Unifying Grain Boundary Networks and Crystal Graphs: A HyperGraph and SuperHyperGraph Perspective in Material Sciences

Abstract

Graph theory provides a foundation for modeling relationships among discrete elements via vertices and edges (Diestel, 2000, 2005). HyperGraphs extend this framework by allowing edges—hyperedges—to join more than two vertices, while superhypergraphs introduce nested powerset layers to capture hierarchical and selfreferential connections. Network analogues—hypernetworks and superhypernetworks—apply these ideas to empirical data structures. In materials science, graph-based models such as Grain Boundary Networks represent grains as vertices and their interfaces as edges (Schuh et al., 2003; Rohrer, 2011; Frary and Schuh, 2005), whereas Crystal Graphs encode atoms and bonds within lattice structures (Xie and Grossman, 2018; Park and Wolverton, 2020; Schmidt et al., 2021). These representations, however, lack the capacity to describe multi-scale and hierarchical features inherent in complex microstructures. This paper investigates the theoretical foundations of hypergraphs and superhypergraphs in materials science, which generalize classical graphs by enabling hyperedges, superedges, or supervertices to simultaneously connect multiple vertices. It further examines the relevance of graph-theoretical approaches in material sciences by introducing and formalizing the concepts of Grain Boundary HyperNetworks, Grain Boundary SuperHyperNetworks, and Crystal SuperHyperGraphs. For each structure, we provide precise mathematical definitions, construct detailed examples based on polycrystalline and crystalline material systems, and analyze fundamental properties such as multi-level connectivity, nesting depth, and combinatorial complexity. By integrating hyperstructure theory with the modeling of material architectures, this work establishes a robust framework for multi-scale and hierarchical analysis in materials science.