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Asymptotic Analysis of k-Hop Connectivity in the 1D Unit Disk Random Graph Model

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  • Nicolas Privault

    (Nanyang Technological University)

Abstract

We propose an algorithm for the closed-form recursive computation of joint moments and cumulants of all orders of 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 and cumulant methods 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 of any orders are provided as an online resource.

Suggested Citation

  • Nicolas Privault, 2024. "Asymptotic Analysis of k-Hop Connectivity in the 1D Unit Disk Random Graph Model," Methodology and Computing in Applied Probability, Springer, vol. 26(4), pages 1-26, December.
    • Handle: RePEc:spr:metcap:v:26:y:2024:i:4:d:10.1007_s11009-024-10115-9
    DOI: 10.1007/s11009-024-10115-9
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    References listed on IDEAS

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    1. de Jong, Peter, 1990. "A central limit theorem for generalized multilinear forms," Journal of Multivariate Analysis, Elsevier, vol. 34(2), pages 275-289, August.
    2. Hanna Döring & Peter Eichelsbacher, 2013. "Moderate Deviations via Cumulants," Journal of Theoretical Probability, Springer, vol. 26(2), pages 360-385, June.
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