Multiple phases in modularity-based community detection

Detecting communities in a network, based only on the adjacency matrix, is a problem of interest to several scientific disciplines. Recently, Zhang and Moore have introduced an algorithm in [P. Zhang and C. Moore, Proceedings of the National Academy of Sciences 111, 18144 (2014)], called mod-bp, that avoids overfitting the data by optimizing a weighted average of modularity (a popular goodness-of-fit measure in community detection) and entropy (i.e. number of configurations with a given modularity). The adjustment of the relative weight, the “temperature” of the model, is crucial for getting a correct result from mod-bp. In this work we study the many phase transitions that mod-bp may undergo by changing the two parameters of the algorithm: the temperature $T$ and the maximum number of groups $q$. We introduce a new set of order parameters that allow to determine the actual number of groups $\hat{q}$, and we observe on both synthetic and real networks the existence of phases with any $\hat{q} \in {1,q}$, which were unknown before. We discuss how to interpret the results of mod-bp and how to make the optimal choice for the problem of detecting significant communities.

💡 Research Summary

The paper provides a comprehensive study of the modularity‑based community detection algorithm known as mod‑bp, originally introduced by Zhang and Moore (2014). Mod‑bp treats modularity Q as an energy function (E = −mQ) and samples from the Gibbs distribution at a finite temperature T (β = 1/T). Belief propagation (BP) is used to iteratively update messages ψ_{i→k}(t) for each node i and each possible group t (t = 1,…,q). After convergence, node marginals ψ_i(t) are obtained, and the most probable group for each node is assigned via argmax_t ψ_i(t).

The authors focus on the two free parameters of the algorithm: the temperature T and the maximum number of groups q. While the original work identified three regimes—paramagnetic (high T, no community structure), recovery (intermediate T, meaningful communities), and spin‑glass (low T, non‑convergence)—this study reveals that the recovery regime can be subdivided into up to q − 1 additional phases. To detect these finer transitions, the authors introduce a new order parameter: the distance between any two groups k and l, defined as

 d_{kl} = (1/N) ∑_{i=1}^N