Topology and spectral interconnectivities of higher-order multilayer networks

Multilayer networks have permeated all the sciences as a powerful mathematical abstraction for interdependent heterogenous complex systems such as multimodal brain connectomes, transportation, ecological systems, and scientific collaboration. But describing such systems through a purely graph-theoretic formalism presupposes that the interactions that define the underlying infrastructures and support their functions are only pairwise-based; a strong assumption likely leading to oversimplifications. Indeed, most interdependent systems intrinsically involve higher-order intra- and inter-layer interactions. For instance, ecological systems involve interactions among groups within and in-between species, collaborations and citations link teams of coauthors to articles and vice versa, interactions might exist among groups of friends from different social networks, etc. While higher-order interactions have been studied for monolayer systems through the language of simplicial complexes and hypergraphs, a broad and systematic formalism incorporating them into the realm of multilayer systems is still lacking. Here, we introduce the concept of crossimplicial multicomplexes as a general formalism for modelling interdependent systems involving higher-order intra- and inter-layer connections. Subsequently, we introduce cross-homology and its spectral counterpart, the cross-Laplacian operators, to establish a rigorous mathematical framework for quantifying global and local intra- and inter-layer topological structures in such systems. When applied to multilayer networks, these cross-Laplacians provide powerful methods for detecting clusters in one layer that are controlled by hubs in another layer. We call such hubs spectral cross-hubs and define spectral persistence as a way to rank them according to their emergence along the cross-Laplacian spectra.