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Register a new userAsymptotic adaptive threshold for connectivity in a random geometric social network
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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 $tgeq 0$. 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 $i eq j$. We prove that when $a_mathsf{f}^star=sqrt{frac{(s_t)^{l-1}}{ppi}}$ for $lin(0,1)$, the number of isolated points is governed by a Poisson approximation as $s_ttoinfty$. 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.
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We consider an even probability distribution on the $d$-dimensional Euclidean space with the property that it assigns measure zero to any hyperplane through the origin. Given $N$ independent random vectors with this distribution, under the condition that they do not positively span the whole space, the positive hull of these vectors is a random polyhedral cone (and its intersection with the unit sphere is a random spherical polytope). It was first studied by Cover and Efron. We consider the expected face numbers of these random cones and describe a threshold phenomenon when the dimension $d$ and the number $N$ of random vectors tend to infinity. In a similar way, we treat the solid angle, and more generally the Grassmann angles. We further consider the expected numbers of $k$-faces and of Grassmann angles of index $d-k$ when also $k$ tends to infinity.
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