Asymptotic adaptive threshold for connectivity in a random geometric social network

Abstract

Consider a dynamic random geometric social network identified by $s_t$ independent points $x_t^1,\dots ,x_t^{s_t}$ in the unit square $[0,1{]}^2$ that interact in continuous time $t\ge 0$. The generative model of the random points is a Poisson point measures. Each point $x_t^i$ can be active or not in the network with a Bernoulli probability $p$. Each pair being connected by affinity thanks to a step connection function if the interpoint distance $\Vert x_t^i-x_t^j\Vert \le a_{\mathsf{f}}^{\star }$ for any $i\ne j$. We prove that when $a_{\mathsf{f}}^{\star }=\sqrt{\frac{(s_t{)}^{l-1}}{p\pi }}$ for $l\in (0,1)$, the number of isolated points is governed by a Poisson approximation as $s_t\to \infty$. This offers a natural threshold for the construction of a $a_{\mathsf{f}}^{\star }$-neighborhood procedure tailored to the dynamic clustering of the network adaptively from the data.