A Representation of Hypergraphs in the Euclidean Space

This paper introduces a graph space that shows concisely the relative weights among combinations of vertices of a given hypergraph. (A hypergraph is a graph in which one edge may connect two or more vertices.) The hypergraph is represented by a collection of points in graph space such that the distance between vertices in graph space reflects the weights of the edges between vertices of the original hypergraph. Vertices of the hypergraph that are connected by edges with large weights are mapped to nearby points in graph space. Thus, graph space reveals properties of the connectivity of vertices in the hypergraph. A natural application of graph space is the placement of modules in computer systems since strongly coupled modules are transformed into nearby points in graph space. The graph of the airlines network in the United States is taken as an example of a hypergraph, and the paper illustrates the corresponding graph space.