Asymptotic analysis of k-hop connectivity in the 1D unit disk random graph model

TL;DR

This paper introduces a recursive algorithm for calculating joint moments and cumulants of k-hop counts in a 1D Poisson random graph, and establishes their normal convergence at high densities.

Contribution

It presents a novel recursive method for exact moment and cumulant computation and applies Stein's method for asymptotic normality analysis in 1D random graphs.

Findings

Recursive formulas for moments and cumulants derived

Berry-Esseen bounds established for normal convergence

Computer code provided for symbolic computations

Abstract

We propose an algorithm for the closed-form recursive computation of joint moments and cumulants of all orders for k-hop counts in the 1D unit disk random graph model with Poisson distributed vertices. Our approach uses decompositions of k-hop counts into multiple Poisson stochastic integrals. As a consequence, using the Stein method we derive Berry-Esseen bounds for the asymptotic convergence of renormalized k-hop path counts to the normal distribution as the density of Poisson vertices tends to infinity. Computer codes for the recursive symbolic computation of moments and cumulants are provided in appendix.