Large degrees in scale-free inhomogeneous random graphs

in Annals of Applied Probability (2022), 32(1), 696-720

We consider a class of scale-free inhomogeneous random graphs, which includes some long-range percolation models. We study the maximum degree in such graphs in a growing observation window and show that ... [more ▼]

We consider a class of scale-free inhomogeneous random graphs, which includes some long-range percolation models. We study the maximum degree in such graphs in a growing observation window and show that its limiting distribution is Frechet. We achieve this by proving convergence of the underlying point process of the degrees to a certain Poisson process. Estimating the index of the power-law tail for the typical degree distribution is an important question in statistics. We prove consistency of the Hill estimator for the inverse of the tail exponent of the typical degree distribution. [less ▲]

Gaussian approximation in random minimal directed spanning trees

in Random Structures and Algorithms (2021)

We study the total $\alpha$-powered length of the rooted edges in a random minimal directed spanning tree - first introduced in \cite{BR04} - on a Poisson process with intensity $s \ge 1$ on the unit cube ... [more ▼]

We study the total $\alpha$-powered length of the rooted edges in a random minimal directed spanning tree - first introduced in \cite{BR04} - on a Poisson process with intensity $s \ge 1$ on the unit cube $[0,1]^d$ for $d \ge 3$. While a Dickman limit was proved in \cite{PW04} in the case of $d=2$, in dimensions three and higher, \cite{BLP06} showed a Gaussian central limit theorem when $\alpha=1$, with a rate of convergence of the order $(\log s)^{-(d-2)/4} (\log \log s)^{(d+1)/2}$. In this paper, we extend these results and prove a central limit theorem in any dimension $d \ge 3$ for any $\alpha>0$. Moreover, making use of recent results in Stein's method for region-stabilizing functionals, we provide presumably optimal non-asymptotic bounds of the order $(\log s)^{-(d-2)/2}$ on the Wasserstein and the Kolmogorov distances between the distribution of the total $\alpha$-powered length of rooted edges, suitably normalized, and that of a standard Gaussian random variable. [less ▲]