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

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.


Introduction
The Erdős-Rényi 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).Several large deviation (LD) results for this graphs have been found (See, for example, O 'Connell, 1998;Biggins & Penman, 2009;Doku-Amponsah & Moerters, 2010;Doku-Amponsah, 2006;Bordenave & Caputo, 2013;Mukherjee, 2013).O'Connell (1998) proved an LDP for the relative size of the largest connected component and the number of isolated vertices in the random graph G(n, p) with p = O(1/n).O'Connell (1998) also presented an LDP and a related result for the number of isolated vertices in the random graph G(n, c/n).i.e. the near-critical or sparse case.An LDP for the proportion of edges to the number of potential vertices of the supercritical case has been found by Biggins and Penman (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.Boucheron et al. (2002) attempted to prove from the LDP for the empirical occupancy process an LDP for the degree distribution of the random graph G(n, nc).But Doku-Amponsah et al. (2010) conjectured that the prove of this LDP does not hold.Recently, Bordenave and Caputo (2013) obtained LDPs for the empirical neighbourhood distribution in G(n, c/n) and G(n, nc/2).The LDP for the empirical degree distribution in G(n, c/n) has been proved in (Mukherjee, 2013).
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.Using this result we shorten or simplify the proof of the LDP for empirical Neighbourhood distribution of symbolled random graphs conditioned on a given empirical symbol measure and empirical pair distribution (See example, Doku-Amponsah et al., 2010 or Doku-Amponsah, 2012, Theorem 2.5.1).
Further, 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).Note that the LDP for the proportion of isolated vertices in the graph G(n, nc/2) is new in the literature (See, O'Connell, 1998 for similar result).
The main technique used in this article is the method of types (see Dembo & Zeitouni, 1998, Section 2.1).The method of types is applied to an exponential approximate model for the symbolled random graph model which we shall obtain by randomly allocating symbolled balls in to symbolled bins.
The symbol random graphs (see Penman, 1998) or Inhomogeneous random graphs (see Van Der Hofstad, 2009), which has Erdős-Rényi graph with one symbol as an example permit a dependence between symbol and connectivity of the nodes.In the next subsection, we review the symbolled random graph model as in (Doku-Amponsah et al., 2010).

The Symbolled Random Graph Model
We begin by fixing the following notations.Let Y be a symbol or colour set.Let V = {1, . . ., n} be a fixed set of n vertices.Denote by E the edge set, i.e.
where we have used the formal ordering of links to simply describe unoriented edges.
Let p n : Y × Y → [0, 1] be a symmetric function and μ: Y → [0, 1] a probability law.We may describe the simply symbolled random graph Y with n vertices in the following manner: Any node v ∈ V gets symbol Y(v) independently and identically according to the symbol law λ.Given the colours, we join any two nodes u, v ∈ V with an edge, independently of everything else, with a edge probability p n (Y(u), Y(v)) otherwise we keep them disconnected.We always look at Y = ((Y(v): v ∈ V), E) under the combine distribution of graph and symbol.We interpret Y as symbolled random graph and consider Y(v) as the symbol of the node v (See Cannings & Penman, 2003;Penman, 1998).
Our interest in this article is on the symbolled random graph models in the near-critical regimes.Thus, we look at cases when the edge probability p n (a, b) By W(X) we denote the space of probability measures on a finite or countable set X, and by W(X) we denote the subspace of finite measures defined on X, and we endow both with the weak topology.Further, we denote by W * (X × X) the subspace of symmetric measures.By convection, we write For any symbolled graph We observe that by definition L 2 Y is finite symmetric measure with total mass L 2 Y equal to 2|E|/n.Finally we recall the definition of the empirical Neighbourhood distribution where Observe that Δ 1 is a continuous function but Δ 2 is discontinuous in the weak topology.In particular, in the summation l(b)∈N μ 2 (a, l(b))l(b) the function l(b) may be unbounded and so the functional μ → Δ 2 (μ) would not be continuous in the weak topology.We call a pair of measures (π, and consistent if equality holds in (1).For any n ∈ N we define the following sets:

The Conditional Symbolled Random Graph Models
In the remaining part of this article we may assume that ν(a) > 0 for all a ∈ Y.Note that the law of the symbolled random graph given the empirical symbol distribution ν n and empirical pair distribution π n , may be described as follows: • We assign symbols to the nodes by drawing from the pool of n symbols which contains any symbol a ∈ Y, nν n (a) times without replacement; • For each unordered pair {a, b} of symbols we construct (exactly) m n (b, a) edges by drawing without replacement from the collection of potential edges linking nodes of symbol a and b, where By Y n we denote the conditional symbolled random graph with empirical symbol measure ν n and empirical pair measure π n .

Main Results
The main theorem in this section is an LDP for the proportion of isolated nodes in the random graph G(n, nc/2).
We recall from (Doku-Amponsah & Moerters, 2010), the empirical degree measure D Y ∈ W(N) of the symbolled random as Theorem 1 Suppose D Y is the empirical degree measure of the random graph G(n, nc/2).Then, as n → ∞, the proportion of isolated nodes D Y (0), obeys an LDP with good, convex rate function Theorem 2 below and the contraction principle imply the LDP for the proportion of isolated vertices, i.e.Theorem 1. O'Connell (1998) obtained similar large deviation result for the number of isolated nodes in the random graph G(n, c/n).
Theorem 2 (Doku-Amponsah, 2006, 2012) Suppose D Y is the empirical degree measure of the random graph G(n, nc/2).Then, as n → ∞, D Y obeys an LDP in the space W(N) with good, convex rate function where q x is a Poisson distribution with parameter x.
Note, the degree distribution D Y is a continuous function of M Y , and so Theorem 3 below and the contraction principle gives the LDP for D Y .In fact the LDP for D Y (above) is a special case of Theorem 3 which was first proved in Doku-Amponsah and Moerters (2010) or Doku-Amponsah (2012) by approximating a given symbolled random graph from below by another symbolled random graphs with the degree of each nodes growing in order of o(n 1/3 ) (See Doku-Amponsah & Moerters, 2010, Lemma 4.10, pp.26-29).Note, that in the special case of classical Erdős-Rényi graph G(n, nc/2) we have that M = D, the degree distribution and Let Y be a symbolled random graph with n nodes conditioned on the event with good rate function

Proof of Theorem 3: Exponential Approximation by Random Allocation
In order to improve (shorten) the proof of Theorem 3, we pass to a simple random allocation model, which turns out to be equivalent.This model is best described in term of symbolled balls being placed randomly into symbolled bins.In the resulting constellation we denote, for any bin v ∈ {1, . . ., n}, by Ỹ(v) its symbol, and by l v (b) the number of balls of symbol b ∈ Y it contains.Now define the empirical occupancy measure of the constellation by where L(u) = ( u (a), a ∈ Y) is the symbol distribution in bin v.In our first theorem we establish exponential equivalence of the law of the empirical occupancy measure M Ỹ under the random allocation model P(ν n ,π n ) and the law of the empirical Neighbourhood distribution M under the law of the symbolled random graph conditioned to have symbol law ν n and edge distribution π n .Recall the definition of exponential equivalence (see Dembo & Zeitouni, 1998, Definition 4.2.10).

Lemma 4
The law of M Ỹ under P(ν n ,π n ) is exponentially equivalent to the law of M Y under P (ν n ,π n ) .
Proof.Define the metric d of total variation by As this metric produces the weak topology, the proof of Lemma 4 is equivalent to showing that for every ε > 0, where P indicates a suitable coupling between the random allocation model and the symbolled graph.
To begin, denote by V(a) the collection of nodes(bins) which have symbol a ∈ Y and observe that For every a, b ∈ Y, begin: At each step k = 1, . . ., m n (b, a), we randomly pick two nodes 1 and one ball of symbol a in V k 2 , and link or the two nodes are already linked.If one of these two things happen, then we simply choose an edge randomly from the set of all potential edges linking symbols a and b, which are not yet present in the graph.This completes the construction of a graph with where B n (b, a) is the total number of steps k ∈ {1, . . ., m n (b, a)} at which there is discrepancy between the vertices V k 1 , V k 2 drawn and the nodes which formed the k th edge connecting a and b in the random graph construction.Given a, b ∈ Y,the probability that V k 1 = V k 2 or the two nodes are already linked is equal to (b,a) 1l {b=a} B n (b, a) is a sum of independent Bernoulli random variables X 1 , ..., X nm n (b,a) with 'success' probabilities equal to and We Define h(x) = (1 + x) log(1 + x) − x, for x ≥ 0 and use Bennett's inequality (see Bennett, 1962), to obtain, for sufficiently large n Hence, Let 0 ≤ δ 2 ≤ 1.Then, for sufficiently large n we have This completes the proof of the lemma.

Proof of Theorem 3: Large Deviation Probabilities by the Method of Types
Now Lemma 4 and the large deviation principle for M Ỹ under P(ν n ,π n ) implies the same large deviation principle for M Y under P (ν n ,π n ) in the weak topology (See, for example, Dembo & Zeitouni, 1998, Theorem 4.2.13).
Consequently, the proof of Theorem 3 is equivalent to showing that for every We begin the proof of Lemma 5 by recalling the definition of Δ a function on W(Y × N Y ) given by or some random allocations process on n bins .
We denote by S(μ) the support of μ and write for μ n ∈ K (n) (ν n , π n ), where We write where We prove Lemma 5 above from the following lemma which uses the the idea of the method of types (see Dembo & Zeitouni, 1998, Chapter 2).
u (b) is the number of nodes of symbol b linked to node v.For any μ ∈ W(Y × N Y )we denote by μ 1 the Y− marginal of μ and for every (b, a) ∈ Y × Y, let μ 2 be the law of the pair (a, l(b)) under the measure μ.Define the measure (finite), μ

Fix
n ≥ 1, a symbol law ν n ∈ W n (Y) and an edge law π n ∈ Wn (Y × Y).The bins V = {1, . . ., n} are now symbolled by drawing without replacement from the pool of symbols, which contains the symbol a ∈ Y exactly nν n (a) times.For each ordered pair (b, a) ∈ Y × Y of symbols, we independently and identically place nm n (b, a) balls of symbol b into the nν n (a) bins of symbol a by drawing without replacement.We denote by P(ν n ,π n ) the distribution of the random allocation model with symbol law ν n ∈ W n (Y) and an edge law π n ∈ Wn (Y × Y).