23,221 research outputs found

    Critical random graphs : limiting constructions and distributional properties

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    We consider the Erdos-Renyi random graph G(n, p) inside the critical window, where p = 1/n + lambda n(-4/3) for some lambda is an element of R. We proved in Addario-Berry et al. [2009+] that considering the connected components of G(n, p) as a sequence of metric spaces with the graph distance rescaled by n(-1/3) and letting n -> infinity yields a non-trivial sequence of limit metric spaces C = (C-1, C-2,...). These limit metric spaces can be constructed from certain random real trees with vertex-identifications. For a single such metric space, we give here two equivalent constructions, both of which are in terms of more standard probabilistic objects. The first is a global construction using Dirichlet random variables and Aldous' Brownian continuum random tree. The second is a recursive construction from an inhomogeneous Poisson point process on R+. These constructions allow us to characterize the distributions of the masses and lengths in the constituent parts of a limit component when it is decomposed according to its cycle structure. In particular, this strengthens results of Luczak et al. [1994] by providing precise distributional convergence for the lengths of paths between kernel vertices and the length of a shortest cycle, within any fixed limit component

    Critical random graphs: limiting constructions and distributional properties

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    We consider the Erdos-Renyi random graph G(n,p) inside the critical window, where p = 1/n + lambda * n^{-4/3} for some lambda in R. We proved in a previous paper (arXiv:0903.4730) that considering the connected components of G(n,p) as a sequence of metric spaces with the graph distance rescaled by n^{-1/3} and letting n go to infinity yields a non-trivial sequence of limit metric spaces C = (C_1, C_2, ...). These limit metric spaces can be constructed from certain random real trees with vertex-identifications. For a single such metric space, we give here two equivalent constructions, both of which are in terms of more standard probabilistic objects. The first is a global construction using Dirichlet random variables and Aldous' Brownian continuum random tree. The second is a recursive construction from an inhomogeneous Poisson point process on R_+. These constructions allow us to characterize the distributions of the masses and lengths in the constituent parts of a limit component when it is decomposed according to its cycle structure. In particular, this strengthens results of Luczak, Pittel and Wierman by providing precise distributional convergence for the lengths of paths between kernel vertices and the length of a shortest cycle, within any fixed limit component.Comment: 30 pages, 4 figure

    RDF Querying

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    Reactive Web systems, Web services, and Web-based publish/ subscribe systems communicate events as XML messages, and in many cases require composite event detection: it is not sufficient to react to single event messages, but events have to be considered in relation to other events that are received over time. Emphasizing language design and formal semantics, we describe the rule-based query language XChangeEQ for detecting composite events. XChangeEQ is designed to completely cover and integrate the four complementary querying dimensions: event data, event composition, temporal relationships, and event accumulation. Semantics are provided as model and fixpoint theories; while this is an established approach for rule languages, it has not been applied for event queries before

    Probabilistic Graphical Model Representation in Phylogenetics

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    Recent years have seen a rapid expansion of the model space explored in statistical phylogenetics, emphasizing the need for new approaches to statistical model representation and software development. Clear communication and representation of the chosen model is crucial for: (1) reproducibility of an analysis, (2) model development and (3) software design. Moreover, a unified, clear and understandable framework for model representation lowers the barrier for beginners and non-specialists to grasp complex phylogenetic models, including their assumptions and parameter/variable dependencies. Graphical modeling is a unifying framework that has gained in popularity in the statistical literature in recent years. The core idea is to break complex models into conditionally independent distributions. The strength lies in the comprehensibility, flexibility, and adaptability of this formalism, and the large body of computational work based on it. Graphical models are well-suited to teach statistical models, to facilitate communication among phylogeneticists and in the development of generic software for simulation and statistical inference. Here, we provide an introduction to graphical models for phylogeneticists and extend the standard graphical model representation to the realm of phylogenetics. We introduce a new graphical model component, tree plates, to capture the changing structure of the subgraph corresponding to a phylogenetic tree. We describe a range of phylogenetic models using the graphical model framework and introduce modules to simplify the representation of standard components in large and complex models. Phylogenetic model graphs can be readily used in simulation, maximum likelihood inference, and Bayesian inference using, for example, Metropolis-Hastings or Gibbs sampling of the posterior distribution

    Counting connected hypergraphs via the probabilistic method

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    In 1990 Bender, Canfield and McKay gave an asymptotic formula for the number of connected graphs on [n][n] with mm edges, whenever nn and the nullity māˆ’n+1m-n+1 tend to infinity. Asymptotic formulae for the number of connected rr-uniform hypergraphs on [n][n] with mm edges and so nullity t=(rāˆ’1)māˆ’n+1t=(r-1)m-n+1 were proved by Karo\'nski and \L uczak for the case t=o(logā”n/logā”logā”n)t=o(\log n/\log\log n), and Behrisch, Coja-Oghlan and Kang for t=Ī˜(n)t=\Theta(n). Here we prove such a formula for any rā‰„3r\ge 3 fixed, and any t=t(n)t=t(n) satisfying t=o(n)t=o(n) and tā†’āˆžt\to\infty as nā†’āˆžn\to\infty. This leaves open only the (much simpler) case t/nā†’āˆžt/n\to\infty, which we will consider in future work. ( arXiv:1511.04739 ) Our approach is probabilistic. Let Hn,prH^r_{n,p} denote the random rr-uniform hypergraph on [n][n] in which each edge is present independently with probability pp. Let L1L_1 and M1M_1 be the numbers of vertices and edges in the largest component of Hn,prH^r_{n,p}. We prove a local limit theorem giving an asymptotic formula for the probability that L1L_1 and M1M_1 take any given pair of values within the `typical' range, for any p=p(n)p=p(n) in the supercritical regime, i.e., when p=p(n)=(1+Ļµ(n))(rāˆ’2)!nāˆ’r+1p=p(n)=(1+\epsilon(n))(r-2)!n^{-r+1} where Ļµ3nā†’āˆž\epsilon^3n\to\infty and Ļµā†’0\epsilon\to 0; our enumerative result then follows easily. Taking as a starting point the recent joint central limit theorem for L1L_1 and M1M_1, we use smoothing techniques to show that `nearby' pairs of values arise with about the same probability, leading to the local limit theorem. Behrisch et al used similar ideas in a very different way, that does not seem to work in our setting. Independently, Sato and Wormald have recently proved the special case r=3r=3, with an additional restriction on tt. They use complementary, more enumerative methods, which seem to have a more limited scope, but to give additional information when they do work.Comment: Expanded; asymptotics clarified - no significant mathematical changes. 67 pages (including appendix

    Nielsen equivalence in a class of random groups

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    We show that for every nā‰„2n\ge 2 there exists a torsion-free one-ended word-hyperbolic group GG of rank nn admitting generating nn-tuples (a1,ā€¦,an)(a_1,\ldots ,a_n) and (b1,ā€¦,bn)(b_1,\ldots ,b_n) such that the (2nāˆ’1)(2n-1)-tuples (a1,ā€¦,an,1,ā€¦,1āŸnāˆ’1times)Ā andĀ (b1,ā€¦,bn,1,ā€¦,1āŸnāˆ’1times)(a_1,\ldots ,a_n, \underbrace{1,\ldots ,1}_{n-1 \text{times}})\hbox{ and }(b_1,\ldots, b_n, \underbrace{1,\ldots ,1}_{n-1 \text{times}}) are not Nielsen-equivalent in GG. The group GG is produced via a probabilistic construction.Comment: 34 pages, 2 figures; a revised final version, to appear in the Journal of Topolog

    Biologically inspired distributed machine cognition: a new formal approach to hyperparallel computation

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    The irresistable march toward multiple-core chip technology presents currently intractable pdrogramming challenges. High level mental processes in many animals, and their analogs for social structures, appear similarly massively parallel, and recent mathematical models addressing them may be adaptable to the multi-core programming problem

    Learning Large-Scale Bayesian Networks with the sparsebn Package

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    Learning graphical models from data is an important problem with wide applications, ranging from genomics to the social sciences. Nowadays datasets often have upwards of thousands---sometimes tens or hundreds of thousands---of variables and far fewer samples. To meet this challenge, we have developed a new R package called sparsebn for learning the structure of large, sparse graphical models with a focus on Bayesian networks. While there are many existing software packages for this task, this package focuses on the unique setting of learning large networks from high-dimensional data, possibly with interventions. As such, the methods provided place a premium on scalability and consistency in a high-dimensional setting. Furthermore, in the presence of interventions, the methods implemented here achieve the goal of learning a causal network from data. Additionally, the sparsebn package is fully compatible with existing software packages for network analysis.Comment: To appear in the Journal of Statistical Software, 39 pages, 7 figure
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