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For a finite typed graph on $n$ nodes and with type law $\mu,$ we define theso-called spectral potential $\rho_{\lambda}(\,\cdot,\,\mu),$ of the graph.Fromthe $\rho_{\lambda}(\,\cdot,\,\mu)$ we obtain Kullback action or the deviationfunction, $\mathcal{H}_{\lambda}(\pi\,\|\,\nu),$ with respect to an empiricalpair measure, $\pi,$ as the Legendre dual.  For the finite typed random graph conditioned to have an empirical linkmeasure $\pi$ and empirical type measure $\mu$, we prove a Local largedeviation principle (LLDP), with rate function$\mathcal{H}_{\lambda}(\pi\,\|\,\nu)$ and speed $n.$ We deduce from this LLDP,a full conditional large deviation principle and a weak variant of theclassical McMillian Theorem for the typed random graphs. Given the typicalempirical link measure, $\lambda\mu\otimes\mu,$ the number of typed randomgraphs is approximately equal$e^{n\|\lambda\mu\otimes\mu\|H\big(\lambda\mu\otimes\mu/\|\lambda\mu\otimes\mu\|\big)}.$Note that we do not require any topological restrictions on the space of finitegraphs for these LLDPs.

Local Large deviation: A McMillian Theorem for Coloured Random Graph Processes