Exponential Approximation, Method of Types for Empirical Neighbourhood Distributions of Random Graphs by Random Allocations

K Doku-Amponsah

doi:10.5539/ijsp.v3n2p110

Abstract

In this article we find exponential good approximation of the empirical neigbourhood distribution of symbolled random graphs conditioned to a given empirical symbol distribution and empirical pair distribution. Using this approximation we shorten or simplify the proof of (Doku-Amponsah \& Morters, 2010, Theorem~2.5); the large deviation principle (LDP) for empirical neigbourhood distribution of symbolled random graphs. We also show that the LDP for the empirical degree measure of the classical Erd\H{o}s-R\'{e}nyi graph is a special case of (Doku-Amponsah \& Moerters, 2010, Theorem~2.5). From the LDP for the empirical degree measure, we derive an LDP for the the proportion of isolated vertices in the classical Erd\H{o}s-R\'{e}nyi graph.

Highlights

The Erdos-Renyi graph G(n, p) or G(n, nc/2) is the simplest imaginable random graph, which arises by taking n vertices, and placing an edge between any two of distinct nodes or vertices with a fixed probability 0 < p < 1 or inserting a fixed number nc edges at random among the n vertices (See Van Der Hofstad, 2009)
In this article we find exponential good approximation of the empirical neigbourhood distribution of symbolled random graphs conditioned to a given empirical symbol distribution and empirical pair distribution
We show that the large deviation principle (LDP) for the empirical degree measure of the classical Erdos-Renyi graph is a special case of (Doku-Amponsah & Moerters, 2010, Theorem 2.5)

Summary

Introduction

The Erdos-Renyi graph G(n, p) or G(n, nc/2) is the simplest imaginable random graph, which arises by taking n vertices, and placing an edge between any two of distinct nodes or vertices with a fixed probability 0 < p < 1 or inserting a fixed number nc edges at random among the n vertices (See Van Der Hofstad, 2009). Doku-Amponsah and Moerters (2010) and Doku-Amponsah (2006, 2012) obtained several LDPs, including the LDP for empirical degree distribution for near-critical or sparse case. Our main aim in this article is to obtain an exponential approximation result, see Lemma 4, for the empirical Neighbourhood distribution of symbolled random graphs. We show that the large deviation principle for the empirical degree measure of G(n, nc/2) is a special case of (Doku-Amponsah, 2012, Theorem 2.5.1) From this result we find an LDP for the proportion of isolated vertices in the graph G(n, nc/2). We review the symbolled random graph model as in (Doku-Amponsah et al, 2010)

The Symbolled Random Graph Model

The Conditional Symbolled Random Graph Models

Main Results

Proof of Theorem 3

Proof of Theorem 1