A universal error bound in the CLT for counting monochromatic edges in uniformly colored graphs

Let $\{G_n: n\geq 1\}$ be a sequence of simple graphs. Suppose $G_n$ has $m_n$ edges and each vertex of $G_n$ is colored independently and uniformly at random with $c_n$ colors. Recently, Bhattacharya, Diaconis and Mukherjee (2013) proved universal limit theorems for the number of monochromatic edges in $G_n$. Their proof was by the method of moments, and therefore was not able to produce rates of convergence. By a non-trivial application of Stein's method, we prove that there exists a universal error bound for their central limit theorem. The error bound depends only on $m_n$ and $c_n$, regardless of the graph structure.


INTRODUCTION
Let {G n : n 1} be a sequence of simple graphs, that is, graphs that contain no loops and no multiple edges. Suppose G n has m n edges and each vertex of G n is colored independently and uniformly at random with c n colors. Let Y n be the number of monochromatic edges in G n . Using the coupling approach in Stein's method for Poisson approximation, Barbour, Holst and Janson (1992) (page 105, Theorem 5.G) proved that where d T V denotes the total variation distance and P oi(λ) denotes the Poisson distribution with mean λ. The bound (1.1) implies that if c n → ∞ and m n /c n → λ > 0, the distribution of Y n converges to the Poisson distribution with mean λ. Recently, Bhattacharya, Diaconis and Mukherjee (2013) reproved this Poisson limit theorem by the method of moments. By the same method, they also showed that in the case c n → ∞ and m n /c n → ∞, the distribution of W n , after proper standardization, converges to the standard normal distribution. These limit theorems were called universal limit theorems because they do not require any assumption on the graph structure. For applications of this and related problems, we refer to Bhattacharya, Diaconis and Mukherjee (2013) and the references therein.
In this note, we prove the following result.
Theorem 1.1. Let Y be the number of monochromatic edges in a simple graph with m edges where each vertex is colored independently and uniformly at random with c colors. Let .
We have where d W denotes the Wasserstein distance and N (0, 1) denotes the standard normal distribution.
The bound (1.2) provides a universal error bound for the central limit theorem for W n as c n → ∞ and m n /c n → ∞. A corollary for fixed c n is also obtained in Remark 2.5. The bound (1.2) is obtained by a non-trivial application of Stein's method for normal approximation. Stein's method was introduced by Stein (1972) for normal approximation. Stein's method for Poisson approximation was first studied by Chen (1975) and popularized by Arratia, Goldstein and Gordon (1990). We refer to Barbour and Chen (2005) for an introduction to Stein's method. Stein's method has been widely used to prove limit theorems with error bounds in graph theory. For example, Arratia, Goldstein and Gordon (1990) and Chatterjee, Diaconis and Meckes (2005) used Stein's method to prove Poisson limit theorems for monochromatic cliques in a uniformly colored complete graph. Cerquetti and Fortini (2006) considered more general monochromatic subgraphs counts when the distribution of colors was exchangeable. Janson and Nowicki (1991) studied the asymptotic distribution of the number of copies of a given graph in various random graph models. All of these results are obtained by exploiting the local dependence structure within random variables. In addition to the local dependence structure, we also exploit the uncorrelatedness within W n . This technique of exploiting the uncorrelatedness within random variables was also used in Fang and Röllin (2014) to obtain rates of convergence for the central limit theorem for subgraph counts in random graphs.
In the next section, we give the proof of Theorem 1.1. For each edge i, we denote by v i1 , v i2 the two vertices it connects, i.e., i = (v i1 , v i2 ). Without loss of generality, assume deg(v i1 ) deg(v i2 ) where deg(v) denotes the degree of vertex i. Using the above notation, the standardized number of monochromatic edges can be expressed as

NORMAL APPROXIMATION
. (2.1) Observing that X i and X j are uncorrelated if i = j, we have W = 0, Var(W ) = 1. We will need the following lemmas in the proof of Theorem 1.1.
Lemma 2.1. We have the following bounds on the moments of X i : Proof. The proof is elementary and therefore omitted.

Lemma 2.3 (Lemma 3.2 of Bhattacharya, Diaconis and Mukherjee
The following proposition is the key ingredient in proving Theorem 1.1.

Proposition 2.4. For any function f with bounded first and second derivatives, we have with
where ||g|| := sup x |g(x)| for any function g.

Proof. For each edge
. . , m} to be consisted of all the edges connects to v i1 . Let Since the color of v i1 is independent of W i , we have X i is independent of W i . Therefore, by X i = 0, X 2 i = 1/m, the Taylor expansion and adding and subtracting corresponding terms, we have where U is an independent random variable distributed uniformly in [0, 1] and , First of all, by the Taylor expansion, Therefore, By the same argument and the fact that {X j : j ∈ N i } are jointly independent, For R 3 , by the Taylor expansion, Again by (2.2) and (2.3) and the fact that Therefore, Finallly we bound |R 4 |. By the Cauchy-schwartz inequality, the fact that {X j : j ∈ N i } are jointly independent and X i = 0, Observe that if j ∈ N i \{i} and l ∈ N k \{k}, Cov(X i X j , X k X l ) = 0 unless {i, j} = {k, l} or {i, j, k, l} forms a triangle. For the case {i, j} = {k, l}, For the case {i, j, k, l} forms a triangle, with distinct i, j, k, where the last inequality is by straightforward calculation. Therefore, by (2.3), Lemma 2.3, and observing that each triangle in G gives rise to 3 ordered pairs of (i, j) such that j ∈ N i \{i}, we have, The bound (2.4) follows from (2.5) and the bounds on |R 1 | − |R 4 |.
The bound (1.2) is proved by (2.6) and applying the above bounds in (2.4).
Remark 2.5. The following bound can be obtained following the proof of Theorem 1.1: where C 0 is an absolute constant, K m = m i=1 d i and d i is defined in Lemma 2.2. For fixed c, the above error bound goes to zero if m → ∞ and K m ≪ m 3/2 . This rules out complete graphs where K m ∼ m 3/2 . Proposition 6.1 of Bhattacharya, Diaconis and Mukherjee (2013) gives this counter-example.