Forcing generalised quasirandom graphs efficiently

We study generalised quasirandom graphs whose vertex set consists of $q$ parts (of not necessarily the same sizes) with edges within each part and between each pair of parts distributed quasirandomly; such graphs correspond to the stochastic block model studied in statistics and network science. Lovasz and Sos showed that the structure of such graphs is forced by homomorphism densities of graphs with at most $(10q)<^>q+q$ vertices; subsequently, Lovasz refined the argument to show that graphs with $4(2q+3)<^>8$ vertices suffice. Our results imply that the structure of generalised quasirandom graphs with $q\ge 2$ parts is forced by homomorphism densities of graphs with at most $4q<^>2-q$ vertices, and, if vertices in distinct parts have distinct degrees, then $2q+1$ vertices suffice. The latter improves the bound of $8q-4$ due to Spencer.