Exchangeable graph-valued Feller processes

Abstract

The transition law of every exchangeable Feller process on the space of countable graphs is determined by a $$\sigma $$ -finite measure on the space of $$\{0,1\}\times \{0,1\}$$ -valued arrays. In discrete time, this characterization gives rise to a construction from an independent, identically distributed sequence of exchangeable random functions. In continuous time, the behavior is enriched by a Levy–Ito–Khintchine-type decomposition of the jump measure into mutually singular components that govern global, vertex-level, and edge-level dynamics. Every such process almost surely projects to a Feller process in the space of graph limits.