Combinatorial metaplexes for identifying higher-order interactions and their associated centrality indices in complex networks

Higher-order interactions (HoIs) are ubiquitous in complex systems along with pairwise interactions. Graphs are often used to describe complex systems involving pairwise interactions, where the system is represented as a network. However, simplicial complexes offer a combinatorial representation of HoIs as simplices of varying dimensions and model complex systems as higher-order networks. Many real-world systems have vertices with internal structure, an aspect that remains largely unaddressed in most existing studies. To address this, we introduce the combinatorial metaplex, where each vertex is endowed with a combinatorial structure. Unlike clique complexes, which treat every clique as a higher-order interaction (HoI), the combinatorial metaplex identifies only those simplices as HoIs whose boundary simplices meet a prescribed weight threshold with the help of an induced vertex concentration map. We define a one-parameter family of degree, betweenness, closeness, and harmonic centralities for non-facet simplices in the combinatorial metaplex, using facet-mediated adjacency and weighted walks. These centralities interpolate between those determined solely by the simplicial complex and the combinatorial metaplex. The framework is illustrated through an ecological example comparing the HoIs identified by the clique complex and the combinatorial metaplex, followed by an analysis of edge centralities.