Modeling complex Hierarchical systems with weighted and signed superhypergraphs: Foundations and applications

Classical graph theory represents pairwise relationships using vertices and edges, while hypergraphs extend this model by allowing hyperedges to join any number of vertices, enabling complex multi‐way connections. SuperHyperGraphs further generalize hypergraphs through iterated powerset constructions, capturing hierarchical relationships at multiple layers. Weighted and signed graph models assign numerical weights or positive/negative signs to edges, respectively, and these concepts have been lifted to hypergraphs and, more recently, to SuperHyperGraphs. In this paper, we systematically develop the definitions and core properties of weighted SuperHyperGraphs and signed SuperHyperGraphs. We provide detailed examples to illustrate their structure and discuss potential applications in modeling layered networks with quantitative and polarity annotations. Our results lay a foundation for future theoretical and algorithmic advances in this emerging area.