Transformations on hypergraph families

TL;DR

This paper introduces a comprehensive mathematical framework for function-based transformations on finite hypergraph families, enabling structured modifications, analysis, and simplification of hypergraphs with potential applications in modeling complex systems.

Contribution

It develops a general theory of hypergraph transformations, including quotient transformations, and demonstrates fundamental examples involving hyperedge modifications.

Findings

Framework formalizes hypergraph modifications consistently.

Includes quotient hypergraph transformations based on equivalence relations.

Provides fundamental examples of hyperedge addition and deletion.

Abstract

We present a new general theory of function-based hypergraph transformations on finite families of finite hypergraphs. A function-based hypergraph transformation formalises the action of structurally modifying hypergraphs from a family in a consistent manner. The mathematical form of the transformations facilitates their analysis and incorporation into larger mathematical structures, and concurs with the function-based nature of modelling in the physical world. Since quotients of hypergraphs afford their simplification and comparison, we also discuss the notion of a quotient hypergraph transformation induced by an equivalence relation on the vertex set of a hypergraph family. Finally, we demonstrate function-based hypergraph transformations with two fundamental classes of examples involving the addition or deletion of hyperedges or hypergraphs.