On weakly optimal partitions in modular networks

Modularity was introduced as a measure of goodness for the community structure induced by a partition of the set of vertices in a graph. Then, it also became an objective function used to find good partitions, with high success. Nevertheless, some works have shown a scaling limit and certain instabilities when finding communities with this criterion. Modularity has been studied proposing several formalisms, as hamiltonians in a Potts model or laplacians in spectral partitioning. In this paper we present a new probabilistic formalism to analyze modularity, and from it we derive an algorithm based on weakly optimal partitions. This algorithm obtains good quality partitions and also scales to large graphs.

💡 Research Summary

The paper “On weakly optimal partitions in modular networks” revisits the widely used modularity measure for community detection and proposes a novel probabilistic formalism that leads to a new optimization concept called “weakly optimal partitions”. Traditional modularity, introduced by Newman and Girvan, quantifies the difference between the observed intra‑community edge density and the expected density under a random null model. While modularity has been extremely successful as an objective function, it suffers from well‑known drawbacks: a resolution limit that hides small communities in large graphs, the presence of many local maxima, and instability of the optimization landscape. Existing remedies—such as multi‑resolution parameters, Potts‑model Hamiltonians, or spectral Laplacian methods—have mitigated some issues but often at the cost of increased computational complexity or reduced scalability.

Probabilistic Re‑formulation
The authors reinterpret modularity as an information‑gain term derived from a probabilistic model of edge formation. For a community (c), the expected fraction of edges under the configuration model is ((2k_c/2m)^2), where (k_c) is the sum of degrees of vertices in (c) and (m) is the total number of edges. The modularity contribution of (c) becomes
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