Hypergraph Neural Networks

A hypergraph is a generalization of a simple graph \(G = (V, E)\), where \(V\) is a set of vertices and \(E\) is a set of edges (hyperedges) connecting an arbitrary number of vertices.

Representation of hyperedges

When we encode input data (graph) in the form of logic data format (i.e., ground relations), we can represent regular edges, for example, as Relation.edge(1, 2).

This form of representation can be simply extended to express hyperedges by adding terms for each connected vertex by the hyperedge. For example, graph \(G = (V, E)\), where \(V = \{1, 2, 3, 4, 5, 6\}\) and \(E = \{\{1, 2\}, \{3, 4, 5\}, \{1, 2, 4, 6\}\}\) can be represented as:

Relation.edge(1, 2),
Relation.edge(3, 4, 5),
Relation.edge(1, 2, 4, 6),

Propagation on hyperedges

The propagation through standard edges can be similarly extended to support propagation through hyperedges.

Relation.h(Var.X) <= (Relation.feature(Var.Y), Relation.edge(Var.Y, Var.X))

The propagation through standard edges above, where Relation.feature might represent vertex features, and Relation.edge represents an edge, might be extended to support hyperedges (for hyperedge connecting three vertices) as follows:

Relation.h(Var.X) <= (
    Relation.feature(Var.Y),
    Relation.feature(Var.Z),
    Relation.edge(Var.Y, Var.Z, Var.X),
)