Combinatorial metaplexes and centrality indices for identifying higher-order interactions

TL;DR

This paper introduces the combinatorial metaplex framework to better capture higher-order interactions in complex systems by integrating internal vertex properties with simplicial complexes, enhancing the analysis of multi-unit interactions.

Contribution

It proposes a novel combinatorial metaplex model combining simplicial complexes and concentration layers, allowing for more detailed higher-order interaction analysis considering vertex properties.

Findings

The framework effectively distinguishes higher-order interactions influenced by vertex properties.

Comparison shows the combinatorial metaplex provides richer interaction insights than traditional clique complexes.

Centrality measures reveal new insights into the importance of higher-order structures.

Abstract

Complex systems consist of interacting units whose interactions may be pairwise, involving two units, or higher-order, involving more than two units simultaneously. Graphs capture pairwise interactions and represent such systems as networks, whereas simplicial complexes can capture higher-order interactions (HoIs) and represent them as higher-order networks comprising simplices. In the clique complex construction, HoIs arise whenever vertices form a clique in the underlying graph. In classical graph-theoretic and simplicial-complex models, vertices are treated as structurally indistinguishable objects. However, in many real-world systems vertices possess internal structure, and their intrinsic properties influence the HoIs present in the system. To address this limitation, we introduce the combinatorial metaplex, consisting of two interacting components: an underlying simplicial complex that serves as an admissibility structure specifying boundary-compatible higher-order simplex candidates, and a concentration layer defined by a concentration map assigning a value to each vertex and extending the map to simplices so that the resulting distribution satisfies a conservation relation between vertex weights and facet weights. This concentration layer provides a deterministic threshold rule governing the inclusion of true HoIs. Using facet-mediated adjacency and weighted walks, we define one-parameter families of degree, closeness, and harmonic centralities for non-facet simplices, interpolating between those determined solely by the simplicial complex and those determined by concentration-induced coupling. The framework is illustrated through a representative example, including a comparison of HoIs obtained from the clique complex and the combinatorial metaplex, followed by an analysis of edge centralities within the combinatorial metaplex.