Topology and spectral interconnectivities of higher-order multilayer networks - Moutuou, Elkaïoum M.; Ali, Obaï B. K.; Benali, Habib (Frontiers in Complex Systems, 2023)
Multilayer networks have permeated all the sciences as an abstraction for interdependent heterogenous complex systems. But describing such systems through a purely graph-theoretic formalism presupposes that the interactions that define the underlying infrastructures 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 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. Using synthetic and empirical datasets, we show that the spectra of the cross-Laplacians of a multilayer network detects different types of 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 spectra. Our framework is broad and can especially be used to study structural and functional connectomes combining connectivities of different types and orders.