Potential grouping of nodes induced by higher-order structures in complex networks

Complex networks display an organization of elements into nontrivial structures at versatile inherent scales, imposing challenges on a more complete understanding of their behavior. The interest of the research presented here is in the characterization of potential mesoscale structures as building blocks of generalized communities in complex networks, with an integrated property that goes beyond the pairwise collections of nodes. For this purpose, a simplicial complex is obtained from a mathematical graph, and indirectly from time series, producing the so-called clique complex from the complex network. As the higher-order organizational structures are naturally embedded in the hierarchical strata of a simplicial complex, the relationships between aggregation of nodes are stored in the higher-order combinatorial Laplacian. Based on the postulate that aggregation of nodes represents integrated configuration of information, the observability parameter is defined for the characterization of potential configurations, computed from the entries of the combinatorial Laplacian matrix. The framework introduced here is used to characterize nontrivial inherent organizational patterns embedded in two real-world complex networks and three complex networks obtained from heart rate time series recordings of three different subject's meditative states.