Study of higher-order interactions in unweighted, undirected networks using persistent homology

Abstract

Persistent homology has been studied to better understand the structural properties and topology features of weighted networks. It can reveal hidden layers of information about the higher-order structures formed by non-pairwise interactions in a network. Studying of higher-order interactions (HoIs) of a system provides a more comprehensive understanding of the complex system; moreover, it is a more precise depiction of the system as many complex systems, such as ecological systems and biological systems, etc., demonstrate HoIs. In this study, the weighted simplicial adjacency matrix has been constructed using the concept of adjacency strength of simplices in a clique complex obtained from an unweighted, undirected network. This weighted simplicial adjacency matrix is thus used to calculate the global measure, which is called generalised weighted betweenness centrality, which further helps us in calculating the persistent homology on the given simplicial complex by constructing a filtration on it. Moreover, a local measure called maximal generalised degree centrality has also been established for better understanding of the network topology of the studied simplicial complex. All the generalizations given in this work can be reduced to the graph-theoretic case. i.e., for a simplicial complex of dimension 1. Three different filtration schemes for constructing the sequence of simplicial complexes have been given with the help of both global and local measures, and by using these measures, the topology of higher-order structures of the studied network due to the interactions of their vertices has been compared. Further, the illustration of established definitions has been given using a real-life network by calculating Betti numbers up to dimension two.