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.