A SAMPLE-PATH LARGE DEVIATION PRINCIPLE FOR DYNAMIC ERDO ?S-ReNYI RANDOM GRAPHS

We consider a dynamic Erdo ?s-Renyi random graph on n vertices in which each edge switches on at rate & lambda; and switches off at rate & mu;, indepen-dently of other edges. The focus is on the analysis of the evolution of the associated empirical graphon in the limit as n & RARR; & INFIN;. Our main result is a large deviation principle (LDP) for the sample path of the empirical graphon observed until a fixed time horizon. The rate is (n), the rate function is a spe-2 cific action integral on the space of graphon trajectories. We apply the LDP to identify (i) the most likely path that starting from a constant graphon creates a graphon with an atypically large density of d-regular subgraphs, and (ii) the mostly likely path between two given graphons. It turns out that bifurcations may occur in the solutions of associated variational problems.