Relational exchangeability

We define a relationally exchangeable structure as a random combinatorial structure whose law is invariant with respect to relabeling its relations, instead of its elements. Examples of relationally exchangeable structures include edge exchangeable random graphs and hypergraphs and also exchangeable random set partitions (when viewed from a certain perspective). Relationally exchangeable models arise most naturally in certain network science applications, particularly network datasets generated processes of interactions. We prove a representation theorem for the class of relationally exchangeable structures and discuss some consequences and open problems. The representation refines Kingman's paintbox correspondence for exchangeable random partitions.