AbstractWe consider a random graph that evolves in time by adding new edges at random times (different edges being added at independent and identically distributed times). A functional limit theorem is proved for a class of statistics of the random graph, considered as stochastic processes. the proof is based on a martingale convergence theorem. the evolving random graph allows us to study both the random graph modelKn, p, by fixing attention to a fixed time, and the modelKn, N, by studying it at the random time it contains exactlyNedges. in particular, we obtain the asymptotic distribution asn→ ∞ of the number of subgraphs isomorphic to a given graphG, both forKn, p(pfixed) andKn, N(N/(n2)→p). the results are strikingly different; both models yield asymptotically normal distributions, but the variances grow as different powers ofn(the variance grows slower forKn, N; the powers ofnusually differ by 1, but sometimes by 3). We also study the number of induced subgraphs of a given type and obtain similar, but more complicated, results. in some exceptional cases, the limit distribution is not n