modularity(G, communities, weight='weight', resolution=1)
Modularity is defined in 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, $A$ is the adjacency matrix of G, $k_i$ is the 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 (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 . That said, it is very common to simply use gamma=1. More on the choice of gamma is in .
The second formula is the one actually used in calculation of the modularity. For directed graphs the second formula replaces $k_c$ with $k^{in}_c k^{out}_c$.
These node sets must represent a partition of G's nodes.
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.
If resolution is less than 1, modularity favors larger communities. Greater than 1 favors smaller communities.
If :None:None:`communities` is not a partition of the nodes of G.
The modularity of the paritition.
Returns the modularity of the given partition of the graph.
>>> import networkx.algorithms.community as nx_comm
... G = nx.barbell_graph(3, 0)
... nx_comm.modularity(G, [{0, 1, 2}, {3, 4, 5}]) 0.35714285714285715
>>> nx_comm.modularity(G, nx_comm.label_propagation_communities(G)) 0.35714285714285715See :
The following pages refer to to this document either explicitly or contain code examples using this.
networkx.algorithms.community.modularity_max.greedy_modularity_communities
networkx.algorithms.community.modularity_max.naive_greedy_modularity_communities
networkx.algorithms.community.quality.modularity
networkx.algorithms.community.modularity_max._greedy_modularity_communities_generator
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