Applications of Pythagorean Fuzzy Superhypergraph Entropy in Uncertainty Optimization
Abstract
This paper introduces the Pythagorean fuzzy supervertices, superedges, links, and Pythagorean fuzzy superhypergraphs (PFSh). We further present the class of Pythagorean fuzzy quasi-superhypergraphs (PFQS), classifying them into smooth, soft, rough, and coverable subcategories. Crucially, we prove these structures are a direct generalization of traditional Pythagorean fuzzy graphs.
We analyze the properties of these models, investigating the impact of Pythagorean $(\alpha, \beta)$-level subsets and establishing key relationships between Pythagorean fuzzy sets and quasi-superhypergraphs. The concept of a strong superhypergraph is introduced, with conditions for its formation from cut-level sets provided.
To quantify uncertainty, we pioneer definitions for Pythagorean fuzzy entropy of supervertices, superedges, and links, proving the resulting superhypergraph entropy is a valid measure. Algorithmically, we adapt the Fuzzy Kruskal's Algorithm to minimize superedge entropy via a spanning superhypertree, denoted $E_{min}(\mathcal{E})$. The framework's utility is demonstrated through applications in business superhypernetworks and social complex hypernetworks, bridging fuzzy set theory and network science.
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