Hypergraph: A Unified and Uniform Definition with Application to Chemical Hypergraph and More
This work provides a foundational framework for machine learning in chemistry by enabling more accurate modeling of complex molecular structures and reactions, though it is incremental in extending existing hypergraph concepts.
The paper addresses the lack of standard definitions for directed and nested hypergraphs by proposing a unified definition that uses hyperedges to represent high-order correlations, and applies this to chemical systems to enable direct representation of features like multicenter bonds and reactions.
The conventional definition of hypergraph has two major issues: (1) there is not a standard definition of directed hypergraph and (2) there is not a formal definition of nested hypergraph. To resolve these issues, we propose a new definition of hypergraph that unifies the concepts of undirected, directed and nested hypergraphs, and that is uniform in using hyperedge as a single construct for representing high-order correlations among things, i.e., nodes and hyperedges. Specifically, we define a hyperedge to be a simple hyperedge, a nesting hyperedge, or a directed hyperedge. With this new definition, a hypergraph is nested if it has nesting hyperedge(s), and is directed if it has directed hyperedge(s). Otherwise, a hypergraph is a simple hypergraph. The uniformity and power of this new definition, with visualization, should facilitate the use of hypergraph for representing (hierarchical) high-order correlations in general and chemical systems in particular. Graph has been widely used as a mathematical structure for machine learning on molecular structures and 3D molecular geometries. However, graph has a major limitation: it can represent only pairwise correlations between nodes. Hypergraph extends graph with high-order correlations among nodes. This extension is significant or essential for machine learning on chemical systems. For molecules, this is significant as it allows the direct, explicit representation of multicenter bonds and molecular substructures. For chemical reactions, this is essential since most chemical reactions involve multiple participants. We propose the use of chemical hypergraph, a multilevel hypergraph with simple, nesting and directed hyperedges, as a single mathematical structure for representing chemical systems. We apply the new definition of hypergraph to chemical hypergraph and, as simplified versions, molecular hypergraph and chemical reaction hypergraph.