Multidimensional HyperNetwork and Multidimensional SuperHyperNetwork

Abstract

Graph theory studies the mathematical structures of vertices and edges to model relationships and connectivity [1, 2]. Hypergraphs extend this framework by allowing hyperedges to connect arbitrarily many vertices at once [3], and superhypergraphs further generalize hypergraphs via iterated powerset constructions to capture hierarchical linkages among edges [4,5]. Weighted multidimensional networks model nodes connected by edges in multiple layers, assigning each edge a weight to quantify relationship strength in its specific dimension(cf. [6,7]). In this paper, we extend these ideas using hypergraphs and superhypergraphs to introduce and formalize multidimensional hypernetworks and multidimensional superhypernetworks.