Abstract This paper studies estimation of stochastic block models with Rissanen’s minimum description length (MDL) principle in the dense graph asymptotics. We focus on the problem of model specification, i.e., identification of the number of blocks. Refinements of the true partition always decrease the code part corresponding to the edge placement, and thus a respective increase of the code part specifying the model should overweight that gain in order to yield a minimum at the true partition. The balance between these effects turns out to be delicate. We show that the MDL principle identifies the true partition among models whose relative block sizes are bounded away from zero. The results are extended to models with Poisson-distributed edge weights.
{"title":"On model selection for dense stochastic block models","authors":"I. Norros, H. Reittu, F. Bazsó","doi":"10.1017/apr.2021.29","DOIUrl":"https://doi.org/10.1017/apr.2021.29","url":null,"abstract":"Abstract This paper studies estimation of stochastic block models with Rissanen’s minimum description length (MDL) principle in the dense graph asymptotics. We focus on the problem of model specification, i.e., identification of the number of blocks. Refinements of the true partition always decrease the code part corresponding to the edge placement, and thus a respective increase of the code part specifying the model should overweight that gain in order to yield a minimum at the true partition. The balance between these effects turns out to be delicate. We show that the MDL principle identifies the true partition among models whose relative block sizes are bounded away from zero. The results are extended to models with Poisson-distributed edge weights.","PeriodicalId":53160,"journal":{"name":"Advances in Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2022-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45613688","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The invariant Galton–Watson (IGW) tree measures are a one-parameter family of critical Galton–Watson measures invariant with respect to a large class of tree reduction operations. Such operations include the generalized dynamical pruning (also known as hereditary reduction in a real tree setting) that eliminates descendant subtrees according to the value of an arbitrary subtree function that is monotone nondecreasing with respect to an isometry-induced partial tree order. We show that, under a mild regularity condition, the IGW measures are attractors of arbitrary critical Galton–Watson measures with respect to the generalized dynamical pruning. We also derive the distributions of height, length, and size of the IGW trees.
{"title":"Invariant Galton–Watson trees: metric properties and attraction with respect to generalized dynamical pruning","authors":"Yevgeniy Kovchegov, Guochen Xu, I. Zaliapin","doi":"10.1017/apr.2022.39","DOIUrl":"https://doi.org/10.1017/apr.2022.39","url":null,"abstract":"Abstract The invariant Galton–Watson (IGW) tree measures are a one-parameter family of critical Galton–Watson measures invariant with respect to a large class of tree reduction operations. Such operations include the generalized dynamical pruning (also known as hereditary reduction in a real tree setting) that eliminates descendant subtrees according to the value of an arbitrary subtree function that is monotone nondecreasing with respect to an isometry-induced partial tree order. We show that, under a mild regularity condition, the IGW measures are attractors of arbitrary critical Galton–Watson measures with respect to the generalized dynamical pruning. We also derive the distributions of height, length, and size of the IGW trees.","PeriodicalId":53160,"journal":{"name":"Advances in Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2022-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49561840","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"APR volume 53 issue 4 Cover and Front matter","authors":"","doi":"10.1017/apr.2021.58","DOIUrl":"https://doi.org/10.1017/apr.2021.58","url":null,"abstract":"","PeriodicalId":53160,"journal":{"name":"Advances in Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2021-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49002043","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"APR volume 53 issue 4 Cover and Back matter","authors":"","doi":"10.1017/apr.2021.59","DOIUrl":"https://doi.org/10.1017/apr.2021.59","url":null,"abstract":"","PeriodicalId":53160,"journal":{"name":"Advances in Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2021-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44227803","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We study shot noise processes with cluster arrivals, in which entities in each cluster may experience random delays (possibly correlated), and noises within each cluster may be correlated. We prove functional limit theorems for the process in the large-intensity asymptotic regime, where the arrival rate gets large while the shot shape function, cluster sizes, delays, and noises are unscaled. In the functional central limit theorem, the limit process is a continuous Gaussian process (assuming the arrival process satisfies a functional central limit theorem with a Brownian motion limit). We discuss the impact of the dependence among the random delays and among the noises within each cluster using several examples of dependent structures. We also study infinite-server queues with cluster/batch arrivals where customers in each batch may experience random delays before receiving service, with similar dependence structures.
{"title":"Shot noise processes with randomly delayed cluster arrivals and dependent noises in the large-intensity regime","authors":"Bo Li, G. Pang","doi":"10.1017/apr.2021.16","DOIUrl":"https://doi.org/10.1017/apr.2021.16","url":null,"abstract":"Abstract We study shot noise processes with cluster arrivals, in which entities in each cluster may experience random delays (possibly correlated), and noises within each cluster may be correlated. We prove functional limit theorems for the process in the large-intensity asymptotic regime, where the arrival rate gets large while the shot shape function, cluster sizes, delays, and noises are unscaled. In the functional central limit theorem, the limit process is a continuous Gaussian process (assuming the arrival process satisfies a functional central limit theorem with a Brownian motion limit). We discuss the impact of the dependence among the random delays and among the noises within each cluster using several examples of dependent structures. We also study infinite-server queues with cluster/batch arrivals where customers in each batch may experience random delays before receiving service, with similar dependence structures.","PeriodicalId":53160,"journal":{"name":"Advances in Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2021-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46472493","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Following Bradonjić and Saniee, we study a model of bootstrap percolation on the Gilbert random geometric graph on the 2-dimensional torus. In this model, the expected number of vertices of the graph is n, and the expected degree of a vertex is $alog n$ for some fixed $a>1$ . Each vertex is added with probability p to a set $A_0$ of initially infected vertices. Vertices subsequently become infected if they have at least $ theta a log n $ infected neighbours. Here $p, theta in [0,1]$ are taken to be fixed constants. We show that if $theta < (1+p)/2$ , then a sufficiently large local outbreak leads with high probability to the infection spreading globally, with all but o(n) vertices eventually becoming infected. On the other hand, for $ theta > (1+p)/2$ , even if one adversarially infects every vertex inside a ball of radius $O(sqrt{log n} )$ , with high probability the infection will spread to only o(n) vertices beyond those that were initially infected. In addition we give some bounds on the $(a, p, theta)$ regions ensuring the emergence of large local outbreaks or the existence of islands of vertices that never become infected. We also give a complete picture of the (surprisingly complex) behaviour of the analogous 1-dimensional bootstrap percolation model on the circle. Finally we raise a number of problems, and in particular make a conjecture on an ‘almost no percolation or almost full percolation’ dichotomy which may be of independent interest.
继bradonjiki和Saniee之后，我们研究了二维环面上Gilbert随机几何图上的自举渗流模型。在该模型中，对于某个固定的$a>1$，图的期望顶点数为n，顶点的期望度数为$alog n$。每个顶点以p的概率添加到初始感染顶点的集合$A_0$中。如果顶点至少有$ theta a log n $被感染的邻居，那么它们随后就会被感染。这里$p, theta in [0,1]$被认为是固定常数。我们表明，如果$theta < (1+p)/2$，那么一个足够大的局部爆发很可能导致感染蔓延到全球，除了o(n)个顶点外，所有顶点最终都被感染。另一方面，对于$ theta > (1+p)/2$，即使一个人对抗性地感染了半径为$O(sqrt{log n} )$的球内的每个顶点，感染很可能只会传播到最初被感染的顶点以外的o(n)个顶点。此外，我们给出了$(a, p, theta)$区域的一些边界，以确保出现大规模的局部爆发或存在从未被感染的顶点岛。我们还给出了圆上类似的一维自举渗透模型(令人惊讶的复杂)行为的完整图像。最后，我们提出了一些问题，特别是对“几乎没有渗透或几乎完全渗透”的二分法提出了一个猜想，这可能是一个独立的兴趣。
{"title":"Bootstrap percolation in random geometric graphs","authors":"Victor Falgas‐Ravry, Amites Sarkar","doi":"10.1017/apr.2023.5","DOIUrl":"https://doi.org/10.1017/apr.2023.5","url":null,"abstract":"\u0000 Following Bradonjić and Saniee, we study a model of bootstrap percolation on the Gilbert random geometric graph on the 2-dimensional torus. In this model, the expected number of vertices of the graph is n, and the expected degree of a vertex is \u0000 \u0000 \u0000 \u0000$alog n$\u0000\u0000 \u0000 for some fixed \u0000 \u0000 \u0000 \u0000$a>1$\u0000\u0000 \u0000 . Each vertex is added with probability p to a set \u0000 \u0000 \u0000 \u0000$A_0$\u0000\u0000 \u0000 of initially infected vertices. Vertices subsequently become infected if they have at least \u0000 \u0000 \u0000 \u0000$ theta a log n $\u0000\u0000 \u0000 infected neighbours. Here \u0000 \u0000 \u0000 \u0000$p, theta in [0,1]$\u0000\u0000 \u0000 are taken to be fixed constants.\u0000 We show that if \u0000 \u0000 \u0000 \u0000$theta < (1+p)/2$\u0000\u0000 \u0000 , then a sufficiently large local outbreak leads with high probability to the infection spreading globally, with all but o(n) vertices eventually becoming infected. On the other hand, for \u0000 \u0000 \u0000 \u0000$ theta > (1+p)/2$\u0000\u0000 \u0000 , even if one adversarially infects every vertex inside a ball of radius \u0000 \u0000 \u0000 \u0000$O(sqrt{log n} )$\u0000\u0000 \u0000 , with high probability the infection will spread to only o(n) vertices beyond those that were initially infected.\u0000 In addition we give some bounds on the \u0000 \u0000 \u0000 \u0000$(a, p, theta)$\u0000\u0000 \u0000 regions ensuring the emergence of large local outbreaks or the existence of islands of vertices that never become infected. We also give a complete picture of the (surprisingly complex) behaviour of the analogous 1-dimensional bootstrap percolation model on the circle. Finally we raise a number of problems, and in particular make a conjecture on an ‘almost no percolation or almost full percolation’ dichotomy which may be of independent interest.","PeriodicalId":53160,"journal":{"name":"Advances in Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2021-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47618727","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We obtain series expansions of the q-scale functions of arbitrary spectrally negative Lévy processes, including processes with infinite jump activity, and use these to derive various new examples of explicit q-scale functions. Moreover, we study smoothness properties of the q-scale functions of spectrally negative Lévy processes with infinite jump activity. This complements previous results of Chan et al. (Prob. Theory Relat. Fields 150, 2011) for spectrally negative Lévy processes with Gaussian component or bounded variation.
{"title":"On q-scale functions of spectrally negative Lévy processes","authors":"Anita Behme, David Oechsler, R. Schilling","doi":"10.1017/apr.2022.10","DOIUrl":"https://doi.org/10.1017/apr.2022.10","url":null,"abstract":"Abstract We obtain series expansions of the q-scale functions of arbitrary spectrally negative Lévy processes, including processes with infinite jump activity, and use these to derive various new examples of explicit q-scale functions. Moreover, we study smoothness properties of the q-scale functions of spectrally negative Lévy processes with infinite jump activity. This complements previous results of Chan et al. (Prob. Theory Relat. Fields 150, 2011) for spectrally negative Lévy processes with Gaussian component or bounded variation.","PeriodicalId":53160,"journal":{"name":"Advances in Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2021-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48825785","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"APR volume 53 issue 3 Cover and Front matter","authors":"","doi":"10.1017/apr.2021.47","DOIUrl":"https://doi.org/10.1017/apr.2021.47","url":null,"abstract":"","PeriodicalId":53160,"journal":{"name":"Advances in Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2021-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44431026","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-09-01Epub Date: 2021-10-08DOI: 10.1017/apr.2021.2
Reza Rastegar, Alexander Roitershtein
We study propagation of avalanches in a certain excitable network. The model is a particular case of the one introduced in [24], and is mathematically equivalent to an endemic variation of the Reed-Frost epidemic model introduced in [28]. Two types of heuristic approximation are frequently used for models of this type in applications, a branching process for avalanches of a small size at the beginning of the process and a deterministic dynamical system once the avalanche spreads to a significant fraction of a large network. In this paper we prove several results concerning the exact relation between the avalanche model and these limits, including rates of convergence and rigorous bounds for common characteristics of the model.
{"title":"AVALANCHES IN A SHORT-MEMORY EXCITABLE NETWORK.","authors":"Reza Rastegar, Alexander Roitershtein","doi":"10.1017/apr.2021.2","DOIUrl":"https://doi.org/10.1017/apr.2021.2","url":null,"abstract":"<p><p>We study propagation of avalanches in a certain excitable network. The model is a particular case of the one introduced in [24], and is mathematically equivalent to an endemic variation of the Reed-Frost epidemic model introduced in [28]. Two types of heuristic approximation are frequently used for models of this type in applications, a branching process for avalanches of a small size at the beginning of the process and a deterministic dynamical system once the avalanche spreads to a significant fraction of a large network. In this paper we prove several results concerning the exact relation between the avalanche model and these limits, including rates of convergence and rigorous bounds for common characteristics of the model.</p>","PeriodicalId":53160,"journal":{"name":"Advances in Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2021-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8547492/pdf/nihms-1673403.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"39566894","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Queueing networks are stochastic systems formed by interconnected resources routing and serving jobs. They induce jump processes with distinctive properties, and find widespread use in inferential tasks. Here, service rates for jobs and potential bottlenecks in the routing mechanism must be estimated from a reduced set of observations. However, this calls for the derivation of complex conditional density representations, over both the stochastic network trajectories and the rates, which is considered an intractable problem. Numerical simulation procedures designed for this purpose do not scale, because of high computational costs; furthermore, variational approaches relying on approximating measures and full independence assumptions are unsuitable. In this paper, we offer a probabilistic interpretation of variational methods applied to inference tasks with queueing networks, and show that approximating measure choices routinely used with jump processes yield ill-defined optimization problems. Yet we demonstrate that it is still possible to enable a variational inferential task, by considering a novel space expansion treatment over an analogous counting process for job transitions. We present and compare exemplary use cases with practical queueing networks, showing that our framework offers an efficient and improved alternative where existing variational or numerically intensive solutions fail.
{"title":"Variational inference for Markovian queueing networks","authors":"Iker Perez, G. Casale","doi":"10.1017/apr.2020.72","DOIUrl":"https://doi.org/10.1017/apr.2020.72","url":null,"abstract":"Abstract Queueing networks are stochastic systems formed by interconnected resources routing and serving jobs. They induce jump processes with distinctive properties, and find widespread use in inferential tasks. Here, service rates for jobs and potential bottlenecks in the routing mechanism must be estimated from a reduced set of observations. However, this calls for the derivation of complex conditional density representations, over both the stochastic network trajectories and the rates, which is considered an intractable problem. Numerical simulation procedures designed for this purpose do not scale, because of high computational costs; furthermore, variational approaches relying on approximating measures and full independence assumptions are unsuitable. In this paper, we offer a probabilistic interpretation of variational methods applied to inference tasks with queueing networks, and show that approximating measure choices routinely used with jump processes yield ill-defined optimization problems. Yet we demonstrate that it is still possible to enable a variational inferential task, by considering a novel space expansion treatment over an analogous counting process for job transitions. We present and compare exemplary use cases with practical queueing networks, showing that our framework offers an efficient and improved alternative where existing variational or numerically intensive solutions fail.","PeriodicalId":53160,"journal":{"name":"Advances in Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2021-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47427125","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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