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

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.

Normal approximation of subgraph counts in the random-connection model

“…For our analysis of cumulant behavior we identify the leading terms in the sum (5.3) over connected partition diagrams. When G is a connected graph with r := |V (G)| vertices, satisfying Assumption 6.1 in the dilute regime (6.1) where λ −1 ≪ c λ ≤ 1, the dominant terms correspond to connected partition diagrams with the highest number of blocks, as found in [Pri22] in the case of k-hop counting in the one-dimensional random-connection model. In Theorem 6.1 this yields the cumulant bounds In the sparse regime (6.2) where c λ ≤ λ −α for some α ≥ 1, the maximal rate λ α−(α−1)r is attained for G a tree-like graph, and in Theorem 6.2 we obtain the cumulant bounds…”

“…Convergence rates in the Kolmogorov distances may be improved into classical Berry-Esseen rates when the connection function H(x, y) is {0, 1}-valued, e.g. in disk models as in [Pri22], by representing subgraph counts as multiple Poisson stochastic integrals and using the fourth moment theorem for U-statistics and sums of multiple stochastic integrals Corollary 4.10 in [ET14], see also Theorem 3 in [LRR16] or Theorem 6.3 in [PS22] for Hoeffding decompositions. In the general case where H(x, y) is [0, 1]-valued this method no longer applies, this is why we rely on the Statulevičius condition which in turn may yield suboptimal convergence rates.…”