Gaussian fluctuations of generalized $U$-statistics and subgraph counting in the binomial random-connection model

TL;DR

The paper develops a unified cumulant-based framework for normal approximation of generalized $U$-statistics with an additional random layer from $Y_{i,j}$, and applies it to subgraph counts in the binomial random-connection model. By leveraging partition-diagram techniques, it obtains explicit moment identities and cumulant bounds, yielding Kolmogorov distance bounds and moderate deviation results for the standardized statistics. It then provides detailed cumulant growth bounds, variance lower bounds, and growth-rate analyses for strongly balanced subgraphs, leading to threshold phenomena and distributional limits across regimes where the connection probability vanishes with $n$. Compared with Erdős–Rényi and Poisson-connected models, these results cover the binomial RCM with $p_n$ possibly tending to zero, broadening the scope of normal and non-normal asymptotics in inhomogeneous random graphs.

Abstract

We derive normal approximation bounds for generalized $U$-statistics of the form \begin{equation*} S_{n,k}(f):=\sum_{ 1 \leq β(1),\dots,β(k) \leq n \atop β(i)\neβ(j), \ 1\leq i\ne j \leq k} f\big(X_{β(1)},\dots,X_{β(k)},Y_{β(1),β(2)},\dots,Y_{β(k-1),β(k)}\big), \end{equation*} where $\{X_i\}_{i=1}^n$ and $\{Y_{i,j}\}_{1\le i<j\le n}$ are independent sequences of i.i.d. random variables. Our approach relies on moment identities and cumulant bounds that are derived using partition diagram arguments. Normal approximation bounds in the Kolmogorov distance and moderate deviation results are then obtained by the cumulant method. Those results are applied to subgraph counting in the binomial random-connection model, which is a generalization of the Erdős-Rényi model.

Figures (6)

• Figure 1: Examples of partition diagrams with $n=3$ and $r=4$.
• Figure 2: Examples of partition diagrams with $n=5$ and $r=4$.
• Figure 3: Example for the mapping $\sqcap$ with $j=5$ and $k=4$.
• Figure 4: Example of graph $\rho_G$ with $j=3$ and $k=3$.
• Figure 5: Set $\Sigma_n(C_3)$ and upper boundary of its convex hull (in red) for $n=3,4$.

Theorems & Definitions (28)

• Definition 2.1
• Example 2.2
• Definition 2.3
• Example 2.4
• Lemma 2.5
• Definition 3.1
• Theorem 3.2
• Corollary 3.3
• Remark 3.4
• Theorem 4.1