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

TL;DR

This paper establishes a local large deviation principle for finite typed random graphs, deriving a McMillian-type theorem and explicit asymptotic counts without topological restrictions.

Contribution

It introduces a spectral potential and deviation function to prove a local large deviation principle for typed random graphs, extending classical results.

Findings

Proves LLDP with rate function _{\u03bb}(\u03c08,a8) for typed graphs.

Derives a full conditional large deviation principle and a McMillian-type theorem.

Provides asymptotic enumeration formula for typed graphs given empirical measures.

Abstract

For a finite typed graph on $n$ nodes and with type law $\mu,$ we define the so-called spectral potential $\rho_{\lambda}(\,\cdot,\,\mu),$ of the graph.From the $\rho_{\lambda}(\,\cdot,\,\mu)$ we obtain Kullback action or the deviation function, $\mathcal{H}_{\lambda}(\pi\,\|\,\nu),$ with respect to an empirical pair measure, $\pi,$ as the Legendre dual. For the finite typed random graph conditioned to have an empirical link measure $\pi$ and empirical type measure $\mu$, we prove a Local large deviation 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 the classical McMillian Theorem for the typed random graphs. Given the typical empirical link measure, $\lambda\mu\otimes\mu,$ the number of typed random graphs 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 finite graphs for these LLDPs.