Q = \frac{1}{2m} \sum_{ij} \left( A_{ij} - \gamma\frac{k_ik_j}{2m}\right) \delta(c_i,c_j)\]

  where \(m\) is the number of edges (or sum of all edge weights as in [5]), \(A\) is the adjacency matrix of `G`, \(k_i\) is the (weighted) degree of \(i\), \(\gamma\) is the resolution parameter, and \(\delta(c_i, c_j)\) is 1 if \(i\) and \(j\) are in the same community else 0.

  According to [2] (and verified by some algebra) this can be reduced to

  \[Q = \sum_{c=1}^{n} \left[ \frac{L_c}{m} - \gamma\left( \frac{k_c}{2m} \right) ^2 \right]\]

  where the sum iterates over all communities \(c\), \(m\) is the number of edges, \(L_c\) is the number of intra-community links for community \(c\), \(k_c\) is the sum of degrees of the nodes in community \(c\), and \(\gamma\) is the resolution parameter.

  The resolution parameter sets an arbitrary tradeoff between intra-group edges and inter-group edges. More complex grouping patterns can be discovered by analyzing the same network with multiple values of gamma and then combining the results [3]. That said, it is very common to simply use gamma=1. More on the choice of gamma is in [4].

  The second formula is the one actually used in calculation of the modularity. For directed graphs the second formula replaces \(k_c\) with \(k^{in}_c k^{out}_c\).

  Parameters:
    G NetworkX Graph
    communities list or iterable of set of nodes

      These node sets must represent a partition of G’s nodes.

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

      The edge attribute that holds the numerical value used as a weight. If None or an edge does not have that attribute, then that edge has weight 1.

    resolution float (default=1)

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

  Returns:
    Q float

      The modularity of the partition.

  Raises:
    NotAPartition

      If `communities` is not a partition of the nodes of `G`.

  References

  [1]

  M. E. J. Newman “Networks: An Introduction”, page 224. Oxford University Press, 2011.

  [2]

  Clauset, Aaron, Mark EJ Newman, and Cristopher Moore. “Finding community structure in very large networks.” Phys. Rev. E 70.6 (2004). <https://arxiv.org/abs/cond-mat/0408187>

  [3]

  Reichardt and Bornholdt “Statistical Mechanics of Community Detection” Phys. Rev. E 74, 016110, 2006. https://doi.org/10.1103/PhysRevE.74.016110

  [4]

  M. E. J. Newman, “Equivalence between modularity optimization and maximum likelihood methods for community detection” Phys. Rev. E 94, 052315, 2016. https://doi.org/10.1103/PhysRevE.94.052315

  [5]

  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

  Examples

      >>> G = nx.barbell_graph(3, 0)
      >>> nx.community.modularity(G, [{0, 1, 2}, {3, 4, 5}])
      0.35714285714285715
      >>> nx.community.modularity(G, nx.community.label_propagation_communities(G))
      0.35714285714285715