Normal to Poisson phase transition for subgraph counting in the random-connection model

TL;DR

This work analyzes subgraph counts in the Poisson random-connection model as the underlying Poisson intensity grows and the connection probability decays as $\lambda^{-\alpha}$. It introduces a cumulant-diagram framework, using planar representations to identify leading contributions and establish precise cumulant growth rates. The authors prove a normal limit with Kolmogorov bounds for $0<\alpha<\alpha^*_m(G)$ and a Poisson limit at the threshold $\alpha=\alpha^*_m(G)$, with the threshold $\alpha^*_m(G)=\min\left(\frac{r}{e(G)},\frac{1}{a_m(G)}\right)$ and a detailed normal-approximation theory including moderate deviations and Cramér corrections. The results extend Erdős–Rényi-type phase-transition phenomena to the more general random-connection model, accommodate rooted/subgraph counts, and provide a unified diagrammatic method for cumulant analysis with explicit rates.

Abstract

We consider the limiting behavior of the count of subgraphs isomorphic to a graph $G$ with $m\geq 0$ fixed endpoints (or roots) in the random-connection model, as the intensity $λ$ of the underlying Poisson point process tends to infinity. When connection probabilities are of order $λ^{-α}$ we identify a phase transition phenomenon depending on a critical decay rate $α^\ast_m (G)>0$ such that normal approximation for subgraph counts holds when $α\in (0,α^\ast_m (G) )$, and a Poisson limit result holds if $α= α^\ast_m (G)$. Our approach relies on cumulant growth rates derived by the convex analysis of planar diagrams that enumerate the partitions involved in cumulant identities. As a result, by the cumulant method we obtain normal approximation results with convergence rates in the Kolmogorov distance, and a Poisson limit theorem, for subgraph counts.

Figures (13)

• Figure 1: Examples of triangles with endpoints, $r=3$.
• Figure 2: Non-flat connected partition of $[3]\times[4]$.
• Figure 3: Connected graph $G$ on five vertices including one endpoint, with $r=4$ and $m=1$.
• Figure 4: Example of graph $\rho_G$ with $n=3$, $r=4$, and $m=1$.
• Figure 5: Set $\Sigma_n(C_3,0)$ and upper boundary of its convex hull (in red) for $n=3,4$.

Theorems & Definitions (34)

• Remark 1.1
• Definition 2.1
• Remark 2.2
• Definition 2.4
• Remark 2.5
• Definition 3.1
• Theorem 3.3
• Corollary 3.4
• Theorem 3.5
• Theorem 3.6