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

Abstract. For a finite typed graph on n nodes and with type law μ, we define the socalled spectral potential ρλ( ·, μ), of the graph.From the ρλ( ·, μ) we obtain Kullback action or the deviation function, Hλ(π ‖ ν), with respect to an empirical pair measure, π, as the Legendre dual. For the finite typed random graph conditioned to have an empirical link measure π and empirical type measure μ, we prove a Local large deviation principle (LLDP), with rate function Hλ(π ‖ ν) and speed n. We deduce from this LLDP, a full conditional large deviation principle and a weak variant of the classical McMillian Theorem for the typed random graphs. Given the typical empirical link measure, λμ⊗μ, the number of typed random graphs is approximately equal e (