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Asymptotic adaptive threshold for connectivity in a random geometric social network. (arXiv:1810.06479v1 [math.PR])
来源于:arXiv
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. 查看全文>>