Efficient Bayesian Community Detection using Non-negative Matrix Factorisation

Identifying overlapping communities in networks is a challenging task. In this work we present a novel approach to community detection that utilises the Bayesian non-negative matrix factorisation (NMF) model to produce a probabilistic output for node memberships. The scheme has the advantage of computational efficiency, soft community membership and an intuitive foundation. We present the performance of the method against a variety of benchmark problems and compare and contrast it to several other algorithms for community detection. Our approach performs favourably compared to other methods at a fraction of the computational costs.

💡 Research Summary

The paper introduces a novel community‑detection algorithm that combines Bayesian inference with non‑negative matrix factorisation (NMF). Traditional NMF approximates an adjacency matrix A by the product of two non‑negative factor matrices W (node‑to‑community strengths) and H (community‑to‑node strengths). While effective for uncovering latent structure, classic NMF provides only a deterministic point estimate and lacks a probabilistic interpretation of community membership.

To address this, the authors place Gaussian‑Gamma priors on the entries of W and H, turning them into random variables. The posterior distribution over the factors, conditioned on the observed network, captures uncertainty and naturally yields soft (overlapping) community assignments. Exact Bayesian inference is intractable, so the authors employ Variational Bayes (VB) to derive a tractable lower bound on the model evidence (the ELBO). By assuming a factorised variational distribution, they obtain closed‑form update equations for the means and variances of W and H. These updates are iterated until the ELBO converges, resulting in an efficient O(NK) algorithm where N is the number of nodes and K the number of communities.

After convergence, the soft membership of node i in community k is computed as

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