Stochastic blockmodels and community structure in networks

Stochastic blockmodels have been proposed as a tool for detecting community structure in networks as well as for generating synthetic networks for use as benchmarks. Most blockmodels, however, ignore variation in vertex degree, making them unsuitable for applications to real-world networks, which typically display broad degree distributions that can significantly distort the results. Here we demonstrate how the generalization of blockmodels to incorporate this missing element leads to an improved objective function for community detection in complex networks. We also propose a heuristic algorithm for community detection using this objective function or its non-degree-corrected counterpart and show that the degree-corrected version dramatically outperforms the uncorrected one in both real-world and synthetic networks.

💡 Research Summary

The paper addresses a fundamental limitation of traditional stochastic block models (SBMs) when applied to real‑world networks: the assumption that all vertices have comparable expected degrees. Most empirical networks exhibit broad, often power‑law, degree distributions, and ignoring this heterogeneity can severely bias community detection results. To remedy this, the authors introduce a degree‑corrected stochastic block model (DC‑SBM) that explicitly incorporates a vertex‑specific parameter θ_i representing the expected degree of node i. In the DC‑SBM, the probability of an edge between vertices i and j, belonging to groups g_i and g_j respectively, is given by

P(A_{ij}=1) = θ_i θ_j ω_{g_i g_j},

where ω_{rs} encodes the connectivity strength between groups r and s. This formulation yields a log‑likelihood

L = Σ_{i<j}