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Author: Mathew Penrose
Publisher: Oxford University Press
Published: 2003
Total Pages: 345
ISBN-13: 0198506260
DOWNLOAD EBOOKThis monograph provides and explains the mathematics behind geometric graph theory. Applications of this theory are used on the study of neural networks, spread of disease, astrophysics and spatial statistics.
Author: Mathew Penrose
Publisher:
Published: 2003
Total Pages:
ISBN-13:
DOWNLOAD EBOOKAuthor: Alan Frieze
Publisher: Cambridge University Press
Published: 2016
Total Pages: 483
ISBN-13: 1107118506
DOWNLOAD EBOOKThe text covers random graphs from the basic to the advanced, including numerous exercises and recommendations for further reading.
Author: Rick Durrett
Publisher: Cambridge University Press
Published: 2010-05-31
Total Pages: 203
ISBN-13: 1139460889
DOWNLOAD EBOOKThe 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: Michael Krivelevich
Publisher: Cambridge University Press
Published: 2016-04-25
Total Pages: 129
ISBN-13: 1316552942
DOWNLOAD EBOOKThe theory of random graphs is a vital part of the education of any researcher entering the fascinating world of combinatorics. However, due to their diverse nature, the geometric and structural aspects of the theory often remain an obscure part of the formative study of young combinatorialists and probabilists. Moreover, the theory itself, even in its most basic forms, is often considered too advanced to be part of undergraduate curricula, and those who are interested usually learn it mostly through self-study, covering a lot of its fundamentals but little of the more recent developments. This book provides a self-contained and concise introduction to recent developments and techniques for classical problems in the theory of random graphs. Moreover, it covers geometric and topological aspects of the theory and introduces the reader to the diversity and depth of the methods that have been devised in this context.
Author: Remco van der Hofstad
Publisher: Cambridge University Press
Published: 2017
Total Pages: 341
ISBN-13: 110717287X
DOWNLOAD EBOOKThis classroom-tested text is the definitive introduction to the mathematics of network science, featuring examples and numerous exercises.
Author: Marc Barthelemy
Publisher: Springer
Published: 2017-12-30
Total Pages: 342
ISBN-13: 331920565X
DOWNLOAD EBOOKThis book develops a morphodynamical approach of spatial networks with a particular emphasis on infrastructure networks such as streets, roads and transportation networks (subway, train). The author presents the mathematical tools needed to characterize these structures and how they evolve in time. The book discusses the most important empirical results and stylized facts, and will present the most important models of spatial networks. The target audience primarily comprises research scientists interested in this rapidly evolving and highly interdisciplinary field, but the book may also be beneficial for graduate students interested in large networks.
Author: Bo'az Klartag
Publisher: Springer Nature
Published: 2020-06-20
Total Pages: 346
ISBN-13: 3030360202
DOWNLOAD EBOOKContinuing the theme of the previous volumes, these seminar notes reflect general trends in the study of Geometric Aspects of Functional Analysis, understood in a broad sense. Two classical topics represented are the Concentration of Measure Phenomenon in the Local Theory of Banach Spaces, which has recently had triumphs in Random Matrix Theory, and the Central Limit Theorem, one of the earliest examples of regularity and order in high dimensions. Central to the text is the study of the Poincaré and log-Sobolev functional inequalities, their reverses, and other inequalities, in which a crucial role is often played by convexity assumptions such as Log-Concavity. The concept and properties of Entropy form an important subject, with Bourgain's slicing problem and its variants drawing much attention. Constructions related to Convexity Theory are proposed and revisited, as well as inequalities that go beyond the Brunn–Minkowski theory. One of the major current research directions addressed is the identification of lower-dimensional structures with remarkable properties in rather arbitrary high-dimensional objects. In addition to functional analytic results, connections to Computer Science and to Differential Geometry are also discussed.
Author: Wolfgang Woess
Publisher: Cambridge University Press
Published: 2000-02-13
Total Pages: 350
ISBN-13: 0521552923
DOWNLOAD EBOOKThe main theme of this book is the interplay between the behaviour of a class of stochastic processes (random walks) and discrete structure theory. The author considers Markov chains whose state space is equipped with the structure of an infinite, locally finite graph, or as a particular case, of a finitely generated group. The transition probabilities are assumed to be adapted to the underlying structure in some way that must be specified precisely in each case. From the probabilistic viewpoint, the question is what impact the particular type of structure has on various aspects of the behaviour of the random walk. Vice-versa, random walks may also be seen as useful tools for classifying, or at least describing the structure of graphs and groups. Links with spectral theory and discrete potential theory are also discussed. This book will be essential reading for all researchers working in stochastic process and related topics.
Author: Geoffrey Grimmett
Publisher: Cambridge University Press
Published: 2018-01-25
Total Pages: 279
ISBN-13: 1108542999
DOWNLOAD EBOOKThis 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. This new edition features accounts of major recent progress, including the exact value of the connective constant of the hexagonal lattice, and the critical point of the random-cluster model on the square lattice. The choice of topics is strongly motivated by modern applications, and focuses on areas that merit further research. 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.
