A note on the distribution of the extreme degrees of a random graph via the Stein-Chen method

We offer an alternative proof, using the Stein-Chen method, of Bollob\'{a}s' theorem concerning the distribution of the extreme degrees of a random graph. Our proof also provides a rate of convergence of the extreme degree to its asymptotic distribution. The same method also applies in a more general setting where the probability of every pair of vertices being connected by edges depends on the number of vertices.


Introduction
Consider a random graph with n labeled vertices {1, 2, . . ., n} in which each edge E ij , 1 ≤ i < j ≤ n is chosen independently and with a fixed probability p, 0 < p < 1. Denote by d i degree of the vertex i, i = 1, . . ., n of a graph G ∈ G(n, p), where G(n, p) is the probability space of graphs, and by d 1:n ≥ d 2:n ≥ • • • ≥ d n:n the degree sequence arranged in decreasing order.The asymptotic distribution of d n:n was established in Erdős and Rényi (1967).Ivchenko (1973) studied the asymptotic distribution of d m:n with fixed m and p = p(n) → 0 as n → ∞.For a fixed p (0 < p < 1), Bollobás (1980) found the asymptotic distribution of d m:n and proved: Theorem 1 (Bollobás (1980)).Suppose p is fixed, 0 < p < 1, q = 1 − p.Then, for every fixed natural number m independent of n, and for a fixed real number t, where We provide a proof sketch of Theorem 1.
Proof sketch of Bollobás (1980) : Let c be a fixed positive constant and let x = x(n, c) be defined by Set K = pn + x(npq) 1/2 and denote by X = X(n, c) the number of vertices of at least K degree.For the natural number r, Y r = X r .Thus, Bollobás established the inequality n r Then, by applying the de Moivre-Laplace limit theorem to the left-hand side (LHS) and right-hand side (RHS) of (3) and using the relation 1 − Φ(x) ∼ 1 x (2π) −1/2 e −x 2 /2 as x → ∞ (Feller (1968)) p. 175, Lemma 2), where Φ() is the CDF of a standard normal random variable, Bollobás proved that (4) implies that for every fixed r, the rth moment of X(n, c) tends to the rth moment of the Poisson distribution with mean c, and therefore from the Carleman theorem (Feller (1971), pp. 227-228) follows that i.e., X(n, c) tends to the Poisson distribution with mean c (see also Bollobás (1981)).Since Then, Bollobás (1980) showed that when n is large the solution of ( 2)

Alternative proof and Rate of Convergence
We present an alternative proof of Bollobás' theorem via the Stein-Chen method.It allows us to obtain a convergence rate.
Denote normalized vertex degrees (zero expectation and unit variance) as . ., n, and their corresponding decreasing sequence as , with a n and b n as defined in (1). Set 1 .
We will need five Assertions.
where the sign ∼ means that the ratio of the quantities on the LHS and the RHS tends to 1 for n → ∞.
Proof.Follows from Feller (1968)(pp. 192-193, Chapter VII.6) since for a fixed real number Assertion 2. For a fixed real number t, nπ Proof.From Assertion 1 combined with the result on page 374 of Cramér (1946), it follows that lim Assertion 3.For a fixed real number t, if npq → ∞ as n → ∞, where p + q = 1, then Proof.By B n denote the random variable Bin(n, p), and set Under this notation, π (n) 1 = P (B n−1 > y), and then using the formula of total probability we obtain The de Moivre-Laplace theorem (Rényi (1970), p. 204, Theorem 4.5.1.)states that uniformly in k, Therefore, by direct calculation, it follows that for a fixed real t, The condition x n (t) (n − 1)pq = o(n 2/3 ) is satisfied since for a fixed real t, x n (t) ∼ √ 2 log n and pq ≤ 1/4.Also, by direct calculation it follows that Combining ( 6), (7), and (8), we obtain Assertion 3.
where d T V (L(W n ), P oi(λ n )) is the total variation distance (TVD) between distributions of W n and the Poisson distribution with mean λ n .
Proof.Let e ij be the indicator random variable for the event {E ij = 1}.The indicators n are increasing functions of the independent edge indicators {e ij , 1 ≤ i < j ≤ n}.
Let I (n) * i be an independent copy of I (n) i .Thus, for all nondecreasing functions f and g for which expectations (from the expansion of the RHS of the expectation below) exist, 2Cov f (I n are associated random variables (Esary et al., 1967).As such, by Theorem 2.G, and hence by Corollary 2.C.4. in Barbour et al. (1992), we obtain a particular form of an upper bound for the TVD, i.e., Hence (9) follows since d * 1 , . . ., d * n are identically distributed, and therefore .
Remark 1.Another upper bound of d T V (L(W n ), P oi(λ n )) is given in Theorem 5.E in Barbour et al. (1992).
From these assertions, we further obtain the following: Theorem 2. For p ∈ (0, 1) and a fixed value of k, with a rate of convergence of W n to the Poisson distribution with mean e −t of order log n n .
We obtain an immediately corollary.
Corollary 1. Suppose p is fixed, 0 < p < 1.Then, for a fixed natural number m and a fixed real number t, Proof.Noticing that P (d * m:n ≤ x n (t)) = P (W n ≤ m − 1), and applying Theorem 2, we obtain (11).
Comment 2. The method and the results can be extended to the case where p depends on n.In that case, Assertions 3 and 5 hold if p(1 − p)n → ∞ as n → ∞.This can be compared with the condition in Theorem 3.3' of Bollobás (2001).
Comment 3. If ξ 1 , . . ., ξ n are independent and identically distributed standard normal random variables, with corresponding ξ 1:n ≥ ξ 2:n ≥ • • • ≥ ξ n:n sequence arranged in decreasing order, then the limit distribution P (ξ m:n ≤ a n t + b n ) is identical to the RHS of (1), with the same a n , b n (see for example Galambos (1987)).However, (1) is not obvious since d 1 , . . ., d n are dependent and their joint distribution depends on n.
Comment 4. The same asymptotic distribution as in (1), for the ordered normalized scores, holds for a round-robin tournament model with n players (Malinovsky, 2022).The difference is that the round-robin tournament is a complete directed graph and the total scores (degrees) of the players are negatively correlated.
{W n } converges in the TVD to the Poisson distribution with mean e −t .Since lim n→∞ λ n = e −t (Assertion 2), TVD convergence occurs if and only if {W n } in the distribution converges to the Poisson distribution with mean e −t