Critical epidemics, random graphs, and Brownian motion with a parabolic drift

This paper deals with the large deviations behavior of a stochastic process called thinned Lévy process. This process appeared recently as a stochastic-process limit in the context of critical inhomogeneous random graphs [3]. The process has a strong negative drift, while we are interested in the rare event of the process being positive at large times. To characterize this rare event, we identify a tilted measure. This presents some challenges inherent to the power-law nature of the thinned Lévy process. General principles prescribe that the tilt should follow from a variational problem, but in the case of the thinned Lévy process this involves a Riemann sum that is hard to control. We choose to approximate the Riemann sum by its limiting integral, derive the first-order correction term, and prove that the tilt that follows from the corresponding approximate variational problem is sufficient to establish the large deviations results.

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“…This form is reminiscent of results for the Erdős-Rényi graph obtained in [13,18]. The Erdős-Rényi graph on the vertex set [n] := {1, .…”

Section: 22)mentioning

confidence: 69%

Large deviations for power-law thinned Lévy processes

Aidékon

Hofstad

Kliem

et al. 2016

Stochastic Processes and their Applications

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“…In [16] asymptotic expressions are determined for the tail probabilities P(T β Wq (0) > x) for x large. The most relevant result is the following: Theorem 4 (Tail busy period length for large x [16]). For bounded q/σ 2 > 0,…”

Section: Numerical Examplesmentioning

confidence: 99%

Heavy-Traffic Analysis Through Uniform Acceleration of Queues with Diminishing Populations

2019

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We consider a single server queue that serves a finite population of n customers that will enter the queue (require service) only once, also known as the ∆ (i) /G/1 queue. This paper presents a method for analyzing heavy-traffic behavior by using uniform acceleration, which simultaneously lets n and the service rate grow large, while the initial resource utilization approaches one. A key feature of the model is that, as time progresses, more customers have joined the queue, and fewer customers can potentially join. This diminishing population gives rise to a class of reflected stochastic processes that vanish over time, and hence do not have a stationary distribution. We establish that, when the arrival times are exponentially distributed, by suitably rescaling space and time, the queue length process converges to a Brownian motion with parabolic drift, a stochastic-process limit that captures the effect of a diminishing population by a negative quadratic drift. When the arrival times are generally distributed, our techniques provide information on the typical queue length and the first busy period. arXiv:1412.5329v2 [math.PR] 30 Nov 2015

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“…A large body of literature aims at understanding the properties of random graphs that experience this phase-transition in the sizes of the large connected components for various models. The behavior is well understood for the Erdős-Rényi random graphs, thanks to a plethora of results [2,19,26,31]. However, these graphs are often inadequate for modeling real-world networks [11,14,28,29] since the real-world network data often show a power-law behavior of the asymptotic degrees whereas the degree distribution of the Erdős-Rényi random graphs has exponentially decaying tails.…”

Section: Introductionmentioning

confidence: 99%

Critical window for the configuration model: finite third moment degrees

Dhara¹,

Hofstad²,

Leeuwaarden³

et al. 2017

Electron. J. Probab.

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We investigate the component sizes of the critical configuration model, as well as the related problem of critical percolation on a supercritical configuration model. We show that, at criticality, the finite third moment assumption on the asymptotic degree distribution is enough to guarantee that the sizes of the largest connected components are of the order $n^{2/3}$ and the re-scaled component sizes (ordered in a decreasing manner) converge to the ordered excursion lengths of an inhomogeneous Brownian Motion with a parabolic drift. We use percolation to study the evolution of these component sizes while passing through the critical window and show that the vector of percolation cluster-sizes, considered as a process in the critical window, converge to the multiplicative coalescent process in the sense of finite dimensional distributions. This behavior was first observed for Erd\H{o}s-R\'enyi random graphs by Aldous (1997) and our results provide support for the empirical evidences that the nature of the phase transition for a wide array of random-graph models are universal in nature. Further, we show that the re-scaled component sizes and surplus edges converge jointly under a strong topology, at each fixed location of the scaling window.Comment: 33 pages. Minor improvement

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Critical epidemics, random graphs, and Brownian motion with a parabolic drift

Cited by 10 publications

References 19 publications

Large deviations for power-law thinned Lévy processes

Large deviations for power-law thinned Lévy processes

Heavy-Traffic Analysis Through Uniform Acceleration of Queues with Diminishing Populations

Critical window for the configuration model: finite third moment degrees

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