Combinatorial metaplexes and centrality indices for identifying higher-order interactions
Hiren J. Dhameliya, Udit Raj, Sudeepto Bhattacharya
TL;DR
The combinatorial metaplex is introduced, consisting of an underlying simplicial complex serving as an admissibility structure that specifies boundary-compatible higher-order simplex candidates, and a concentration layer that assigns concentration to vertices, extends it to simplices, and induces a threshold-based rule governing the realisation and weighting of higher-dimensional simplices.
Abstract
Simplicial complexes provide a combinatorial framework for modelling higher-order interactions (HoIs), that is, interactions involving more than two units. In this representation, simplices encode interactions and inclusion relations determine the structural organisation of the system. In classical graph-theoretic and simplicial complex-based models, vertices are treated as structurally homogeneous entities, and their intrinsic properties do not influence the formation of higher-dimensional simplices. However, in many real-world systems, HoIs are influenced by discrete intrinsic properties of the participating units. To address this limitation, we introduce the combinatorial metaplex, consisting of two interacting components: an underlying simplicial complex serving as an admissibility structure that specifies boundary-compatible higher-order simplex candidates, and a concentration layer that assigns concentration to vertices, extends it to simplices, and induces a threshold-based rule governing the realisation and weighting of higher-dimensional simplices. We investigate how simplex concentration influences the emergence of HoIs and identify the true HoIs as those satisfying both combinatorial admissibility and concentration-based realisation. Using facet-mediated adjacency, we define degree and closeness centrality for non-facet simplices. We further introduce a one-parameter family of centralities that interpolates between purely structural connectivity and concentration-induced coupling. This provides a framework for analysing structurally enriched higher-order networks.