Large deviation principles for graphon sampling

We investigate possible large deviation principles (LDPs) for the $n$-vertex sampling from a given graphon with various speeds $s(n)$ and resolve all the cases except when the speed $s(n)$ is of order $n^2$. For quadratic speed $s=(c+o(1))n^2$, we establish an LDP for an arbitrary $k$-step graphon, which extends a result of Chatterjee and Varadhan [Europ. J. Combin., 32 (2011) 1000-1017] who did this for $k=1$ (that is, for the homogeneous binomial random graphs). This is done by reducing the problem to the LDP for stochastic $k$-block models established recently by Borgs, Chayes, Gaudio, Petti and Sen ["A large deviation principle for block models", arxiv:2007.14508, 2020]. Also, we improve some results by Borgs et al.