  We use an inequality of Sidorenko to show a general relation between local
and global subgraph counts and degree moments for locally weakly convergent
sequences of sparse random graphs. This yields an optimal criterion to check
when the asymptotic behaviour of graph statistics such as the clustering
coefficient and assortativity is determined by the local weak limit. As an
application we obtain new facts for several common models of sparse random
intersection graphs where the local weak limit, as we see here, is a simple
random clique tree corresponding to a certain two-type Galton-Watson branching
process.
