Asymptotic adaptive threshold for connectivity in a random geometric social network

Consider a dynamic random geometric social network identified by $s_t$ independent points $x_t^1,\ldots,x_t^{s_t}$ in the unit square $[0,1]^2$ that interact in continuous time $t\geq 0$. 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 $\|x_t^i-x_t^j\|\leq a_\mathsf{f}^\star$ for any $i\neq j$. 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.