Defining groups (communities/modules) in networks

Determine a precise and operational formal definition of a "group" (also called a community or module) in the mesoscale organization of complex networks, including multilayer networks, that specifies the structural or dynamical criteria a set of nodes must satisfy to constitute a group.

Background

Community detection seeks to uncover mesoscale structure by grouping nodes with stronger intra-group than inter-group connections or by other generative or flow-based principles. Multiple prominent approaches exist, including multilayer modularity maximization, non-negative tensor factorization, Bayesian stochastic block modeling, and description-length minimization via the map equation. Each relies on an implicit notion of what constitutes a group.

Despite extensive methodological development, there is no single, universally accepted definition of a group. Different criteria (e.g., link density relative to null models, block structure probabilities, or flow persistence) can yield different partitions on the same data. In multilayer systems, added complexity from inter-layer coupling further complicates what a group should mean across layers.

A precise, broadly applicable definition would clarify objectives for algorithm design and evaluation, enable principled comparisons across methods, and guide the interpretation of mesoscale structures identified in empirical multilayer systems.

References

However, the definition of what a group exactly corresponds to is an open problem.

— Multilayer Network Science: from Cells to Societies  (2401.04589 - Artime et al., 2024) in Section 3.4, Communities and modules (Mesoscale Organization)

A key direction for future research is to determine how network motifs contribute to the emergence of functional communities within biologically plausible neural circuits. In particular, it remains to be established which motifs form the structural and dynamical cores of such communities, and how their spatial and functional organization influences community detectability and structure.

— Stochastic dynamics and synchronization in motif-based neuronal networks  (2610.00597 - Jagdev et al., 30 Sep 2026) in Section 5, Discussion and future work, subsection “Future work”

However, efforts to find such modularity have proven difficult, partially because it is not clear in what form we should expect such modularity to appear.

— RouterInterp: Understanding Superposed Specialisation in Mixture of Experts Routing  (2610.11775 - Lasy et al., 8 Oct 2026) in Appendix, Section Modularity in Neural Networks