Review of Probabilistic HyperGraph and Probabilistic SuperHyperGraph

Takaaki Fujita

doi:10.31181/sor31202651

Authors

Takaaki Fujita Independent Researcher, Shinjuku, Shinjuku-ku, Tokyo, Japan Author https://orcid.org/0000-0002-9380-386X

DOI:

https://doi.org/10.31181/sor31202651

Keywords:

SuperHyperGraph, HyperGraph, Probabilistic HyperGraph, Probabilistic SuperHyperGraph, Probabilistic Graph, Probability

Abstract

Uncertainty pervades many real‐world networks, yet existing models such as probabilistic graphs and hypergraphs capture only pairwise or fixed‐order interactions. We introduce the novel concept of \emph{Probabilistic \(n\)-SuperHyperGraphs}, which unify nested higher-order relationships with edge‐level uncertainty by assigning probabilities to “superedges” at multiple powerset levels. We present a rigorous formal framework, derive fundamental properties—including degree–sum identities and closure under substructure—and show that our model subsumes classical probabilistic graphs and hypergraphs as special cases. These results pave the way for more expressive and scalable methods in modeling and analyzing complex, uncertainty-laden systems across diverse application domains.

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