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

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 Erdős-Rényi 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 Erdős-Rényi graph.

Large Deviation Result for the Empirical Locality Measure of Typed Random Geometric Graphs

“…Using Doku-Amponsah [5], under the condition P z ∈ B p andp = q we have that e −nH(pn qn)+nH(q qn)−o(n) ≤ dQ (ηn,πn) (z) dQ (ηn,πn) (z) ≤ e −nH(pn qn)+nH(q qn)+o(n) ,…”

“…Some large deviation principles for this random graph have been found. See, Bordenave& Caputo [1],Doku-Amponsah [5], and Doku-Amponsah and Moerters [8] Doku-Amponsah and Moerters [8] provided LDPs for the near-critical or sparse typed random graphs with this model as a special case. Bordenave and Caputo [1] obtained large deviation principle for the empirical neigbhourhood measure of the model G(n, nc/2).…”

“…Bordenave and Caputo [1] obtained large deviation principle for the empirical neigbhourhood measure of the model G(n, nc/2). Large deviation for the empirical distribution was presented in Doku-Amponsah [5] using the method of types and the coupling argument presented by Boucheron et. al [2].…”

“…To state our main results in Section 2 we need some notions, concepts and notations from [8] and [4] in the background section below. For any typed graph Z( with n nodes), recall from [5] the definitions of the empirical type distribution P 1 ∈ L(Z), the empirical link distribution P 2 ∈L(Z × Z) and the empirical locality distribution Mathematics Subject Classification : 94A15, 94A24, 60F10, 05C80 Keywords:…”

Lossy Asymptotic Equipartition Property for Networked Data Structures

“…Lemma 4.1 (Doku-Amponsah, 2014) For any process level empirical measure v n with v n,1 , v n,2 ∈ Σ (n) (σ n , π n ), we have: …”