On vertex, edge, and vertex-edge random graphs

TL;DR

This paper explores three classes of random graphs—edge, vertex, and vertex-edge—demonstrating that the most general model can be approximated by the vertex model, yet they remain fundamentally distinct.

Contribution

It introduces and compares three classes of random graphs, showing that vertex-edge graphs can be approximated by vertex graphs but are not equivalent.

Findings

Vertex-edge random graphs can be approximated by vertex random graphs.

Vertex, edge, and vertex-edge graphs are fundamentally distinct categories.

Abstract

We consider three classes of random graphs: edge random graphs, vertex random graphs, and vertex-edge random graphs. Edge random graphs are Erdos-Renyi random graphs, vertex random graphs are generalizations of geometric random graphs, and vertex-edge random graphs generalize both. The names of these three types of random graphs describe where the randomness in the models lies: in the edges, in the vertices, or in both. We show that vertex-edge random graphs, ostensibly the most general of the three models, can be approximated arbitrarily closely by vertex random graphs, but that the two categories are distinct.