IntComplex for High-Order Interactions

Abstract

Graphs serve as powerful tools for modeling pairwise interactions in diverse fields such as biology, material science, and social networks. However, they inherently overlook interactions involving more than two entities. Simplicial complexes and hypergraphs have emerged as prominent frameworks for modeling many-body interactions. Despite their strengths, these models are limited in their ability to represent high-order interactions that possess internal structures. For instance, consider a three-body interaction among a father, mother, and son within a family. The relationship between the father and mother inherently influences their son, but this interaction cannot be adequately captured by either simplicial complexes or hypergraphs. Addressing this gap, we propose IntComplex as an innovative framework to characterize such high-order interactions comprehensively. Our framework leverages homology theory to provide a quantitative representation of the topological structure inherent in such interactions. IntComplex is defined as a collection of interactions, each of which can be equivalently represented by a binary tree. Drawing inspiration from GLMY homology, we introduce homology for the detailed analysis of structural patterns arising from interactions within a specific layer, between adjacent layers, and across multiple dimensions. Furthermore, we introduce persistent homology through a filtration process and establish its stability to ensure robust quantitative analysis of these complex interactions. The proposed IntComplex establishes a foundational framework for the analysis of topological properties in such high-order interactions, presenting potential to drive forward the advancements in the domain of complex network analysis.