# Poisson approximation of random sub-graphs

I add the edges of $G(n)$ the complete graph on $n$ vertices one by one, at random and without replacement, and denote by $G(n,m)$ the resulting Erdos Renyi random graph process. At step $m$ in the process, $0 \leq m \leq {n \choose 2}$, the fraction $\rho^n_k(m)$ of vertices in components with size $k$ is given by: $$\rho^n_k(m)= \sum_{v \in V(G_n)} \frac{1_{\{|\mathcal C_m(v)|=k\}}}{n}, k \in \{1, \ldots,n\},$$ with $|\mathcal C_m(v)|$the size of the component that contains the vertex $v \in V(G_n)$ at step $m$. As $n$ go to $\infty$, the (random) sequence of functions $(\rho^n(\lfloor t.n \rfloor ), t \geq 0)$ is known to converge to $(\rho(t), t \geq 0)$ the unique (deterministic) solution of the (discrete) Smoluchowski (coagulation) equation: $$\rho'_k(t)= k \bigg[ (\rho \star \rho)_k(t) - 2 \rho_k(t) \sum_{k \in \mathbb N} \rho_k(t) \bigg]$$ started at $\rho_k(0)=1_{\{k =1\}}$. I used $(\rho \star \rho)_k = \sum_{m+n=k} \rho_m \rho_n$ to denote the convolution. (The solution of this equation is explicit and involves the Borel Tanner distribution, that describes the size of a Poisson Galton Watson tree).

Question:

I now replace $G(n)$ by a sequence of (deterministic or random) graphs $H(n)$ with $n$ vertices. These $H(n)$ are supposed to have enough edges so that the problem is meaningful (that is $\omega(n)$ edges). Also, in case $H(n)$ is random, the alea should be independent of the construction of the process.

I wonder what are the relevant (geometric, probabilistic, ...) conditions on the sequence of graphs $H(n)$ that ensure the convergence of $(\rho^n(\lfloor t.n \rfloor), t \geq 0)$ to $(\rho(t), t \geq 0)$ the solution of the Smoluchowski equation?

Intuition:

An example where convergence does not hold is the discrete torus $H(n)=(\mathbb Z/m \mathbb Z)^d$, for fixed $d$, with $n= m^d$. If $d=d(n) \to \infty$ however this should work.

A condition like "all degrees in $H(n)$ are equal, and diverge with $n$" is relevant as a sufficient condition. It is however in no way a necessary condition. For instance we may take $H(n)=G(n,m)$ defined above, for $m=\omega(n)$.

just a comment: Looking up for conditions that would simply guarantee that the difference in, say, total variation between the distribution of the statistic you consider on $G(n)$ and the one on $H(n)$ goes to zero as $n$ goes to infinity at first instance, how does this relate to your question? Hope this is not too bad of a trade off to your question and might help you to find something relevant in the literature. Chau