A novel framework for the Cartesian product of clique simplicial complexes and its applications

This article develops a theoretical framework for the Cartesian product of simplicial complexes, with a focus on clique complexes due to their increasing importance in applications such as social network analysis, biological network modelling, data mining, and community detection. We define the Cartesian product for clique simplicial complexes as a natural extension of the graph Cartesian product and derive formulas for key structural metrics—including dimension, cardinality, degree, adjacency matrix, Katz centrality, and eigenvector centrality from nodes to higher-dimensional simplices. Two algorithms for constructing and visualising the product are presented and demonstrated through Python examples. We also outline several potential applications, highlighting in particular how interdependent disease properties can be modeled using the Cartesian product of psychological and physiological clique simplicial complexes, supported by detailed mathematical analysis and visualisation.