Edge-exchangeable graphs and sparsity

A known failing of many popular random graph models is that the Aldous–Hoover Theorem guarantees these graphs are dense with probability one; that is, the number of edges grows quadratically with the number of nodes. This behavior is considered unrealistic in observed graphs. We define a notion of edge exchangeability for random graphs in contrast to the established notion of infinite exchangeability for random graphs—which has traditionally relied on exchangeability of nodes (rather than edges) in a graph. We show that, unlike node exchangeability, edge exchangeability encompasses models that are known to provide a projective sequence of random graphs that circumvent the Aldous–Hoover Theorem and exhibit sparsity, i.e., sub-quadratic growth of the number of edges with the number of nodes. We show how edge-exchangeability of graphs relates naturally to existing notions of exchangeability from clustering (a.k.a. partitions) and other familiar combinatorial structures.