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Register a new userPoisson Approximation and Connectivity in a Scale-free Random Connection Model
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We study an inhomogeneous random connection model in the connectivity regime. The vertex set of the graph is a homogeneous Poisson point process $mathcal{P}_s$ of intensity $s>0$ on the unit cube $S=left(-frac{1}{2},frac{1}{2}right]^{d},$ $d geq 2$ . Each vertex is endowed with an independent random weight distributed as $W$, where $P(W>w)=w^{-beta}1_{[1,infty)}(w)$, $beta>0$. Given the vertex set and the weights an edge exists between $x,yin mathcal{P}_s$ with probability $left(1 - expleft( - frac{eta W_xW_y}{left(d(x,y)/rright)^{alpha}} right)right),$ independent of everything else, where $eta, alpha > 0$, $d(cdot, cdot)$ is the toroidal metric on $S$ and $r > 0$ is a scaling parameter. We derive conditions on $alpha, beta$ such that under the scaling $r_s(xi)^d= frac{1}{c_0 s} left( log s +(k-1) loglog s +xi+logleft(frac{alphabeta}{k!d} right)right),$ $xi in mathbb{R}$, the number of vertices of degree $k$ converges in total variation distance to a Poisson random variable with mean $e^{-xi}$ as $s to infty$, where $c_0$ is an explicitly specified constant that depends on $alpha, beta, d$ and $eta$ but not on $k$. In particular, for $k=0$ we obtain the regime in which the number of isolated nodes stabilizes, a precursor to establishing a threshold for connectivity. We also derive a sufficient condition for the graph to be connected with high probability for large $s$. The Poisson approximation result is derived using the Steins method.
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We study the random connection model driven by a stationary Poisson process. In the first part of the paper, we derive a lace expansion with remainder term in the continuum and bound the coefficients using a new version of the BK inequality. For our main results, we consider thr
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.
We study geometric random graphs defined on the points of a Poisson process in $d$-dimensional space, which additionally carry independent random marks. Edges are established at random using the marks of the endpoints and the distance between points in a flexible way. Our framework includes the soft Boolean model (where marks play the role of radii of balls centred in the vertices), a version of spatial preferential attachment (where marks play the role of birth times), and a whole range of other graph models with scale-free degree distributions and edges spanning large distances. In this versatile framework we give sharp criteria for absence of ultrasmallness of the graphs and in the ultrasmall regime establish a limit theorem for the chemical distance of two points. Other than in the mean-field scale-free network models the boundary of the ultrasmall regime depends not only on the power-law exponent of the degree distribution but also on the spatial embedding of the graph, quantified by the rate of decay of the probability of an edge connecting typical points in terms of their spatial distance.
We consider Gaussian approximation in a variant of the classical Johnson-Mehl birth-growth model with random growth speed. Seeds appear randomly in $mathbb{R}^d$ at random times and start growing instantaneously in all directions with a random speed. The location, birth time and growth speed of the seeds are given by a Poisson process. Under suitable conditions on the random growth speed and a weight function $h:mathbb{R}^d to [0,infty)$, we provide sufficient conditions for a Gaussian convergence of the sum of the weights at the exposed points, which are those seeds in the model that are not covered at the time of their birth. Moreover, using recent results on stabilization regions, we provide non-asymptotic bounds on the distance between the normalized sum of weights and a standard Gaussian random variable in the Wasserstein and Kolmogorov metrics.
We study the critical behavior for percolation on inhomogeneous random networks on $n$ vertices, where the weights of the vertices follow a power-law distribution with exponent $tau in (2,3)$. Such networks, often referred to as scale-free networks, exhibit critical behavior when the percolation probability tends to zero at an appropriate rate, as $ntoinfty$. We identify the critical window for a host of scale-free random graph models such as the Norros-Reittu model, Chung-Lu model and generalized random graphs. Surprisingly, there exists a finite time inside the critical window, after which, we see a sudden emergence of a tiny giant component. This is a novel behavior which is in contrast with the critical behavior in other known universality classes with $tau in (3,4)$ and $tau >4$. Precisely, for edge-retention probabilities $pi_n = lambda n^{-(3-tau)/2}$, there is an explicitly computable $lambda_c>0$ such that the critical window is of the form $lambda in (0,lambda_c),$ where the largest clusters have size of order $n^{beta}$ with $beta=(tau^2-4tau+5)/[2(tau-1)]in[sqrt{2}-1, tfrac{1}{2})$ and have non-degenerate scaling limits, while in the supercritical regime $lambda > lambda_c$, a unique `tiny giant component of size $sqrt{n}$ emerges. For $lambda in (0,lambda_c),$ the scaling limit of the maximum component sizes can be described in terms of components of a one-dimensional inhomogeneous percolation model on $mathbb{Z}_+$ studied in a seminal work by Durrett and Kesten. For $lambda>lambda_c$, we prove that the sudden emergence of the tiny giant is caused by a phase transition inside a smaller core of vertices of weight $Omega(sqrt{n})$.
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