In this article, we prove a local large deviation principle (LLDP) for the empirical locality measure of typed random networks on $n$ nodes conditioned to have a given \emph{ empirical type measure} and \emph{ empirical link measure.} From the LLDP, we deduce a full large deviation principle for the typed random graph, and the classical Erdos-Renyi graphs, where $nc/2$ links are inserted at random among $n$ nodes. No topological restrictions are required for these results.