Some large deviation results for sparse random graphs

by Neil O"Connell

Publisher: Hewlett Packard in Bristol [England]

Written in English
Published: Pages: 15 Downloads: 520
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  • Large deviations,
  • Graph theory,
  • Random graphs

Edition Notes

StatementNeil O"Connell.
Series[Technical report] / HP Laboratories Bristol. Basic Research Institute in the Mathematical Sciences -- HPL-BRIMS-96-22., BRIMS technical report -- HPL-BRIMS-96-22.
ContributionsHewlett-Packard Laboratories.
The Physical Object
Pagination15 p. ;
Number of Pages15
ID Numbers
Open LibraryOL17613355M

We perform a thorough study of various characteristics of the asynchronous push-pull protocol for spreading a rumor on Erdős-Rényi random graphs Gn,p, for any p>cln(n)/n with c>1. In particular, we provide a simple strategy for analyzing the asynchronous push-pull protocol on arbitrary graph topologies and apply this strategy to Gn,p. We prove tight bounds of logarithmic order for the total. In Fig. 2, we show the overall improvements achieved from our implementations on graphs from Table 1. Further results on some more graphs from the UFL Sparse Matrix Collection and on random matrices generated using the R-MAT synthetic graph generator are shown in our previous work. Download: Download high-res image (KB)Cited by: 8. Outlier Detection in Graphs: A Study on the Impact of Multiple Graph Models and to compose an ensemble based on a set of multiple graph models in Section 3 and its implementation in case studies in Section 4. We analyze and discuss the results in Section 5. . A stem and leaf plot breaks each value of a quantitative data set into two pieces: a stem, typically for the highest place value, and a leaf for the other place values. It provides a way to list all data values in a compact form. For example, if you are using this graph to review student test scores of 84, 65, 78, 75, 89, 90, 88, 83, 72, 91, the stems would be 6, 7, 8, and 9.

List coloring of random and pseudo-random graphs random graphs, and the book of Bollob as [8] is an excellent extensive account of the known results where each set is assigned to one of our subgraphs, making sure by some standard large deviation inequalities that the list of colors of each vertex contains su ciently many colors assigned.   Logistic random effects models are a popular tool to analyze multilevel also called hierarchical data with a binary or ordinal outcome. Here, we aim to compare different statistical software implementations of these models. We used individual patient data from patients in centers with moderate and severe Traumatic Brain Injury (TBI) enrolled in eight Randomized Controlled Trials Cited by: Philippe Rigollet works at the intersection of statistics, machine learning, and optimization, focusing primarily on the design and analysis of statistical methods for high-dimensional problems. His recent research focuses on statistical optimal transport. Neighborhoods: Graphs induced by the friends of a single Face-book user ego and the friendship connections among these indi-viduals (excluding the ego). Groups: Graphs induced by the members of a ‘Facebook group’, a Facebook feature for organizing focused conversations between a small or moderate-sized set of users.

  The mini-course is an introduction to local convergence of sparse graph sequences. Initiated by Benjamini and Schramm around 15 years ago, the theory recently advanced in several directions with many remarkable applications. We will discuss some key examples, with special emphasis on the case of sparse random graphs. Research interests. I am currently assistant professor (maître de conférences) at the Department of Mathematics of the University Paris-Sud (Orsay, France), and I am working in the team Probability and is a detailed curriculum vitæ: in French and in is also a research statement: in French and in English. My works focus on the use of techniques from harmonic. Edge-independent random graphs are a model of random graphs in which each potential edge appears independently with an individual probability. Based on the relative entropy method, we determine the upper and lower bounds for the extremal vertex degrees using the edge probability matrix and its largest eigenvalue. Moreover, an application to random graphs with given expected degree sequences is Cited by: 5. Some useful facts about eigenvectors and eigenvalues will assist in the following discussion: The eigenvectors of X + cI (where c is a constant, and I is an identity matrix) are the same as the eigenvectors of other words, adding a constant, c, to the diagonal elements of X does not affect the eigenvectors. However, the eigenvalues of X + c I are shifted by an amount equal to c.

Some large deviation results for sparse random graphs by Neil O"Connell Download PDF EPUB FB2

The standard reference on random graphs is the book of Bolloba´s [1]; the lecture notes of Spencer [5] provide a useful introduction. For an overview of the main results on sparse random graphs, see [2].

The giant component is the subject of recent paper by Janson et al. [3], where some very sharp results are presented. 2 Preliminaries. The results of the previous chapters were derived using tools from graph limit theory.

This theory, however, is inadequate for understanding the behavior of sparse graphs. The goal of this chapter is Cited by: 1.

The standard reference on random graphs is the book ofBollobas [lJ; the lecture notes of Spencer [5J provide a useful introduction. For an overview of the main results on sparse random graphs, see [2J.

The giant component is the subject of recent paper by Janson et al [4]' where some very sharp results are presented. 2 A smidgen of large Cited by: First book-length treatment of large deviations for random graphs, plus a chapter on exponential random graphs Contains a summary of important results from graph limit theory with complete proofs Written in a style for beginning graduate students, self-contained with essentially no need for background knowledge other than some amount of Brand: Springer International Publishing.

Some related asymptotics are also established. The proofs of the large and moderate deviation asymptotics employ methods of idempotent probability theory. As a byproduct of the results, we provide some additional insight into the nature of phase transitions in sparse random : Anatolii A.

Puhalskii. PDF | For a finite typed graph on $n$ nodes and with type law $\\mu,$ we define the so-called spectral potential $\\rho_{\\lambda}(\\,\\cdot,\\,\\mu),$ of the. The method is applied to compute the large deviation rate functions for subgraph counts in sparse random graphs.

Previous technology, based on Szemer\'edi's regularity lemma, works only for dense. ‘The first volume of Remco van der Hofstad's Random Graphs and Complex Networks is the definitive introduction into the mathematical world of random networks.

Written for students with only a modest background in probability theory, it provides plenty of motivation for the topic and introduces the essential tools of probability at a gentle by: The large deviation principle for P ˜ n, p on (W ˜, δ) is much more useful and is the main result of this article.

Theorem For each fixed p ∈ (0, 1), the sequence P ˜ n, p obeys Some large deviation results for sparse random graphs book large deviation principle in the space W ˜ (equipped with the cut metric) with rate function I p defined by: Abstract.

Let H d (n,p) signify a random d-uniform hypergraph with n vertices in which each of the \({n}\choose{d}\) possible edges is present with probability p = p(n) independently, and let H d (n,m) denote a uniformly distributed d-uniform hypergraph with n vertices and m edges.

We establish a local limit theorem for the number of vertices and edges in the largest component of H d (n,p) in Cited by: 6. Large deviation preliminaries -- 5. Large deviations for dense random graphs -- 6. Applications of dense graph large deviations -- 7.

Exponential random graph models -- 8. Large deviations for sparse graphs -- Index.\/span>\"@ en\/a> ; \u00A0\u00A0\u00A0\n schema:description\/a> \" This book addresses the emerging body of literature on the.

What is this book about. High-dimensional probability is an area of probability theory that studies random objects in Rn where the dimension ncan be very large. This book places par-ticular emphasis on random vectors, random matrices, and random projections.

It teaches basic theoretical skills for the analysis of these objects, which include. Of particular interest in the study of sparse random graphs is the phase transition in the size of the largest We aren’t interested in all the edges of the random graph, only in some tree skeleton of each component.

The slogan for the LDP as in Frank den Hollander’s excellent book is: “A large deviation event will happen in the. Large deviations for random graphs: École d'été de probabilités de Saint-Flour XLV - Large deviation preliminaries.- 5.

Large deviations for dense random graphs.- 6. Applications of dense graph large deviations.- 7. Exponential random graph models.- 8. Large deviations for sparse graphs.- Index. Series Title: Lecture notes in. For some results, this is a genuine problem, when it may be easier to work on the torus.

Of particular interest in the study of sparse random graphs is the phase transition in the size of the largest component The slogan for the LDP as in Frank den Hollander’s excellent book is: “A large deviation event will happen in the least.

LOCALIZATION IN RANDOM GEOMETRIC GRAPHS WITH TOO MANY EDGES its from the geometry of Rd to evaluate the upper tail large deviation rate function. In addition, we provide a “structure theorem” to describe the such as sets which are sparse but of large measure (e.g.

generalized Cantor sets) or have boundaries that take up a. Right-convergence of sparse random graphs arXivv1 [] 14 Feb David Gamarnik ∗ Febru Abstract The paper is devoted to the problem of establishing right-convergence of sparse random graphs.

probability graph-theory random-graphs large-deviation-theory. asked Apr 17 at user 5 5 bronze badges. votes. I am reading the book Random Graph Dynamics by Rick Durett and on page 42 they apply the optional stopping theorem, which I have never heard of before, and I can not figure out how it can be applied.

Maximum Cut problem on sparse random hypergraphs. Structural results using the interpolation method and the algorithmic implications. Abstract: We consider a particular version of the Maximum Cut problem (to be defined in the talk) on a sparse random K-uniform hypergraph, when K is even.

The goal is to compute the maximum cut value w.h.p., and. In section 4 we present and prove the key results of the paper, the local and the extended large deviation principles, for the trajectories of univariate random walks in the space ${\mathbb D}$ of functions without discontinuities of the second by: ima, in some sense.

The purpose of this book is to explain a simple idea which enables one to write down, with little e ort, approximate solutions to such questions. Let us try to say this idea in one paragraph. (a) Problems about random extrema can often be translated into prob-lems about sparse random sets in d 1 dimensions.

1 Introduction Computer science as an academic discipline began in the ’s. Emphasis was on programming languages, compilers, operating systems, and the mathematical theory that. This means that with sparse or rare event data, logistic regression will produce biased results. Commonly recommended "solutions" for this problem are to go out and get a larger sample of data or, alternatively, to specifically subsample those segments that are both important to the analysis and sparsely populated.

Books [IV] Random Graphs. (With Tomasz Luczak and Andrzej Rucinski.) Wiley, New York, ISBN: Cover, Table of Contents, Errata, pages (missing in some copies). [III] Gaussian Hilbert Spaces.

Cambridge University Press, Cambridge, UK, ISBN: [II] Orthogonal Decompositions and Functional Limit Theorems for Random Graph Statistics. Eyal Lubetzky and I just uploaded to the arXiv our new paper “On replica symmetry of large deviations in random graphs.” In this paper we answer the following question of Chatterjee and Varadhan.

Question. be a large integer and let be an instance of the Erdős-Rényi random graph conditioned on the rare event that has at least as many triangles as the typical.

Chapter 1. Vectors, Matrices, and Arrays Introduction NumPy is the foundation of the Python machine learning stack. NumPy allows for efficient operations on the data structures often used in - Selection from Machine Learning with Python Cookbook [Book].

Lemma there is some absolute positive value ε independent of n for which e n 2 > ε. So the number of edges behaves like cn2 where c is some positive constant. This is a large number of edges and so the Regularity Lemma does not work well for graphs with few edges (such graphs are called sparse graphs, the telephone graph is an example of such.

In this paper, we present and analyze a simple and robust spectral algorithm for the stochastic block model with k blocks, for any k fixed. Our algorithm works with graphs having constant edge density, under an optimal condition on the gap between the density inside a block and the density between the blocks.

As a co-product, we settle an open question posed by Abbe et. concerning censor. I am (slowly) writing a book on the statistical analysis of complex systems models. Teaching Student: Whenever there is any question, one's mind is confused.

What is the matter. Master Ts'ao-shan: Kill, kill. In fallI will teach /data over space and time, an introduction to time series, spatial, and spatio-temporal.

The addition of two new sections, numerous new results and references means that this represents an up-to-date and comprehensive account of random graph theory.

The theory estimates the number of graphs of a given degree that exhibit certain properties. On the Resilience of Long Cycles in Random Graphs Domingos Dellamonica Jr We state our results in terms of pseudorandom graphs. Using the fact that truly random graphs are asymptotically almost surely, i.e., with probability tending Szemer edi’s Regularity Lemma for sparse graphs Let a graph G = (V;E) and a real number 0.

The main theorem is stated in Section 3, and results on the chromatic number and clique number are given in Section 4. Some important preliminaries on continuity of the mapping μ ↦ Γ μ are dealt with in Section 5, and Section 6 gives the proof of the main theorem.

Section 7 discusses some examples of interval graph limits and the corresponding random interval by:   These are some of the main features of aforementioned real-world networks. We also empirically observed that word association networks have many of the theoretical properties of the DRGG model.

Mathematical results. For any class of random graphs, this should not have a large impact on the results. The same procedure was applied to the Author: Jesse Michel, Sushruth Reddy, Rikhav Shah, Sandeep Silwal, Ramis Movassagh.