Joint large deviation result for empirical measures of the coloured random geometric graphs

Abstract

We prove joint large deviation principle for the empirical pair measure and empirical locality measure of the near intermediate coloured random geometric graph models on n points picked uniformly in a d-dimensional torus of a unit circumference. From this result we obtain large deviation principles for the number of edges per vertex, the degree distribution and the proportion of isolated vertices for the near intermediate random geometric graph models.

Lossy asymptotic equipartition property for geometric networked data structures

“…We shall call the measure µ ∈ M[(X × N (X )) 2 ] consistent if µ 1 , µ 2 are both consistent marginals of µ. Refer to [7,Equation 2.1] for the concept of consistent measures.…”

Asymptotics of the Partition Function of Ising Model on Inhomogeneous Random Graphs

“…In this section, we review some large deviation results of [19], and extend the joint LDP for empirical distributions of coloured random graph to spinned random graphs.i.e. we assume a more general spin law ℓ : R → [0, 1] with all its exponential moments finite and prove an LDP for this model in a topology generated by the total variation norm.…”

“…Thus we study random graphs with a local structure of multitype Galton-Watson branching trees. To be specific, we use the large deviation principle (LDP) techniques developed in [19] and furthered in [20] to prove annealed asymptotic result for the log-partition functions of Ising model on inhomogeneous random graphs. Our annealed asymptotic result may serve as the basis for understanding the thermodynamic limiting behaviour of the free-energy function of the Ising model on inhomogeneous graphs.…”

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

“…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).…”

“…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:…”