Formula
Q = (1/2m) · sum_{ij} (A_ij - k_i k_j / (2m)) · δ(c_i, c_j). Where m = edges, A = adjacency, k = degree, c = community.
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Range
[-0.5, 1]. Random assignment ≈ 0. Well-clustered real networks: 0.3-0.7.
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Resolution limit
Communities smaller than √(2m) not resolvable. Multi-resolution modularity via γ parameter.
Extensions
Weighted modularity (weighted edges). Signed modularity (positive + negative edges). Bipartite modularity.