Graph Hyperembedding and Graph SuperHyperembedding: Extensions via HyperVector and SuperHyperVector Spaces

Graph embedding assigns vectors to vertices (or to entire graphs) in a low-dimensional space so that adjacency information and broader structural relationships are reflected in the resulting representations. Hypervector spaces extend the classical vector-space setting by replacing ordinary scalar multiplication with a set-valued scalar action, allowing each scalar–vector pair to produce a nonempty set of vectors while maintaining familiar distributive and associative behavior in an appropriate sense. Building on this idea, (m, n)-SuperhyperVector spaces further generalize the framework by admitting an m-ary scalar superhyperoperation whose outputs belong to an n-level iterated nonempty powerset associated with the underlying abelian group. In this paper, we introduce Graph Hyperembedding and Graph SuperHyperembedding by using HyperVector and SuperHyperVector spaces, and we investigate fundamental properties of these new embedding models.