Random Graphs and Network Statistics
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Franceschetti M., Meester R. Random Networks for Communication
This page intentionally left blank Random Networks for Communication When is a network (almost) connected? How much information can it carry? How can you find a particular destination within the network? And how do you approach these questions – and others – when the network is random? The analysis of communication networks requires a fascinating synthesis of random graph theory, stochastic geometry and percolation theory to provide models for both structure and information flow. This book is the first comprehensive introduction for graduate students and scientists to techniques and problems in the field of spatial random networks. The selection of material is driven by applications arising in engineering, and the treatment is both readable and mathematically rigorous. Though mainly concerned with information-flow-related questions motivated by wireless data networks, the models developed are also of interest in a broader context, ranging from engineering to social networks, biology, and physics. Massimo Franceschetti is assistant professor of electrical and computer engineering at the University of California, San Diego. His work in communication system theory sits at the interface between networks, information theory, control, and electromagnetics. Ronald Meester is professor of mathematics at the Vrije Universiteit Amsterdam. He has published broadly in percolation theory, spatial random processes, self-organised crit- icality, ergodic theory, and forensic statistics and is the author of Continuum Percolation (with Rahul Roy) and A Natural Introduction to Probability Theory. CAMBRIDGE SERIES IN STATISTICAL AND PROBABILISTIC MATHEMATICS Editorial Board R. Gill (Mathematisch Instituut, Leiden University) B. D. Ripley (Department of Statistics, University of Oxford) S. Ross (Department of Industrial & Systems Engineering, University of Southern California) B. -
Random Graphs in Evolution François Bienvenu
Random graphs in evolution François Bienvenu To cite this version: François Bienvenu. Random graphs in evolution. Combinatorics [math.CO]. Sorbonne Université, 2019. English. NNT : 2019SORUS180. tel-02932179 HAL Id: tel-02932179 https://tel.archives-ouvertes.fr/tel-02932179 Submitted on 7 Sep 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. École Doctorale de Sciences Mathématiques de Paris Centre Laboratoire de Probabilités, Statistiques et Modélisation Sorbonne Université Thèse de doctorat Discipline : mathématiques présentée par François Bienvenu Random Graphs in Evolution Sous la direction d’Amaury Lambert Après avis des rapporteurs : M. Simon Harris (University of Auckland) Mme Régine Marchand (Université de Lorraine) Soutenue le 13 septembre 2019 devant le jury composé de : M. Nicolas Broutin Sorbonne Université Examinateur M. Amaury Lambert Sorbonne Université Directeur de thèse Mme Régine Marchand Université de Lorraine Rapporteuse Mme. Céline Scornavacca Chargée de recherche CNRS Examinatrice M. Viet Chi Tran Université de Lille Examinateur Contents Contents3 1 Introduction5 1.1 A very brief history of random graphs................. 6 1.2 A need for tractable random graphs in evolution ........... 7 1.3 Outline of the thesis........................... 8 Literature cited in the introduction..................... -
Who Would Have Thought Card Shuffling Was So Involved?
Who would have thought card shuffling was so involved? Leah Cousins August 26, 2019 Abstract In this paper we present some interesting mathematical results on card shuffling for two types of shuffling: the famous riffle shuffle and the random-to-top shuffle. A natural question is how long it takes for a deck to be randomized under a particular shuffling technique. Mathematically, this is the mixing time of a Markov chain. In this paper we present these results for both the riffle shuffle and the random-to-top shuffle. For the same two shuffles, we also include natural results for card shuffling which directly relate to well-known combinatorial objects such as Bell numbers and Young tableaux. Contents 1 Introduction 2 2 What is Card Shuffling, Really? 3 2.1 Shuffles are Bijective Functions . .3 2.2 Random Walks and Markov Chains . .4 2.3 What is Randomness? . .5 2.4 There are even shuffle algebras ?.........................6 3 The Riffle Shuffle 7 3.1 The Mixing Time . .8 3.2 Riffle Shuffling for Different Card Games . 11 3.3 Carries and Card Shuffles . 14 4 The Random-to-top Shuffle 17 4.1 Lifting Cards . 17 4.2 Properties of the Lifting Process . 18 4.3 Bell Numbers and the Random-to-top Shuffle . 18 4.4 Young Diagrams . 22 4.5 The Mixing Time . 29 5 Concluding Remarks 36 1 1 Introduction Humans have been shuffling decks of cards since playing cards were first invented in 1000AD in Eastern Asia [25]. It is unclear when exactly card shuffling theory was first studied, but since card shuffling theory began with magicians' card tricks, it is fair to think that it was around the time that magicians were studying card tricks. -
When Is a Deck of Cards Well Shuffled?
Technische Universiteit Delft Faculteit Elektrotechniek, Wiskunde en Informatica Delft Institute of Applied Mathematics When is a deck of cards well shuffled? (Nederlandse titel: Wanneer is een pak kaarten goed geschud?) Verslag ten behoeve van het Delft Institute of Applied Mathematics als onderdeel ter verkrijging van de graad van BACHELOR OF SCIENCE in TECHNISCHE WISKUNDE door RICARDO TEBBENS Delft, Nederland Augustus 2018 Copyright © 2018 door Ricardo Tebbens. Alle rechten voorbehouden. 2 BSc verslag TECHNISCHE WISKUNDE \When is a deck of cards well shuffled?" (Nederlandse titel: \Wanneer is een pak kaarten goed geschud?)" RICARDO TEBBENS Technische Universiteit Delft Begeleider Dr. M.T. Joosten Overige commissieleden Dr. B. van den Dries Prof. Dr.ir. M.C. Veraar Augustus, 2018 Delft Abstract When is a deck of cards shuffled good enough? We have to perform seven Riffle Shuffles to randomize a deck of 52 cards. The mathematics used to calculate this, has some strong connections with permutations, rising sequences and the L1 metric: the variation distance. If we combine these factors, we can get an expression of how good a way of shuffling is in randomizing a deck. We say a deck is randomized, when every possible order of the cards is equally likely. This gives us the cut-off result of seven shuffles. Furthermore, this gives us a window to look at other ways of shuffling, some even used in casinos. It turns out that some of these methods are not randomizing a deck enough. We can also use Markov chains in order to see how we randomize cards by "washing" them over a table. -
Local Cliques in ER-Perturbed Random Geometric Graphs
Local Cliques in ER-Perturbed Random Geometric Graphs Matthew Kahle Department of Mathematics, The Ohio State University, USA [email protected] Minghao Tian Computer Science and Engineering Dept., The Ohio State University, USA [email protected] Yusu Wang Computer Science and Engineering Dept., The Ohio State University, USA [email protected] Abstract We study a random graph model introduced in [20] where one adds Erdős–Rényi (ER) type ∗ perturbation to a random geometric graph. More precisely, assume GX is a random geometric graph sampled from a nice measure on a metric space X = (X, d). An ER-perturbed random geometric ∗ graph Gb(p, q) is generated by removing each existing edge from GX with probability p, while inserting ∗ each non-existent edge to GX with probability q. We consider a localized version of clique number for Gb(p, q): Specifically, we study the edge clique number for each edge in a graph, defined as the size of the largest clique(s) in the graph containing that edge. We show that the edge clique number presents two fundamentally different types of behaviors in Gb(p, q), depending on which “type” of randomness it is generated from. As an application of the above results, we show that by a simple filtering process based on the ∗ edge clique number, we can recover the shortest-path metric of the random geometric graph GX within a multiplicative factor of 3 from an ER-perturbed observed graph Gb(p, q), for a significantly wider range of insertion probability q than what is required in [20]. -
Random Networks for Communication
This page intentionally left blank Random Networks for Communication When is a network (almost) connected? How much information can it carry? How can you find a particular destination within the network? And how do you approach these questions – and others – when the network is random? The analysis of communication networks requires a fascinating synthesis of random graph theory, stochastic geometry and percolation theory to provide models for both structure and information flow. This book is the first comprehensive introduction for graduate students and scientists to techniques and problems in the field of spatial random networks. The selection of material is driven by applications arising in engineering, and the treatment is both readable and mathematically rigorous. Though mainly concerned with information-flow-related questions motivated by wireless data networks, the models developed are also of interest in a broader context, ranging from engineering to social networks, biology, and physics. Massimo Franceschetti is assistant professor of electrical and computer engineering at the University of California, San Diego. His work in communication system theory sits at the interface between networks, information theory, control, and electromagnetics. Ronald Meester is professor of mathematics at the Vrije Universiteit Amsterdam. He has published broadly in percolation theory, spatial random processes, self-organised crit- icality, ergodic theory, and forensic statistics and is the author of Continuum Percolation (with Rahul Roy) and A Natural Introduction to Probability Theory. CAMBRIDGE SERIES IN STATISTICAL AND PROBABILISTIC MATHEMATICS Editorial Board R. Gill (Mathematisch Instituut, Leiden University) B. D. Ripley (Department of Statistics, University of Oxford) S. Ross (Department of Industrial & Systems Engineering, University of Southern California) B. -
Spectral Analysis of Random Geometric Graphs Mounia Hamidouche
Spectral analysis of random geometric graphs Mounia Hamidouche To cite this version: Mounia Hamidouche. Spectral analysis of random geometric graphs. Signal and Image processing. Université Côte d’Azur, 2020. English. NNT : 2020COAZ4019. tel-03135086 HAL Id: tel-03135086 https://tel.archives-ouvertes.fr/tel-03135086 Submitted on 8 Feb 2021 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. THÈSE DE DOCTORAT Analyse Spectrale de Graphes Géométriques Aléatoires Mounia HAMIDOUCHE EURECOM Présentée en vue de l’obtention Devant le jury, composé de : du grade de docteur en automatique, Laura Cottatellucci Directeur traitement du signal et des images Professeur, FAU, Allemagne d’Université Côte d’Azur Konstantin Avrachenkov Directeur Directeur de recherche (Inria), Sophia Dirigée par : Laura Cottatellucci et Antipolis Konstantin Avrachenkov Pierre Borgnat Rapporteur Directeur de recherche (CNRS), ENS de Lyon Soutenue le: 29/05/2020 Sergey Skipetrov Rapporteur Directeur de recherche (CNRS), Grenoble Hocine Cherifi Examinateur Professeur, Université de Boulogne Dirk Slock Examinateur Professeur, Eurecom, Sophia Antipolis Abstract (English) In this thesis we study random geometric graphs (RGGs) using tools from random ma- trix theory and probability theory to tackle key problems in complex networks. -
Introduction
Cambridge University Press 978-0-521-85442-9 - Random Networks for Communication: From Statistical Physics to Information Systems Massimo Franceschetti and Ronald Meester Excerpt More information 1 Introduction Random networks arise when nodes are randomly deployed on the plane and randomly connected to each other. Dependingon the specific rules used to construct them, they create structures that can resemble what is observed in real natural, as well as in artificial, complex systems. Thus, they provide simple models that allow us to use probability theory as a tool to explain the observable behaviour of real systems and to formally study and predict phenomena that are not amenable to analysis with a deterministic approach. This often leads to useful design guidelines for the development and optimal operation of real systems. Historically, random networks has been a field of study in mathematics and statistical physics, although many models were inspired by practical questions of engineering inter- est. One of the early mathematical models appeared in a series of papers startingin 1959 by the two Hungarian mathematicians Paul Erdös and Alfréd Rényi. They investigated what a ‘typical’ graph of n vertices and m edges looks like, by connecting nodes at ran- dom. They showed that many properties of these graphs are almost always predictable, as they suddenly arise with very high probability when the model parameters are chosen appropriately. This peculiar property generated much interest among mathematicians, and their papers marked the starting point of the field of random graph theory. The graphs they considered, however, were abstract mathematical objects and there was no notion of geometric position of vertices and edges. -
Percolation on Directed Graphs with Given Degree Sequence
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by University of Waterloo's Institutional Repository On the Strongly Connected Components of Random Directed Graphs with Given Degree Sequences by Alessandra Graf A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree of Master of Mathematics in Combinatorics and Optimization Waterloo, Ontario, Canada, 2016 c Alessandra Graf 2016 I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii Abstract A strongly connected component of a directed graph G is a maximal subgraph H of G such that for each pair of vertices u and v in H, there is a directed path from u to v and a directed path from v to u in H. A strongly connected component is said to be giant if it has linear size. We determine the threshold at which a random directed graph with a well-behaved degree sequence asymptotically almost surely contains a giant strongly connected compo- nent. This is a new proof of a result by Cooper and Frieze in [12]. In addition, we predict the site percolation threshold for the presence of a giant strongly connected component in a graph with a well-behaved degree sequence. iii Acknowledgements I would like to thank the my supervisor, Jane Gao, for all her support, guidance, and patience. -
Final Copy 2019 06 25 Pratt
This electronic thesis or dissertation has been downloaded from Explore Bristol Research, http://research-information.bristol.ac.uk Author: Pratt, Pete Title: Connectivity and Mobility in Wireless Networks General rights Access to the thesis is subject to the Creative Commons Attribution - NonCommercial-No Derivatives 4.0 International Public License. A copy of this may be found at https://creativecommons.org/licenses/by-nc-nd/4.0/legalcode This license sets out your rights and the restrictions that apply to your access to the thesis so it is important you read this before proceeding. Take down policy Some pages of this thesis may have been removed for copyright restrictions prior to having it been deposited in Explore Bristol Research. However, if you have discovered material within the thesis that you consider to be unlawful e.g. breaches of copyright (either yours or that of a third party) or any other law, including but not limited to those relating to patent, trademark, confidentiality, data protection, obscenity, defamation, libel, then please contact [email protected] and include the following information in your message: •Your contact details •Bibliographic details for the item, including a URL •An outline nature of the complaint Your claim will be investigated and, where appropriate, the item in question will be removed from public view as soon as possible. Connectivity and Mobility in Wireless Networks Peter Pratt A dissertation submitted to the University of Bristol in accordance with the requirements for award of the degree of Doctor of Philosophy in the Faculty of Mathematics June 23, 2019 Word Count: 56785 To my Wife and Son Abstract With the ever continuing increase in the number of devices connecting to one another, forming the Internet of Things (IoT), understanding how the performance of these networks connect is of paramount importance. -
Fall 2009 Version of Course 15-359, Computer Science Department, Carnegie Mellon University
Fall 2009 version of Course 15-359, Computer Science Department, Carnegie Mellon University. Acknowledgments: CMU’s course 15-359, Probability and Computing, was originally conceived and designed by Mor Harchol-Balter and John Lafferty. The choice, order, and presentation of topics in the latter half of the course is strongly informed by the work of Mor Harchol-Balter. Indeed, you might like to buy her book! Performance Modeling and Design of Computer Systems: Queueing Theory in Action, http://www.cs.cmu.edu/~harchol/PerformanceModeling/book.html The choice, order, and presentation of topics in the earlier half of the course is informed by the work of John Lafferty: http://galton.uchicago.edu/~lafferty/ Further, a very great deal of material in these lecture notes was strongly informed by the outstanding book Probability and Computing by Michael Mitzenmacher and Eli Upfal, http://www.cambridge.org/us/academic/subjects/computer-science/algorithmics-complexity- computer-algebra-and-computational-g/probability-and-computing-randomized-algorithms-and- probabilistic-analysis Many thanks to Mor Harchol-Balter, John Lafferty, Michael Mitzenmacher, Eli Upfal (and many other web sources from which I borrowed)! 15-359: Probability and Computing Fall 2008 Lecture 1: Probability in Computing; Verifying matrix multiplication 1 Administrivia The course web site is: http://15-359.blogspot.com. Please read carefully the Policies document there, which will also be handed out in class. This is also a blog, so please add it to your RSS reader, or at least read it frequently. Important announcements | and more! | will appear there. 2 Probability and computing Why is probability important for computing? Why is randomness important for computing? Here are some example applications: Cryptography.