A study of higher-order structures formed by non-pairwise interactions in a network reveals hidden layers with patterns that provide a more comprehensive understanding of the underlying system. Merging pairwise and higher-order approaches to the analysis of complex networks remains a conceptual and practical challenge. Also, persistent homology has been shown to be a computational framework for better understanding the organizational and topological higher-order features of simplicial complexes. To make a consistent methodological framework, the main criterion for the derivations presented in this work is that all generalized quantities can be reduced to the graph-theoretic case, i.e., for a simplicial complex of dimension 1. Thus, the weighted simplicial adjacency matrix has been derived and used to compute global and local measures, thereby playing a role in the filtration parameters for the calculation of persistent homology. For this purpose, three different filtration schemes for constructing the sequence of simplicial complexes have been given, based on the introduced generalized measures. The topology of the higher-order structures of the complex network can be compared using these measures and the persistent homology method, based on the induced interactions among its vertices. The application of the introduced definitions has been demonstrated on a real-world network by computing Betti numbers, thereby illustrating the benefits of the established approach.
Raj et al. (Mon,) studied this question.