Attack Graph, Attack HyperGraph, and Attack SuperHyperGraph in Security

Abstract

Graph theory studies mathematical structures of vertices and edges to model relationships and connectivity. Hypergraphs extend traditional graphs by allowing hyperedges that connect any number of vertices simultaneously. Superhypergraphs further generalize this idea through iterated powerset layers, enabling hierarchical and self referential connections among hyperedges. An Attack Graph is the subgraph of an Attack Model containing exactly those states from which the attacker can reach a security violation, including its transitions and error states. An Attack HyperGraph, which extends the Attack Graph concept from the perspective of hypergraphs, is also well known and has been similarly studied. In this paper, we investigate Attack Superhypergraphs, which extend Attack Graphs to the superhypergraph framework. We expect that these developments will advance research in graph theory and hierarchical security models.