  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
amounts to a construction from an independent, identically distributed sequence
of exchangeable random functions. In continuous-time, the behavior is enriched
by a L\'evy--It\^o-type decomposition of the jump measure into mutually
singular components that govern global, vertex-level, and edge-level dynamics.
Furthermore, every such process almost surely projects to a Feller process in
the space of graph limits.
