MetaHyperGraphs, MetaSuperHyperGraphs, and Iterated MetaGraphs: Modeling Graphs of Graphs, Hypergraphs of Hypergraphs, Superhypergraphs of Superhypergraphs, and Beyond

Graph theory studies mathematical structures composed of vertices and edges to model relationships and connectivity [1,2]. Hypergraphs extend traditional graphs by allowing hyperedges that connect more than two vertices simultaneously [3]. Superhypergraphs further enrich this concept by introducing iterated powerset layers, enabling hierarchical and self-referential connections among hyperedges [4, 5]. A MetaGraph is a graph whose vertices are themselves graphs, with edges representing specified relations between those graphs. In this paper, we formally define the hypergraph analogue (MetaHyperGraph) and the superhypergraph analogue (MetaSuperHyperGraph) of MetaGraphs, and provide a concise discussion of their characteristics and illustrative applications. We also introduce iterative constructions such as the Iterated MetaGraph, representing a “graph of graphs of . . . of graphs,” and briefly explore their properties and potential uses.