1.1. Perturbations of graph sequences

To find laws $Q_n$ for random graphs $Y^n$ that share asymptotic properties with the $X^n\sim P_n$ , we can impose the (sufficient) condition that the Hellinger (or total variational) distance between $P_n$ and $Q_n$ goes to zero in the limit (asymptotic equivalence, see [Reference Janson6, Definition 1.1]); then any property of $(X^n)$ is shared by the graph sequence $(Y^n)$ in the sense that, for any sequence of n-vertex graph events $(A_n)$ , $P_n(A_n) - Q_n(A_n)\to0$ . Asymptotic equivalence occurs if and only if there exists a coupling of $X_n$ and $Y_n$ such that $P(X_n\neq Y_n)$ tends to zero as $n\to\infty$ [Reference Janson6, Theorem 4.2].

Janson argues that asymptotic equivalence is too strong as a condition for the sharing of asymptotic properties: for example, the giant component occurs in a large family of inhomogeneous versions of the ER graph [Reference Bollobás, Janson and Riordan1], much larger than the subset of all asymptotically equivalent graphs. Le Cam’s notion of contiguity [Reference Le Cam8, Reference Le Cam9, Reference Le Cam and Yang10] is a weaker, more appropriate condition for the sharing of asymptotic properties: $(Q_n)$ is said to be contiguous with respect to $(P_n)$ (notation $Q_n\mathop{\vartriangleleft} P_n$ ) if $P_n(A_n)=o(1) \ \Rightarrow\ Q_n(A_n)=o(1)$ for any sequence of n-vertex events $(A_n)$ . Janson applies contiguity to sequences of perturbed ER graphs and demonstrates its wider applicability. (We discuss Janson’s condition for contiguity of inhomogeneous ER graphs in Section 3.)

The main point of this paper is that contiguity is still too strong if the $P_n(A_n)$ are known to converge to zero faster than a certain rate $(a_n)$ (rather than just being o(1)). Given a sequence $a_n\downarrow0$ , we say that $(Q_n)$ is $a_n$ -remotely contiguous with respect to $(P_n)$ (notation $Q_n\mathop{\vartriangleleft} a_n^{-1}P_n$ ) if

(1) \begin{equation} P_n(A_n) = o(a_n) \ \Rightarrow\ Q_n(A_n)=o(1)\end{equation}

for any sequence of n-vertex events $(A_n)$ . Remote contiguity was introduced in [Reference Kleijn7] for the frequentist analysis of Bayesian, posterior-based limits, and argued to offer generalization of contiguity-based statistical arguments, which typically apply in smooth-parametric (e.g. local-asymptotically normal, see [Reference Le Cam8]) models, to a much more general (e.g. non-parametric) setting; see [Reference Kleijn7, Subsection 3.3]. [Reference Falk, Padoan and Rizzelli3] and [Reference Padoan and Rizzelli11] use remote contiguity to generalize a consistency conclusion reached for an idealized sequence of data distributions to the general class of sequences that are realistic for the data in the problem.

To use remote contiguity with sequences of random graphs, we look for sequential models $(P_n)$ for n-vertex graphs with asymptotic properties $(A_n)$ and known $(a_n)$ , as in (1), and analyse the family of those sequential graph distributions $(Q_n)$ that satisfy $Q_n\mathop{\vartriangleleft} a_n^{-1}P_n$ to conclude that random graphs distributed according to $(Q_n)$ share the asymptotic property reflected by the events $(A_n)$ . It is noted that for many asymptotic graph properties, sharp rates $(a_n)$ are known (see, e.g., [Reference van der Hofstad13]).

In Section 2 we briefly review ER random graphs to fix the notation, and we recall Janson’s condition for contiguity of ER graph sequences. In Section 3 we apply remote contiguity to sequences of inhomogeneous ER graphs, compare with Janson’s condition, and formulate a weaker, Lindeberg-type condition that involves the rate sequence $(a_n)$ (which, in most cases, is not only sufficient but also necessary). In Section 4 we define so-called remotely contiguous domains of attraction for all the aforementioned asymptotic connectivity properties of ER graphs. Section 5 summarizes conclusions and discusses possible directions of further research. Details on remote contiguity are discussed in [Reference Kleijn7] and summarized in Appendix A.

The ultimate goal of this paper is to convince the reader that remote contiguity provides a meaningful generalization of the notion of contiguity, applicable to a much wider range of problems. We remark that, like asymptotic equivalence and contiguity, remote contiguity compares any pair of sequences of probability distributions, including (but not limited to) the comparison of ER graph distributions.

3.1. Sufficient conditions for remote contiguity of ER graphs

To extend the results of [Reference Janson6] to remotely contiguous random graphs, consider the likelihood ratio with observation $Y^n$ , which is that of $\frac{1}{2}n(n-1)$ independent Bernoulli experiments:

(4) \begin{align} \frac{{\mathrm{d}} P_n}{{\mathrm{d}} Q_n}(Y^n) & = \prod_{1\leq i < j\leq n}\bigg(\frac{p_{n,ij}}{q_{n,ij}}\bigg)^{Y^n_{ij}} \bigg(\frac{1-p_{n,ij}}{1-q_{n,ij}}\bigg)^{1-Y^n_{ij}} \nonumber \\ & = \prod_{1\leq i < j\leq n}\bigg(\frac{p_{n,ij}}{1-p_{n,ij}}\frac{1-q_{n,ij}}{q_{n,ij}}\bigg)^{Y^n_{ij}} \bigg(\frac{1-p_{n,ij}}{1-q_{n,ij}}\bigg) \nonumber \\ & = \exp\Bigg({-}\sum_{i < j}\big(k_{n,ij}Y^n_{ij} + l_{n,ij}\big)\Bigg), \end{align}

where

\begin{align*} k_{n,ij} = \log\bigg(\frac{q_{n,ij}(1-p_{n,ij})}{p_{n,ij}(1-q_{n,ij})}\bigg), \quad l_{n,ij} = \log\bigg(\frac{1-q_{n,ij}}{1-p_{n,ij}}\bigg).\end{align*}

Before stating the lemma, we note that the Kullback–Leibler divergences of $P_n$ with respect to $Q_n$ equal

\begin{equation*} -\mathbb{E}_{Q_n}\log\frac{{\mathrm{d}} P_n}{{\mathrm{d}} Q_n} = \sum_{i < j}\big(k_{n,ij}q_{n,ij}+l_{n,ij}\big) \geq 0.\end{equation*}

Lemma 1. If we write $\Delta_n=-\mathbb{E}_{Q_n}\log({\mathrm{d}} P_n/{\mathrm{d}} Q_n)+\log(a_n)$ , and for every ${\epsilon}>0$ there is an $M>0$ such that, for large enough n,

(5) \begin{equation} Q_n\Biggl(\sum_{i < j}k_{n,ij}(Y^n_{ij}-q_{n,ij})> \log(M)-\Delta_n\Biggr)<{\epsilon}, \end{equation}

then $Q_n\mathop{\vartriangleleft} a_n^{-1} P_n$ .

Proof. Consider the application of Lemma 8(ii) to the likelihood ratios (4), with $Y^n\sim Q_n$ . For given ${\epsilon}>0$ , let $M>0$ be as in (5); then

\begin{align*} Q_n\Bigg(\sum_{i < j}\big(k_{n,ij}Y^n_{ij} + l_{n,ij}\big)+\log(a_n) > \log(M)\Bigg) < {\epsilon}, \end{align*}

which we may rewrite as $Q_n(a_n(({{\mathrm{d}} P_n}/{{\mathrm{d}} Q_n})(Y^n))^{-1} > M) < {\epsilon}$ for large enough $n\geq1$ .

Clearly, of primary concern is the connection with [Reference Janson6]: the following proposition illustrates how Janson’s Corollary 2.12 (represented in abridged form in Theorem 1) relates to remote contiguity.

Proof. According to (5), remote contiguity revolves around the control of tail probabilities for the sequence $\sum_{i < j} k_{n,ij}(Y_{ij}^n -q_{n,ij})$ . A sufficient condition for uniform tightness is

\begin{align*} \mathbb{E}_{Q_n}\Bigg(\sum_{i < j} k_{n,ij}(Y_{ij}^n-q_{n,ij})\Bigg)^2=O(1), \end{align*}

which we prove below. Note that $\mathbb{E}_{Q_n}\big(\sum_{i < j} k_{n,ij}(Y_{ij}^n -q_{n,ij})\big)^2 = \sum_{i < j} k_{n,ij}^2q_{n,ij}(1-q_{n,ij})$ , and that

\begin{align*} \begin{split} k_{n,ij}^2 & \leq 4\max\bigg\{\bigg|\log\bigg(\frac{q_{n,ij}}{p_{n,ij}}\bigg)\bigg|, \bigg|\log\bigg(\frac{1-q_{n,ij}}{1-p_{n,ij}}\bigg)\bigg|\bigg\}^2 \\ & \leq 4\bigg(\log\bigg(\frac{q_{n,ij}}{p_{n,ij}}\bigg)\bigg)^2 + 4\bigg(\log\bigg(\frac{1-q_{n,ij}}{1-p_{n,ij}}\bigg)\bigg)^2; \end{split} \end{align*}

using the inequality $(\log(y))^2 \leq (y-1)^2/y$ for $y > 0$ either with $y= q_{n,ij}/p_{n,ij}$ or with $y=(1-q_{n,ij})/(1-p_{n,ij})$ , we obtain

\begin{align*} \begin{split} k_{n,ij}^2 & \leq 4 \bigg(\frac{q_{n,ij}}{p_{n,ij}}-1\bigg)^2\frac{p_{n,ij}}{q_{n,ij}} + 4\bigg(\frac{1-q_{n,ij}}{1-p_{n,ij}}-1\bigg)^2\frac{1-p_{n,ij}}{1-q_{n,ij}} \\ & = 4\frac{(p_{n,ij} -q_{n,ij} )^2 }{p_{n,ij}} \frac{1}{q_{n,ij}} + 4\frac{(p_{n,ij} -q_{n,ij} )^2 }{1-p_{n,ij}} \frac{1}{1-q_{n,ij}}. \end{split} \end{align*}

Substituting and using that $\sup_{i < j} p_{n,ij} < C/(1+C)$ , we find

\begin{align*} \begin{split} \sum_{i < j} k_{n,ij}^2 q_{n,ij}(1-q_{n,ij}) & \leq 4 \sum_{i < j} \frac{(p_{n,ij} -q_{n,ij} )^2 }{p_{n,ij}} (1-q_{n,ij}) + 4 \sum_{i < j} \frac{(p_{n,ij} -q_{n,ij} )^2 }{1-p_{n,ij}} q_{n,ij} \\ & \leq 4(1+C) \sum_{i < j} \frac{(p_{n,ij} -q_{n,ij} )^2 }{p_{n,ij}}, \end{split} \end{align*}

and under Janson’s condition (3) the term on the right-hand side is O(1). In addition, in view of the inequality $\log(y) \leq (y-1)$ for $y>0$ , we have

\begin{align*} \begin{split} -\mathbb{E}_{Q_n}\log\frac{{\mathrm{d}} P_n}{{\mathrm{d}} Q_n} & = \sum_{i < j}\log\bigg(\frac{q_{n,ij}}{p_{n,ij}}\bigg)q_{n,ij} + \sum_{i < j}\log\bigg(\frac{1-q_{n,ij}}{1-p_{n,ij}}\bigg)(1-q_{n,ij}) \\ & \leq \sum_{i < j}\bigg(\frac{q_{n,ij}}{p_{n,ij}}-1\bigg)q_{n,ij} + \sum_{i < j}\bigg(\frac{1-q_{n,ij}}{1-p_{n,ij}}-1\bigg)(1-q_{n,ij}) \\ & = \sum_{i < j}\frac{(q_{n,ij} -p_{n,ij})^2}{p_{n,ij}(1-p_{n,ij})} \leq (1+C)\sum_{i < j}\frac{(q_{n,ij} -p_{n,ij})^2}{p_{n,ij}}. \end{split} \end{align*}

Under Janson’s condition (3) the term on the right-hand side is O(1). Consequently, $-\Delta_n\to \infty$ for any $a_n \to 0$ . Together with the uniform tightness of the sequence of sums $\sum_{i < j}(k_{n,ij}Y_{ij}^n +l_{n,ij})$ , this leads to remote contiguity $Q_n\mathop{\vartriangleleft} a_n^{-1}P_n$ for any $a_n\to 0$ .

This contiguity proof, however, does not exploit the presence of a sum of independent components to the full extent. To sharpen the argument, we normalize the sums appropriately and impose sufficient conditions for remote contiguity. In this case the contributions to sums are independent, and the appropriate normalization constants are

\begin{align*}s_n^2=\sum_{i < j}k^2_{n,ij}q_{n,ij}(1-q_{n,ij})\end{align*}

for all $n\geq1$ , as per Lindeberg’s theorem. To illustrate how conditions for remote contiguity weaken those for contiguity, we note that Janson’s condition (3) implies that $s_n^2$ remains bounded, whereas in Lemma 2 $s_n^2$ may diverge.

Lemma 2. Assume that $s_n<\infty$ for every $n\geq1$ , and $s_n\to\infty$ . Suppose that, for every ${\epsilon}>0$ ,

(6) \begin{equation} \frac{1}{s_n^2}\sum_{i < j}\mathbb{E}_{Q_n}\bigl( k_{n,ij}^2(Y^n_{ij}-q_{n,ij})^2 \boldsymbol{1}_{\{k_{n,ij}|Y^n_{ij}-q_{n,ij}| > {\epsilon} s_n\}} \bigr) \to 0. \end{equation}

Then $Q_n\mathop{\vartriangleleft} a_n^{-1} P_n$ for any $(a_n)$ , $a_n\downarrow0$ , such that

(7) \begin{equation} \frac1{s_n}\Bigg(\sum_{i < j}\bigl(k_{n,ij}q_{n,ij}+l_{n,ij}\bigr)+\log(a_n)\Bigg)\to-\infty. \end{equation}

3.2. Necessary conditions for remote contiguity of ER graphs

Next, we argue that, when (6) holds and $s_n \to \infty$ , then (7) is also a necessary condition for remote contiguity at rate $a_n$ .

Proof. For every $n\geq1$ , let $\nu_n$ be a measure that dominates both $P_n$ and $Q_n$ (e.g. $\nu_n= (P_n +Q_n)/2$ ) and define $p_n={\mathrm{d}} P_n/{\mathrm{d}}\nu_n$ and $q_n={\mathrm{d}} Q_n/{\mathrm{d}}\nu_n$ . Let $(a_n)$ , $(b_n)$ be such that $a_n,b_n>0$ , $a_n,b_n\downarrow 0$ , and $s_n^{-1}\log (b_n) \to 0$ . Define $C_n = \lbrace y^n \in \mathscr{X}_n \colon q_n(y^n) > p_n(y^n)/({a_n b_n})\rbrace$ . It is immediate that $P_n(C_n) \leq \int_{C_n} a_n b_n q_n(y^n)\,{\mathrm{d}}\nu_n(y^n) \leq a_n b_n Q_n(C_n)=o(a_n)$ . For any $\eta>0$ and n large enough, we find

\begin{align*} Q_n(C_n) & = Q_n\bigg(a_n b_n\bigg(\frac{{\mathrm{d}} P_n}{{\mathrm{d}} Q_n}(Y_n)\bigg)^{-1} >1\bigg) \\ & = Q_n \Bigg(\frac{\sum_{i < j}k_{n,ij}(Y_{ij}^n-q_{n,ij})}{s_n} > \frac{-\log(b_n)}{s_n}-\frac{\Delta_n}{s_n}\Bigg) \\ & \geq Q_n\Bigg(\frac{\sum_{i < j}k_{n,ij}(Y_{ij}^n-q_{n,ij})}{s_n} > \eta-\frac{\Delta_n}{s_n}\Bigg), \end{align*}

since $-\log (b_n) /s_n \to 0$ . If (7) does not hold, $\lim_{n\to \infty} \Delta_n/s_n>-\infty$ , so that, for some $M>0$ and all n large enough,

\begin{equation*} Q_n(C_n) \geq Q_n \Bigg(\frac{\sum_{i < j}k_{n,ij}(Y_{ij}^n-q_{n,ij})}{s_n} > \eta + M\Bigg). \end{equation*}

By Theorem 5, $\sum_{i < j} k_{n,ij}(Y_{ij}^n -q_{n,ij})/s_n$ converges weakly to a standard normal distribution.

Then $\liminf_{n \to \infty} Q_n(C_n)>0$ and $(Q_n)$ is not $a_n$ -remotely contiguous with respect to $(P_n)$ . We conclude that (7) is necessary for remote contiguity at rate $(a_n)$ .

To estimate whether sequential data $(Y^n)$ , $Y^n\in\mathscr{X}_n$ , was generated by $(P_n)$ or by $(Q_n)$ , statisticians use (randomized tests based on) test functions $\phi_n\colon\mathscr{X}_n\to[0,1]$ . A test function is considered (minimax-)optimal if it minimizes the sum of type-I and type-II errors:

\begin{align*} \pi_n(\phi_n) = \mathbb{E}_{P_n}\phi_n(Y^n) + \mathbb{E}_{Q_n}(1-\phi_n(Y^n)).\end{align*}

As it turns out, there is a general upper bound for $\pi_n$ in terms of the Hellinger affinity $\alpha(P_n,Q_n)$ between $P_n$ and $Q_n$ :

\begin{align*} \inf_\phi \pi_n(\phi) \leq \int_{\mathscr{X}_n}\sqrt{p_n(y^n)q_n(y^n)}\,{\mathrm{d}}\nu_n(y^n)\quad(\,=\!:\,\alpha(P_n,Q_n)),\end{align*}

and the likelihood ratio test function minimizes $\pi_n$ (see also [Reference Le Cam9, Section 16.4]). Clearly, if there exists a sequence $(\phi_n)$ such that $\pi_n(\phi_n)=o(1)$ , then $(P_n)$ and $(Q_n)$ are not contiguous.

To reason likewise regarding $a_n$ -remote contiguity, consider an $a_n$ -weighted version of $\pi_n$ , $\pi'_n(\phi_n) = a_n^{-1}\mathbb{E}_{P_n}\phi_n(Y^n) + \mathbb{E}_{Q_n}(1-\phi_n(Y^n))$ . Reasoning the same as in the contiguous case, we find the following correspondence between testability and remote contiguity.

Proof. For every $n\geq1$ , the likelihood ratio test function, defined for all $y^n\in\mathscr{X}_n$ by $\psi_n(y^n) = \boldsymbol{1}_{\{q_n>a_n\,p_n\}}(y^n)$ , minimizes $\pi'_n$ . Suppose that there exists a sequence $(\phi_n)$ such that $\pi'_n(\phi_n)=o(1)$ . Then $\pi'_n(\psi_n)=o(1)$ , so there are events $A_n=\{y^n\colon q_n(y^n)>a_n\,p_n(y^n)\}$ such that $P_n(A_n)=o(a_n)$ , but $Q_n(A_n)\to1$ , showing that $(P_n)$ and $(Q_n)$ are not $a_n$ -remotely contiguous. We note the following upper bound for $\pi'_n$ :

\begin{align*} \begin{split} \pi'_n(\psi_n) = \inf_{\phi}\pi'_n(\phi) & = \int_{\{q_n>a_n\,p_n\}} p_n(y^n)\,{\mathrm{d}}\nu_n(y^n) + \int_{\{q_n\leq a_n\,p_n\}} q_n(y^n)\,{\mathrm{d}}\nu_n(y^n) \\ & \leq \int_{\{q_n>a_n\,p_n\}} \sqrt{a_n^{-1}\,p_n(y^n)q_n(y^n)}\,{\mathrm{d}}\nu_n(y^n) \\ & \quad + \int_{\{q_n\leq a_n\,p_n\}} \sqrt{a_n\,p_n(y^n)q_n(y^n)}\,{\mathrm{d}}\nu_n(y^n) \leq a_n^{-1/2}\,\alpha(P_n,Q_n), \end{split} \end{align*}

where the last bound holds for large enough n.

In the proof of Lemma 3 we change the argument of Lemma 4 slightly (through the inclusion of a separate sequence $b_n\downarrow0$ ), but the essence is the same: the existence of certain test sequences precludes remote contiguity.

In the case of n-vertex ER graph distributions, the Hellinger affinity is equal to the product of the Hellinger affinities for each of the independent, Bernoulli-distributed random variables $Y^n_{ij}$ :

(8) \begin{equation} \alpha(P_n,Q_n) = \prod_{i < j}\big(\sqrt{p_{n,ij}\,q_{n,ij}\vphantom{)}} + \sqrt{(1-p_{n,ij})(1-q_{n,ij})}\big).\end{equation}

3.3. Perturbations of ER graphs

Lemma 2 formulates a condition that delimits the range of applicability for remote contiguity in terms of a Lindeberg-type condition involving the sequence of Kullback–Leibler divergences. In this section we simplify that condition with sufficient conditions formulated directly in terms of the defining parameters of the ER graphs.

Lemma 5. Choose $q_{n,ij}=\lambda_{n,ij}/n$ , $p_{n,ij}=\mu_{n,ij}/n$ with $0 \leq \lambda_{n,ij},\mu_{n,ij}\leq n$ , and define $P_n=P_{(p_{n,ij}),n}$ , $Q_n=P_{(q_{n,ij}),n}$ for all $n\geq1$ and all $1\leq i,j\leq n$ . Assume that

(9) \begin{equation} r_n \,:\!=\, \sup_{i < j} \frac{|\mu_{n,ij}-\lambda_{n,ij}|}{\mu_{n,ij}} \to 0, \qquad R_n \,:\!=\, \sum_{i < j}\frac{(\lambda_{n,ij}-\mu_{n,ij})^2}{\mu_{n,ij}(n-\mu_{n,ij})} \to \infty. \end{equation}

Then $Q_n\mathop{\vartriangleleft} a_n^{-1}P_n$ if and only if $a_n=o(\exp(\!-R_n))$ . If, instead, $R_n=O(1)$ , then $Q_n\mathop{\vartriangleleft} P_n$ .

Proof. Let $n\geq1$ be given. Using the inequality $\log (1+x) \leq x$ , valid for any $x>-1$ , we estimate the Kullback–Leibler divergence as follows:

\begin{align*} -\mathbb{E}_{Q_n}\log\frac{{\mathrm{d}} P_n}{{\mathrm{d}} Q_n}(Y^n) & = \sum_{i < j}\log\big(\frac{q_{n,ij}}{p_{n,ij}}\bigg)q_{n,ij} + \sum_{i < j}\log\bigg(\frac{1-q_{n,ij}}{1-p_{n,ij}}\bigg)(1-q_{n,ij}) \nonumber\\[-2pt] & = \sum_{i < j}\bigg[\frac{\lambda_{n,ij}}{n}\log\bigg(1+\frac{\lambda_{n,ij}-\mu_{n,ij}}{\mu_{n,ij}}\bigg) \nonumber\\[-2pt] & \qquad\quad + \sum_{i < j}\log\bigg(1-\frac{(\lambda_{n,ij}-\mu_{n,ij})/n}{1-\mu_{n,ij}/n}\bigg) \bigg(1-\frac{\lambda_{n,ij}}{n}\bigg)\bigg] \nonumber\\[-2pt] &\leq \sum_{i < j}\frac{\lambda_{n,ij}-\mu_{n,ij}}{n} \bigg(\frac{\lambda_{n,ij}}{\mu_{n,ij}} - \frac{1-\lambda_{n,ij}/n}{1-\mu_{n,ij}/n}\bigg) \nonumber\\[-2pt] & = \sum_{i < j}\bigg(1+\frac{\mu_{n,ij}/n}{1-\mu_{n,ij}/n}\bigg)\frac{(\lambda_{n,ij}-\mu_{n,ij})^2}{n\mu_{n,ij}} = \sum_{i < j}\frac{(\lambda_{n,ij}-\mu_{n,ij})^2}{\mu_{n,ij}(n-\mu_{n,ij})}. \end{align*}

Note that, by (2), (9), and the expansion $\log (1+x)=x+O(x^2)$ ( $x\to0$ ), we have

\begin{align*} \begin{split} k^2_{n,ij} & = \bigg(\log\bigg(\frac{\lambda_{n,ij}}{\mu_{n,ij}} \frac{1-\mu_{n,ij}/n}{1-\lambda_{n,ij}/n}\bigg)\bigg)^2 \\ & = \bigg(\log\bigg(1+\frac{\lambda_{n,ij}-\mu_{n,ij}}{\mu_{n,ij}}\bigg) - \log\bigg(1-\frac{(\lambda_{n,ij}-\mu_{n,ij})/n}{1-\mu_{n,ij}/n}\bigg)\bigg)^2 \\ & = \bigg(\frac{\lambda_{n,ij}-\mu_{n,ij}}{\mu_{n,ij}}\bigg)^2\bigg(\frac1{1-\mu_{n,ij}/n} + O(r_n)\bigg)^2 \end{split} \end{align*}

for all $1\leq i < j\leq n$ , and

\begin{align*} \begin{split} q_{n,ij}(1-q_{n,ij}) & = \frac{\lambda_{n,ij}}{n}\bigg(1-\frac{\lambda_{n,ij}}{n}\bigg) \\ & = \frac{\mu_{n,ij}}{n}\bigg(1+\frac{\lambda_{n,ij}-\mu_{n,ij}}{\mu_{n,ij}}\bigg) \bigg(1-\frac{\mu_{n,ij}}{n}\bigg(1+\frac{\lambda_{n,ij}-\mu_{n,ij}}{\mu_{n,ij}}\bigg)\bigg) \\ & = \frac{\mu_{n,ij}}{n}\bigg( 1 -\frac{\mu_{n,ij}}{n}+O(r_n)\bigg). \end{split} \end{align*}

As a consequence of these two displays, we have

\begin{align*} \begin{split} s_n^2 & = \sum_{i < j}k^2_{n,ij}q_{n,ij}(1-q_{n,ij}) \\ & = \sum_{i < j}\bigg(\frac{\lambda_{n,ij}-\mu_{n,ij}}{\mu_{n,ij}}\bigg)^2\bigg(\frac1{1-\mu_{n,ij}/n}+O(r_n)\bigg)^2 \frac{\mu_{n,ij}}{n}\bigg(1-\frac{\mu_{n,ij}}{n}+O(r_n) \bigg) \\ & = \sum_{i < j}\frac{(\lambda_{n,ij}-\mu_{n,ij})^2}{\mu_{n,ij}(n -\mu_{n,ij})} \bigg(1+\bigg(1-\frac{\mu_{n,ij}}{n}\bigg) O(r_n)\bigg)^2({1 + O(r_n)} ) \\ & = (1+o(1))\sum_{i < j} \frac{(\lambda_{n,ij}-\mu_{n,ij})^2}{\mu_{n,ij}(n-\mu_{n,ij})}. \end{split} \end{align*}

Condition (7) is then satisfied for any $a_n$ such that $\limsup_{n \to \infty } R_n^{-1}\log (a_n) < -1$ . It remains to verify condition (6). To that end, consider, for any ${\epsilon},\delta>0$ ,

\begin{align*} \frac{1}{s_n^2}\sum_{i < j}\mathbb{E}_{Q_n} & \big(k_{n,ij}^2(Y^n_{ij}-q_{n,ij})^2\boldsymbol{1}_{\{k_{n,ij}|Y^n_{ij}-q_{n,ij}| > {\epsilon} s_n\}} \big) \\ & \leq \frac{1}{{\epsilon}^\delta s_n^{2+\delta}} \sum_{i < j}\mathbb{E}_{Q_n}|k_{n,ij}|^{2+\delta}|Y^n_{ij}-q_{n,ij}|^{2+\delta} \\ & = \frac{1}{{\epsilon}^\delta s_n^{2+\delta}}\sum_{i < j}|k_{n,ij}|^{2+\delta} q_{n,ij}(1-q_{n,ij}) \big((1-q_{n,ij})^{1+\delta} + q_{n,ij}^{1+\delta}\big) \nonumber\\ & \leq \frac{1}{{\epsilon}^\delta s_n^{2+\delta}}\|k_n\|_{\infty,n}^\delta\sum_{i < j}k_{n,ij}^2q_{n,ij}(1-q_{n,ij}) \\ & = \frac1{{\epsilon}^\delta}\bigg(\frac{\|k_n\|_{\infty,n}}{s_n}\bigg)^\delta \leq O\bigg(\frac{r_n}{R_n^{1/2}}\bigg)^\delta \to 0, \end{align*}

implying that condition (6) is satisfied. All the assumptions of Lemma 2 are thus fulfilled and an application of the latter, along with Lemma 3, yields the first result.

If we replace the hypothesis $R_n \to \infty$ with $R_n=O(1)$ , we have

\begin{align*} -\mathbb{E}_{Q_n}\log\frac{{\mathrm{d}} P_n}{{\mathrm{d}} Q_n}(Y^n) = O(R_n) = O(1). \end{align*}

The hypothesis $R_n=O(1)$ also implies that $s_n^2 = (1+o(1))R_n =O(1)$ and, in turn, uniform tightness of the sequence $\sum_{i < j}k_{n,ij}(Y_{ij}^n-q_{n,ij})$ , so that $Q_n\mathop{\vartriangleleft} P_n$ .

The following results give examples of applications of the previous lemma to inhomogeneous and homogeneous perturbations of a homogeneous ER graph.

Corollary 1. (Homogeneous perturbation of the ER graph.) Choose $q_{n,ij}=\lambda_{n}/n$ , $p_{n,ij}=\lambda/n$ with $0 \leq \lambda_{n}\leq n$ , $1\leq i,j\leq n$ , and define $P_n=P_{(p_{n,ij}),n}$ , $Q_n=P_{(q_{n,ij}),n}$ for all $n\geq1$ . Assume that $\lambda_n\to\lambda$ . If $n(\lambda -\lambda_n)^2 \to \infty$ , then $Q_n \mathop{\vartriangleleft} a_n^{-1} P_n$ if and only if

\begin{align*} a_n=o\bigg(\exp\bigg({-}\frac{n}{2\lambda}(\lambda_n-\lambda)^2\bigg)\bigg). \end{align*}

If instead $n(\lambda_n-\lambda)^2=O(1)$ , then $Q_n \mathop{\vartriangleleft} P_n$ .

Proof. We have $r_n=|\lambda_n-\lambda|/\lambda$ and, as $n \to \infty$ ,

\begin{align*} R_n = \binom{n}{2}\frac{(\lambda_n-\lambda)^2}{\lambda(n-\lambda)} = \frac12(1+O(n^{-1}))\frac{n(\lambda_n-\lambda)^2}{\lambda}. \end{align*}

Then the result follows immediately from Lemma 5.

Corollary 2. (Inhomogeneous perturbation of the ER graph.) Choose $q_{n,ij}=\lambda_{n,ij}/n$ , $p_{n,ij}=\lambda/n$ with $0 \leq \lambda_{n}\leq n$ , $1\leq i,j\leq n$ , and define $P_n=P_{(p_{n,ij}),n}$ , $Q_n=P_{(q_{n,ij}),n}$ for all $n\geq1$ . Assume that $\|\lambda_n-\lambda\|_{\infty,n}=o(1)$ . If also $n^{-1}\| \lambda_n - \lambda \|_{2,n}^2 \to \infty$ , then $Q_n\mathop{\vartriangleleft} a_n^{-1}P_n$ if and only if

\begin{align*} a_n = o\bigg(\exp\bigg({-}\frac{\|\lambda_n-\lambda\|_{2,n}^2}{n\lambda}\bigg)\bigg). \end{align*}

If instead $n^{-1}\| \lambda_n - \lambda \|_{2,n}^2 =O(1)$ , then $Q_n\mathop{\vartriangleleft} P_n$ .

Proof. The result follows immediately from Lemma 5 when we note that $r_n=\| \lambda_n - \lambda \|_{n,\infty}/\lambda$ and that

\begin{equation*} R_n = \frac{ \| \lambda_n - \lambda \|_{2,n}^2} {\lambda(n-\lambda)} = (1+O(n^{-1}))\frac{\| \lambda_n - \lambda \|_{2,n}^2}{n \lambda} \quad \text{as $n \to \infty$}. \end{equation*}

4.1. The giant component in the supercritical ER graph

As was first shown in [Reference Erdös and Rényi2], every supercritical ER graph contains a giant component, i.e. a sequence of connected components in $X^n$ containing a non-vanishing fraction of all vertices, with probability growing to 1. More precisely, we have the following theorem.

Theorem 2. For every $\lambda>1$ , $\nu\in\big(\frac12,1\big)$ there is a $\delta(\lambda,\nu)>0$ such that

(10) \begin{equation} P_{\lambda,n}\big(\big||\mathscr{C}_{\mathrm{max}}|-\zeta_\lambda n\big| > n^{\nu}\big) = O\big(n^{-\delta(\lambda,\nu)}\big), \end{equation}

where $\zeta_\lambda$ is the survival probability of a Poisson branching process with mean offspring $\lambda$ .

For a proof of this classical result (and the specific way in which $\delta(\lambda,\nu)$ depends on $\lambda$ and $\nu$ ) see, for example, [Reference van der Hofstad13, Theorem 4.8].

To generalize the occurrence of a giant component, we set conditions for sequences of laws $(Q_n)$ of ER graphs such that, for some $0 < \delta < \delta(\lambda,\nu)$ ,

(11) \begin{equation} Q_n \mathop{\vartriangleleft} n^{\delta} P_{\lambda,n}.\end{equation}

Let $\mathscr{Q}(\lambda,\delta)$ denote the collection of all sequences $(Q_n)$ in $\prod_n M^1(\mathscr{X}_n)$ satisfying (11). In all those cases, a giant component containing an asymptotic fraction $\zeta_\lambda$ of all vertices occurs (to within order $n^\nu$ vertices). For given $\lambda>1$ and $\nu\in\big(\frac12,1\big)$ , the sequences $(Q_n)$ for which (11) holds for some $0 < \delta < \delta(\lambda,\nu)$ is the union $\mathscr{Q}(\lambda,\nu)=\bigcup\{\mathscr{Q}(\lambda,\delta):0<\delta<\delta(\lambda,\nu)\}$ and, for given $\lambda>1$ , the union over all $\nu$ contains the cases in which a giant component containing a fraction $\zeta_\lambda$ occurs (to within some negligible $n^\nu$ -fraction of the vertices). We call $\mathscr{Q}(\lambda)=\cup\big\{\mathscr{Q}(\lambda,\nu)\colon\nu\in\big(\frac12,1\big)\big\}$ the remotely contiguous domain of attraction for the occurrence of a giant component containing $\zeta_\lambda n$ vertices. Ultimately, the union $\mathscr{Q}=\cup\{\mathscr{Q}(\lambda)\colon\lambda>1\}$ forms a class in which a giant component (containing some asymptotically non-vanishing fraction of all vertices) will form with probability growing to 1. We refer to that class as the remotely contiguous domain of attraction for the occurrence of a giant component.

As we have seen in Corollary 1, $R_n={\frac12{n}}\lambda^{-1}(1+O(n^{-1}))(\lambda_n -\lambda)^2$ , so to render the rate $a_n$ in definition (1) high enough to cover the probabilities for occurrence of a giant component, cf. Theorem 2, we choose

(12) \begin{equation} \lambda_n = \bigg(1 + \sqrt{\frac{2\delta}{\lambda}\frac{\log(n)}{n}}\bigg)\lambda, \end{equation}

so that $R_n = (1+O(n^{-1}))\delta\log(n)$ . Then $n (\lambda_n - \lambda)^2 \to \infty$ and $n^{-\delta(\lambda,\nu)}=o(\exp(\!-R_n))$ . Therefore, by Corollary 1, we can conclude that, for all homogeneous $\lambda_n$ -perturbations of the ER graph sequence of the form (12), a giant component containing an asymptotic fraction $\zeta_\lambda$ of the vertices occurs to within order- $n^\nu$ vertices, $P_{\lambda_n,n}\big(\big||\mathscr{C}_{\text{max}}|-\zeta_\lambda n\big| > n^{\nu}\big) = o(1)$ , since (10) ensures the occurrence of such a giant component at $\lambda>1$ . We may therefore characterize the class of homogeneous ER graphs in the remotely contiguous domain of attraction for the occurrence of a giant component containing $\zeta_\lambda n$ vertices as

\begin{align*} \mathscr{H}\cap\mathscr{Q}(\lambda) = \bigcup_{\nu\in(1/2,1)}\bigg\{ (P_{\lambda_n,n})\in\mathscr{H}\colon (\lambda_n-\lambda)^2 < 2\lambda\,\delta(\lambda,\nu){\frac{\log(n)}{n}, \,n\geq1}\bigg\}, \end{align*}

and the class of homogeneous ER graphs in the remotely contiguous domain of attraction for the occurrence of a giant component as $\mathscr{H}\cap\mathscr{Q} = \bigcup\{\mathscr{H}\cap\mathscr{Q}(\lambda)\colon\lambda>1\}$ .

Example 2. (Inhomogeneous perturbation of supercritical ER graphs.) Denote by $\mathscr{I}_{\infty}$ the class of inhomogeneous ER graphs obtained via uniform perturbations of a homogeneous ER graph $P_{\lambda,n}$ , with $\lambda>0$ , i.e. the class of ER graphs with edge probabilities $q_{n,ij}=\lambda_{n,ij}/n$ satisfying $\sup_{i < j}|\lambda_{n,ij}-\lambda|\to0$ for some $\lambda>0$ . In an analogous fashion, resorting to Corollary 2, we can characterize the class of uniformly perturbed inhomogeneous ER graphs in $\mathscr{Q}(\lambda)$ , with $\lambda>1$ , as

\begin{align*} \mathscr{I}_\infty \cap \mathscr{Q}(\lambda) = \bigcup_{\nu\in(1/2,1)}\Bigg\{ (P_{(\lambda_{n,ij}),n})\in\mathscr{I}_\infty\colon \sum_{i < j}(\lambda_{n,ij}- \lambda)^2 < \lambda\,\delta(\lambda,\nu)\,n\log(n), \,n\geq1\Bigg\}, \end{align*}

and the class of uniformly perturbed inhomogeneous ER graphs in the remotely contiguous domain of attraction for the occurrence of a giant component as

\begin{align*} \mathscr{I}_\infty\cap\mathscr{Q} = \bigcup\big\{\mathscr{I}_\infty\cap\mathscr{Q}(\lambda)\colon\lambda>1\big\}. \end{align*}

4.2. Fragmentation in subcritical ER graphs

Define $I_\lambda=\lambda-1-\log(\lambda)$ for $0 < \lambda < 1$ , that is, for the subcritical regime of the ER graph.

For a proof of Theorem 3, see, for example, [Reference van der Hofstad13, Theorems 4.4 and 4.5]. To prove that the largest connected component in other random graph sequences has cardinality lying between two multiples of $\log(n)$ , we again require (11) for some $0<\delta< \min(\delta(a,\lambda), \eta(a', \lambda))\,=\!: \zeta(\lambda, a, a')$ and $0 < a < I_\lambda^{-1} < a'$ . We thus define the remotely contiguous domain of attraction for fragmentation into clusters of maximal cardinality $I_{\lambda}^{-1}\log(n)$ :

\begin{align*} \mathscr{Q}(\lambda,a,a')=\bigcup\big\{\mathscr{Q}(\lambda,\delta)\colon0<\delta<\zeta(\lambda,a, a')\big\}.\end{align*}

The union over all $0 < \lambda < 1$ and $0 < a < I_\lambda^ < a' < \infty$ forms the remotely contiguous domain of attraction for fragmentation into clusters of maximal cardinality of order $\log(n)$ :

\begin{align*} \mathscr{L} = \bigcup\big\{\mathscr{Q}(\lambda,a,a')\colon0 < \lambda < 1,\,0 < a < I_\lambda^{-1} < a'< \infty\big\}.\end{align*}

Example 3. (Homogeneous perturbation of subcritical ER graphs.) Following reasoning similar to Example 1 and applying Corollary 1, we characterize the class of ER graphs with maximal connected component of order $\log(n)$ , which are obtained by homogeneous perturbations of a subcritical graphs with $0 < \lambda < 1$ , as

\begin{align*} \begin{split} \mathscr{L}(\lambda) = \bigcup\bigg\{(P_{\lambda_n,n})\in\mathscr{H} \,\colon & 0 < a < I_\lambda^{-1} < a' < \infty, \\ & (\lambda_n-\lambda)^2 < 2\lambda\zeta(\lambda,a,a')\frac{\log(n)}{n},\, n\geq1\bigg\}. \end{split} \end{align*}

And we define the homogeneous part of the remotely contiguous domain of attraction for fragmentation into clusters of maximal cardinality of order $\log(n)$ by $\mathscr{L}\cap\mathscr{H}=\cup_{0<\lambda<1}\mathscr{L}(\lambda)$ .

4.3. Maximal connected components in the critical ER graph

It is well known that the largest connected components in a sequence of ER graphs at criticality ( $\lambda=1$ ) have cardinalities of order $O(n^{2/3})$ . In fact, there exists a so-called critical window of $O(n^{-1/3})$ homogeneous perturbations around $\lambda=1$ for which this critical behaviour of the largest connected component remains valid.

For a proof, see, for example, [Reference van der Hofstad13, Theorem 5.1].

We now examine to what extent the remotely contiguous domain of attraction for occurrence of a maximal connected component of order (approximating) $n^{2/3}$ around the critical point $\lambda=1$ coincides with the perturbations of order $n^{-1/3}$ in the parameter $\lambda$ that Theorem 4 guarantees.

To reformulate the question: for some $\lambda_n\to1$ , define the homogeneous ER graphs $Y_n$ distributed according to $Q_n=P_{\lambda_n,n}$ , $P_n=P_{1,n}$ , and analyse the requirement $Q_n\mathop{\vartriangleleft} \omega_n P_n$ for any rate $a_n=1/\omega_n$ .

To render the assertion of Theorem 4 at $\lambda=1$ amenable to extension by remote contiguity, we have to make a choice for a sequence $a_n\to0$ : applied to $\lambda=1$ , Theorem 4 guarantees that there exists a constant $b=b(0)>0$ such that

(13) \begin{equation} P_{1,n}\big(|\mathscr{C}_{\text{max}}| < a_n n^{2/3}\ \text{or} \ |\mathscr{C}_{\text{max}}| > a_n^{-1} n^{2/3} \big) \leq {b}\,a_n.\end{equation}

We examine the family of perturbed ER graphs that displays the same $a_n$ -adjusted critical maximal cluster size of order $n^{2/3}$ .

The choice for $(a_n)$ is of great influence on the maximal permitted perturbation $|\lambda_n-1|$ .

Proof. For $Q_n=P_{\lambda_n,n}$ , $P_n=P_{{1},n}$ , the Hellinger affinity (cf. (8)) is given by

\begin{align*} \begin{split} \alpha(P_n,Q_n) & = \bigg(\frac{\lambda_n^{1/2}}{n} + \bigg(1-\frac{1}{n}({\lambda_n}+1)+\frac{\lambda_n}{n^2}\bigg)^{1/2}\bigg)^{\binom{n}{2}} \\ & = \bigg(1+\frac{(1+(\lambda_n-1))^{1/2}}{n}-\frac{\lambda_n+1}{2n}+ O(n^{-2})\bigg)^{\binom{n}{2}} \\ & = \bigg(1+\frac{1+\frac{1}{2}(\lambda_n-1)-\frac{1}{8}(\lambda_n-1)^2}{n}- \frac{\lambda_n+1}{2n}+O(n^{-1}(\lambda_n-1)^3)+O(n^{-2})\biggr)^{\binom{n}{2}} \\ & = \bigg(1-\frac{(\lambda_n-1)^2}{8n}+O(n^{-1}(\lambda_n-1)^3)+O(n^{-2})\bigg)^{\binom{n}{2}} \\ & = \exp\bigg({-}\frac{1}{16}n(\lambda_n-1)^2+O(n(\lambda_n-1)^3)+O(1)\bigg) \\ & = A\exp\bigg({-}\frac{1}{16}n(\lambda_n-1)^2\bigg)+o(1) \end{split} \end{align*}

for some constant $A>0$ .

A slightly more detailed version of this proof shows that, whenever $\lambda_n-1=o(n^{-1/2})$ , the representation of $\alpha(P_{\lambda_n,n}, P_{1,n})$ in Lemma 6 holds with $A=1$ and therefore $P_{\lambda_n,1} \mathop{\vartriangleleft} \,P_{1,n}$ , in which case remote contiguity applies with any $a_n$ decaying to 0. For example, for some small ${\epsilon}>0$ and the choice $a_n=n^{-{\epsilon}}$ , we find that

(14) \begin{equation} P_{\lambda_n,n}\big(|\mathscr{C}_{\text{max}}| < n^{2/3-{\epsilon}}\ \text{or} \ |\mathscr{C}_{\text{max}}| > n^{2/3+{\epsilon}} \big)\to0.\end{equation}

On the other hand, in light of Lemma 3, if $n|\lambda_n-1|^2$ goes to $\infty$ fast enough, that is, if $\alpha_n(P_{\lambda_n,1}, P_{1,n})=o(a_n^{1/2})$ , $(P_{\lambda_n})$ is not $a_n$ -remotely contiguous with respect to $(P_{1,n})$ . For example, for some small ${\epsilon}>0$ and the choice $a_n=n^{-{\epsilon}}$ , we find that if

\begin{align*}\liminf_{n\to\infty}\frac{n(\lambda_n-1)^2}{\log(n)} > 8{\epsilon}\end{align*}

then $(P_{\lambda_n,n})$ is not $n^{-{\epsilon}}$ -remotely contiguous with respect to $(P_{1,n})$ . An application of Corollary 1 allows us to further refine the requirement by imposing $a_n =o(\exp(\!-n(\lambda_n -1)^2/2))$ . For example, if we choose $a_n$ to decrease as $\log(n)^{-1}$ , then the above shows that remote contiguity limits the perturbation to be of smaller order than $\sqrt{\log(\log(n))/n}$ .

Unfortunately, remotely contiguous domains of attraction for near-critical maximal cluster sizes (for example, those intended in (14)) have an extent of order $(n^{-1}\log(n))^{1/2}$ , not the order $n^{-1/3}$ that occurs in Theorem 4. So remote contiguity does not cover the entire range of possible perturbations that preserve near-critical maximal cluster sizes. If we impose perturbations proportional to $n^{-1/3}$ , requiring remote contiguity leads to exponential rates $a_n\sim \exp(\!-n^{1/3})$ , which overwhelms the polynomial factor in assertion (13).

This illustrates a limitation that is important to point out: remote contiguity makes no distinction between asymptotic assertions, other than by rate: as long as the probabilities $P_n(A_n)$ converge to zero fast enough, cf. (1), remote contiguity asserts $Q_n(A_n)=o(1)$ without regard for the further details involved in the definition of the events $A_n$ . In the case at hand, when we ask questions regarding the size of the maximal cluster, there are properties very specific to homogeneous ER graphs at criticality that enable $n^{-1/3}$ -proportionality of the critical window. Lemmas 4 and 6 demonstrate that there are other asymptotic assertions $(B_n)$ with probabilities $P_{1,n}(B_n)$ of order $o(a_n)$ but with probabilities $P_{1+O(n^{-1/3}),n}(B_n)$ that do not go to zero.

4.4. Asymptotic connectedness in ER graphs

Recall that any homogeneous ER graph with edge probability $\lambda_n/n$ is disconnected with high probability if $\limsup_{n \to \infty }\lambda_n <\infty$ (see, e.g. [Reference van der Hofstad13, Section 5.3]). The results of this subsection apply to ER graphs with diverging $(\lambda_n)$ , typically of $O(\log(n))$ .

Proof. The first result is a direct consequence of the first inequality in [Reference van der Hofstad13, Proposition 5.10, (5.3.25), (5.3.26)]. As for the second result, with $\lambda_n^*=\min(\lambda_n, 2 \log (n))$ ,

\begin{equation*} P_{\lambda_n,n}(\mathscr{C}_\mathrm{max}\text{ is disconnected}) \leq 1 - P_{\lambda_n^*,n}(\mathscr{C}_\mathrm{max}\text{ is connected}). \end{equation*}

The conclusion follows by using [Reference van der Hofstad13, (5.3.14), (5.3.21)--(5.3.24), (5.3.27)] with $\lambda=\lambda_n^*$ .

Example 4. (Connectivity in inhomogeneous ER graphs.) Consider an inhomogeneous ER graph with edge probabilities $q_{n,ij}= c_{n,ij}\log(n)/n$ . A sufficient condition for such a graph to be asymptotically connected is the existence of a suitable sequence $(d_n)$ with $d_n>0$ , $\liminf_{n \to \infty}d_n>1$ , and $\lim_{n \to \infty} d_n \log(n)/n <1$ . To see this, also assume that

\begin{align*} \sup\nolimits_{i < j}\bigg|\frac{c_{n,ij}}{d_n}-1\bigg| \to 0, \qquad \limsup_{n \to \infty}\frac{\sum_{i < j}(c_{n,ij}-d_n)^2}{d_n(n- \log (n))}<\frac{1}{4}. \end{align*}

Then the assumptions $r_n=o(1)$ and $a_n=o(\exp(\!-R_n))$ of Lemma 5 are satisfied, with $\mu_{n,ij}=d_n \log(n)$ and $a_n=n^{-\delta}$ for some $\delta <\frac14$ . Hence, $Q_n\mathop{\vartriangleleft} n^{\delta}P_n$ , where $P_n$ is the distribution of the homogeneous ER graph with edge probability $d_n \log(n)/n$ . By Lemma 7, the latter has a probability of not being connected of order $O(n^{-1/4})$ , thus entailing that

\begin{align*} Q_n(\mathscr{C}_\mathrm{max}\text{ is disconnected})=o(1) \end{align*}

as $n\to \infty$ by remote contiguity.