Scaling conjecture for clustering in finite spatial inhomogeneous random graphs

Let $\mathbb{G}^{(n)}$ be the finite spatial inhomogeneous random graph, let $\mathrm{CC}_{\mathbb{G}^{(n)}}(k)$ denote its clustering function, and let $\gamma(k)$ denote the clustering function of the corresponding infinite SIRG. For a sequence $k_n\to\infty$, define

$$\phi(k_n):=\begin{cases} \frac{k_n}{\log k_n} &\text{if }\beta=a+1,\ a\in(1,\infty),\\ k_n &\text{otherwise,} \end{cases}$$

and

$$\psi(n):=\begin{cases} n^{\frac{1}{\beta-1}} &\text{if }\beta\geq (a+1)\vee 2,\ a\in[0,\infty),\\ n^{\frac{2+a-\beta}{\beta-1}} &\text{otherwise.} \end{cases}$$

Scaling conjecture for finite graphs. Consider the finite SIRG $\mathbb{G}^{(n)}$. If $\phi(k_n)=o(\psi(n))$, then

$$\frac{\mathrm{CC}_{\mathbb{G}^{(n)}}(k_n)}{\gamma(k_n)}\xrightarrow{\mathbb{P}}1.$$

If $\phi(k_n)=(1+o(1))\psi(n)$, then, for some $L_0,\lambda>0$,

$$\frac{\mathrm{CC}_{\mathbb{G}^{(n)}}(k_n)}{\gamma(k_n)}\xrightarrow{d}L_0\operatorname{Poi}(\lambda).$$

If $\phi(k_n)=\omega(\psi(n))$, then

$$\frac{\mathrm{CC}_{\mathbb{G}^{(n)}}(k_n)}{\gamma(k_n)}\xrightarrow{\mathbb{P}}0.$$

These conjectured limit laws describe the regimes in which the finite model follows the infinite-model clustering asymptotics, exhibits a critical Poisson-type fluctuation, or has negligible clustering relative to $\gamma(k_n)$. The conjecture is motivated by local convergence of the finite model to the infinite SIRG and the known fixed-$k$ convergence, but the simultaneous growth of $k_n$ with $n$ remains unresolved.