  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.
