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 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.

通过新定义的超图，将无向超图、有向超图和嵌套超图的概念统一起来，以超边作为高阶事物之间的关联的单一结构，方便地表示一般和化学系统中的高阶关联，尤其适用于化学系统的机器学习。

超图：一个统一且一致的定义及其在化学超图中的应用

Hypergraph: A Unified and Uniform Definition with Application to  Chemical Hypergraph

In spectral graph theory, the Cheeger's inequality gives upper and lower
bounds of edge expansion in normal graphs in terms of the second eigenvalue of
the graph's Laplacian operator. Recently this inequality has been extended to
undirected hypergraphs and directed normal graphs via a non-linear operator
associated with a diffusion process in the underlying graph.
In this work, we develop a unifying framework for defining a diffusion
operator on a directed hypergraph with stationary vertices, which is general
enough for the following two applications.
1. Cheeger's inequality for directed hyperedge expansion.
2. Quadratic optimization with stationary vertices in the context of
semi-supervised learning.
Despite the crucial role of the diffusion process in spectral analysis,
previous works have not formally established the existence of the corresponding
diffusion processes. In this work, we give a proof framework that can indeed
show that such diffusion processes are well-defined. In the first application,
we use the spectral properties of the diffusion operator to achieve the
Cheeger's inequality for directed hyperedge expansion. In the second
application, the diffusion operator can be interpreted as giving a continuous
analog to the subgradient method, which moves the feasible solution in discrete
steps towards an optimal solution.

本文在有向超图的定点上定义扩散算子，推广了 Cheeger 不等式，并在半监督学习中实现了二次优化。