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Author : Alan Frieze,Michał Karoński
Publisher : Cambridge University Press
Page : 483 pages
File Size : 41,5 Mb
Release : 2016
Category : Mathematics
ISBN : 9781107118508
The text covers random graphs from the basic to the advanced, including numerous exercises and recommendations for further reading.
Author : Remco van der Hofstad
Publisher : Cambridge University Press
Page : 341 pages
File Size : 54,6 Mb
Release : 2016-12-22
Category : Computers
ISBN : 9781107172876
This classroom-tested text is the definitive introduction to the mathematics of network science, featuring examples and numerous exercises.
Author : Svante Janson,Tomasz Luczak,Andrzej Rucinski
Publisher : John Wiley & Sons
Page : 350 pages
File Size : 42,6 Mb
Release : 2011-09-30
Category : Mathematics
ISBN : 9781118030967
A unified, modern treatment of the theory of random graphs-including recent results and techniques Since its inception in the 1960s, the theory of random graphs has evolved into a dynamic branch of discrete mathematics. Yet despite the lively activity and important applications, the last comprehensive volume on the subject is Bollobas's well-known 1985 book. Poised to stimulate research for years to come, this new work covers developments of the last decade, providing a much-needed, modern overview of this fast-growing area of combinatorics. Written by three highly respected members of the discrete mathematics community, the book incorporates many disparate results from across the literature, including results obtained by the authors and some completely new results. Current tools and techniques are also thoroughly emphasized. Clear, easily accessible presentations make Random Graphs an ideal introduction for newcomers to the field and an excellent reference for scientists interested in discrete mathematics and theoretical computer science. Special features include: * A focus on the fundamental theory as well as basic models of random graphs * A detailed description of the phase transition phenomenon * Easy-to-apply exponential inequalities for large deviation bounds * An extensive study of the problem of containing small subgraphs * Results by Bollobas and others on the chromatic number of random graphs * The result by Robinson and Wormald on the existence of Hamilton cycles in random regular graphs * A gentle introduction to the zero-one laws * Ample exercises, figures, and bibliographic references
Author : Rick Durrett
Publisher : Cambridge University Press
Page : 203 pages
File Size : 45,8 Mb
Release : 2010-05-31
Category : Mathematics
ISBN : 9781139460880
The theory of random graphs began in the late 1950s in several papers by Erdos and Renyi. In the late twentieth century, the notion of six degrees of separation, meaning that any two people on the planet can be connected by a short chain of people who know each other, inspired Strogatz and Watts to define the small world random graph in which each site is connected to k close neighbors, but also has long-range connections. At a similar time, it was observed in human social and sexual networks and on the Internet that the number of neighbors of an individual or computer has a power law distribution. This inspired Barabasi and Albert to define the preferential attachment model, which has these properties. These two papers have led to an explosion of research. The purpose of this book is to use a wide variety of mathematical argument to obtain insights into the properties of these graphs. A unique feature is the interest in the dynamics of process taking place on the graph in addition to their geometric properties, such as connectedness and diameter.
Author : Philipp Blanchard,Dimitri Volchenkov
Publisher : Springer Science & Business Media
Page : 271 pages
File Size : 45,9 Mb
Release : 2011-05-26
Category : Science
ISBN : 9783642195921
Most networks and databases that humans have to deal with contain large, albeit finite number of units. Their structure, for maintaining functional consistency of the components, is essentially not random and calls for a precise quantitative description of relations between nodes (or data units) and all network components. This book is an introduction, for both graduate students and newcomers to the field, to the theory of graphs and random walks on such graphs. The methods based on random walks and diffusions for exploring the structure of finite connected graphs and databases are reviewed (Markov chain analysis). This provides the necessary basis for consistently discussing a number of applications such diverse as electric resistance networks, estimation of land prices, urban planning, linguistic databases, music, and gene expression regulatory networks.
Author : Béla Bollobás
Publisher : Cambridge University Press
Page : 520 pages
File Size : 49,5 Mb
Release : 2001-08-30
Category : Mathematics
ISBN : 0521797225
This is a revised and updated version of the classic first edition.
Author : Sourav Chatterjee
Publisher : Springer
Page : 170 pages
File Size : 51,9 Mb
Release : 2017-08-31
Category : Mathematics
ISBN : 9783319658162
This book addresses the emerging body of literature on the study of rare events in random graphs and networks. For example, what does a random graph look like if by chance it has far more triangles than expected? Until recently, probability theory offered no tools to help answer such questions. Important advances have been made in the last few years, employing tools from the newly developed theory of graph limits. This work represents the first book-length treatment of this area, while also exploring the related area of exponential random graphs. All required results from analysis, combinatorics, graph theory and classical large deviations theory are developed from scratch, making the text self-contained and doing away with the need to look up external references. Further, the book is written in a format and style that are accessible for beginning graduate students in mathematics and statistics.
Author : Edgar M. Palmer
Publisher : Wiley-Interscience
Page : 208 pages
File Size : 52,7 Mb
Release : 1985-03-07
Category : Mathematics
ISBN : UOM:39015039000271
Probability models for graphs; Models a, b and c; Expection; properties of almost all graphs Threshold functions; The evolution randon graphs; A threshold for isolated vertices; A sharper threshold; Threshold for existence; Selected highlights.
Author : Alan Frieze,Michał Karoński
Publisher : Cambridge University Press
Page : 233 pages
File Size : 55,9 Mb
Release : 2023-03-31
Category : Computers
ISBN : 9781009260282
A rigorous yet accessible introduction to the rapidly expanding subject of random graphs and networks.
Author : Anthony C. C. Coolen,Alessia Annibale,Ekaterina Roberts
Publisher : Oxford University Press
Page : 325 pages
File Size : 47,8 Mb
Release : 2017
Category : Mathematics
ISBN : 9780198709893
This book describes how to correctly and efficiently generate random networks based on certain constraints. Being able to test a hypothesis against a properly specified control case is at the heart of the 'scientific method'.
Author : Martin T. Barlow,Gordon Slade
Publisher : Springer Nature
Page : 421 pages
File Size : 43,7 Mb
Release : 2019-12-03
Category : Mathematics
ISBN : 9783030320119
The 2017 PIMS-CRM Summer School in Probability was held at the Pacific Institute for the Mathematical Sciences (PIMS) at the University of British Columbia in Vancouver, Canada, during June 5-30, 2017. It had 125 participants from 20 different countries, and featured two main courses, three mini-courses, and twenty-nine lectures. The lecture notes contained in this volume provide introductory accounts of three of the most active and fascinating areas of research in modern probability theory, especially designed for graduate students entering research: Scaling limits of random trees and random graphs (Christina Goldschmidt) Lectures on the Ising and Potts models on the hypercubic lattice (Hugo Duminil-Copin) Extrema of the two-dimensional discrete Gaussian free field (Marek Biskup) Each of these contributions provides a thorough introduction that will be of value to beginners and experts alike.
Author : Alexander Grigor’yan
Publisher : American Mathematical Soc.
Page : 150 pages
File Size : 54,5 Mb
Release : 2018-08-23
Category : Finite groups
ISBN : 9781470443979
A central object of this book is the discrete Laplace operator on finite and infinite graphs. The eigenvalues of the discrete Laplace operator have long been used in graph theory as a convenient tool for understanding the structure of complex graphs. They can also be used in order to estimate the rate of convergence to equilibrium of a random walk (Markov chain) on finite graphs. For infinite graphs, a study of the heat kernel allows to solve the type problem—a problem of deciding whether the random walk is recurrent or transient. This book starts with elementary properties of the eigenvalues on finite graphs, continues with their estimates and applications, and concludes with heat kernel estimates on infinite graphs and their application to the type problem. The book is suitable for beginners in the subject and accessible to undergraduate and graduate students with a background in linear algebra I and analysis I. It is based on a lecture course taught by the author and includes a wide variety of exercises. The book will help the reader to reach a level of understanding sufficient to start pursuing research in this exciting area.
Author : Geoffrey Grimmett
Publisher : Cambridge University Press
Page : 260 pages
File Size : 40,7 Mb
Release : 2010-06-24
Category : Mathematics
ISBN : 9781139488365
This introduction to some of the principal models in the theory of disordered systems leads the reader through the basics, to the very edge of contemporary research, with the minimum of technical fuss. Topics covered include random walk, percolation, self-avoiding walk, interacting particle systems, uniform spanning tree, random graphs, as well as the Ising, Potts, and random-cluster models for ferromagnetism, and the Lorentz model for motion in a random medium. Schramm–Löwner evolutions (SLE) arise in various contexts. The choice of topics is strongly motivated by modern applications and focuses on areas that merit further research. Special features include a simple account of Smirnov's proof of Cardy's formula for critical percolation, and a fairly full account of the theory of influence and sharp-thresholds. Accessible to a wide audience of mathematicians and physicists, this book can be used as a graduate course text. Each chapter ends with a range of exercises.
Author : Bela Bollobas
Publisher : Springer Science & Business Media
Page : 191 pages
File Size : 42,6 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461299677
From the reviews: "Béla Bollobás introductory course on graph theory deserves to be considered as a watershed in the development of this theory as a serious academic subject. ... The book has chapters on electrical networks, flows, connectivity and matchings, extremal problems, colouring, Ramsey theory, random graphs, and graphs and groups. Each chapter starts at a measured and gentle pace. Classical results are proved and new insight is provided, with the examples at the end of each chapter fully supplementing the text... Even so this allows an introduction not only to some of the deeper results but, more vitally, provides outlines of, and firm insights into, their proofs. Thus in an elementary text book, we gain an overall understanding of well-known standard results, and yet at the same time constant hints of, and guidelines into, the higher levels of the subject. It is this aspect of the book which should guarantee it a permanent place in the literature." #Bulletin of the London Mathematical Society#1
Author : William L. William L. Hamilton
Publisher : Springer Nature
Page : 141 pages
File Size : 41,9 Mb
Release : 2022-06-01
Category : Computers
ISBN : 9783031015885
Graph-structured data is ubiquitous throughout the natural and social sciences, from telecommunication networks to quantum chemistry. Building relational inductive biases into deep learning architectures is crucial for creating systems that can learn, reason, and generalize from this kind of data. Recent years have seen a surge in research on graph representation learning, including techniques for deep graph embeddings, generalizations of convolutional neural networks to graph-structured data, and neural message-passing approaches inspired by belief propagation. These advances in graph representation learning have led to new state-of-the-art results in numerous domains, including chemical synthesis, 3D vision, recommender systems, question answering, and social network analysis. This book provides a synthesis and overview of graph representation learning. It begins with a discussion of the goals of graph representation learning as well as key methodological foundations in graph theory and network analysis. Following this, the book introduces and reviews methods for learning node embeddings, including random-walk-based methods and applications to knowledge graphs. It then provides a technical synthesis and introduction to the highly successful graph neural network (GNN) formalism, which has become a dominant and fast-growing paradigm for deep learning with graph data. The book concludes with a synthesis of recent advancements in deep generative models for graphs—a nascent but quickly growing subset of graph representation learning.