The book gives an accessible account of modern probabilistic methods for analyzing combinatorial structures and algorithms. It will be an useful guide for graduate students and researchers.Special features included: a simple treatment of Talagrand's inequalities and their applications; an overview and many carefully worked out examples of the probabilistic analysis of combinatorial algorithms; a discussion of the "exact simulation" algorithm (in the context of Markov Chain Monte Carlo Methods); a general method for finding asymptotically optimal or near optimal graph colouring, showing how the probabilistic method may be fine-tuned to exploit the structure of the underlying graph; a succinct treatment of randomized algorithms and derandomization techniques.

Topology-based methods are of increasing importance in the analysis and visualization of dataset from a wide variety of scientific domains such as biology, physics, engineering, and medicine. Current challenges of topology-based techniques include the management of time-dependent data, the representation large and complex datasets, the characterization of noise and uncertainty, the effective integration of numerical methods with robust combinatorial algorithms, etc. (see also below for a list of selected issues). While there is an increasing number of high-quality publications in this field, many fundamental questions remain unsolved. New focused efforts are needed in a variety of techniques ranging from the theoretical foundations of topological models, algorithmic issues related to the representation power of computer-based implementations as well as their computational efficiency, user interfaces for presentation of quantitative topological information, and the development of new techniques for systematic mapping of science problems in topological constructs that can be solved computationally. In this forum the editors have brought together the most prominent and best recognized researchers in the field of topology-based data analysis and visualization for a joint discussion and scientific exchange of the latest results in the field. The 2009 workshop in Snowbird, Utah, follows the two successful workshops in 2005 (Budmerice, Slovakia) and 2007 (Leipzig, Germany).

Algorithmic graph theory has been expanding at an extremely rapid rate since the middle of the twentieth century, in parallel with the growth of computer science and the accompanying utilization of computers, where efficient algorithms have been a prime goal. This book presents material on developments on graph algorithms and related concepts that will be of value to both mathematicians and computer scientists, at a level suitable for graduate students, researchers and instructors. The fifteen expository chapters, written by acknowledged international experts on their subjects, focus on the application of algorithms to solve particular problems. All chapters were carefully edited to enhance readability and standardize the chapter structure as well as the terminology and notation. The editors provide basic background material in graph theory, and a chapter written by the book's Academic Consultant, Martin Charles Golumbic (University of Haifa, Israel), provides background material on algorithms as connected with graph theory.

This book uses rigorous mathematical analysis to advance opinion dynamics models for social networks in three major directions. First, a novel model is proposed to capture how a discrepancy between an individual's private and expressed opinions can develop due to social pressures that arise in group situations or through extremists deliberately shaping public opinion. Detailed theoretical analysis of the final opinion distribution is followed by use of the model to study Asch's seminal experiments on conformity, and the phenomenon of pluralistic ignorance. Second, the DeGroot-Friedkin model for evolution of an individual's social power (self-confidence) is developed in a number of directions. The key result establishes that an individual's initial social power is forgotten exponentially fast, even when the network changes over time; eventually, an individual's social power depends only on the (changing) network structure. Last, a model for the simultaneous discussion of multiple logically interdependent topics is proposed. To ensure that a consensus across the opinions of all individuals is achieved, it turns out that the interpersonal interactions must be weaker than an individual's introspective cognitive process for establishing logical consistency among the topics. Otherwise, the individual may experience cognitive overload and the opinion system becomes unstable. Conclusions of interest to control engineers, social scientists, and researchers from other relevant disciplines are discussed throughout the thesis with support from both social science and control literature.

The Complexity Theory Companion is an accessible, algorithmically oriented, research-centered, up-to-date guide to some of the most interesting techniques of complexity theory. The book's thesis is that simple algorithms are at the heart of complexity theory. From the tree-pruning and interval-pruning algorithms that shape the first chapter to the query simulation procedures that dominate the last chapter, the central proof methods of the book are algorithmic. And to more clearly highlight the role of algorithmic techniques in complexity theory, the book is - unlike other texts on complexity - organized by technique rather than by topic. Each chapter of this book focuses on one technique: what it is, and what results and applications it yields. This textbook was developed at the University of Rochester in courses given to graduate students and advanced undergraduates. Researchers also will find this book a valuable source of reference due to the comprehensive bibliography of close to five hundred entries, the thirty-five page subject index, and the appendices giving overviews of complexity classes and reductions.

Graph Theory and Its Applications, Third Edition is the latest edition of the international, bestselling textbook for undergraduate courses in graph theory, yet it is expansive enough to be used for graduate courses as well. The textbook takes a comprehensive, accessible approach to graph theory, integrating careful exposition of classical developments with emerging methods, models, and practical needs. The authors' unparalleled treatment is an ideal text for a two-semester course and a variety of one-semester classes, from an introductory one-semester course to courses slanted toward classical graph theory, operations research, data structures and algorithms, or algebra and topology. Features of the Third Edition Expanded coverage on several topics (e.g., applications of graph coloring and tree-decompositions) Provides better coverage of algorithms and algebraic and topological graph theory than any other text Incorporates several levels of carefully designed exercises that promote student retention and develop and sharpen problem-solving skills Includes supplementary exercises to develop problem-solving skills, solutions and hints, and a detailed appendix, which reviews the textbook's topics About the Authors Jonathan L. Gross is a professor of computer science at Columbia University. His research interests include topology and graph theory. Jay Yellen is a professor of mathematics at Rollins College. His current areas of research include graph theory, combinatorics, and algorithms. Mark Anderson is also a mathematics professor at Rollins College. His research interest in graph theory centers on the topological or algebraic side.

Through three editions, Cryptography: Theory and Practice, has been embraced by instructors and students alike. It offers a comprehensive primer for the subject's fundamentals while presenting the most current advances in cryptography. The authors offer comprehensive, in-depth treatment of the methods and protocols that are vital to safeguarding the seemingly infinite and increasing amount of information circulating around the world. Key Features of the Fourth Edition: New chapter on the exciting, emerging new area of post-quantum cryptography (Chapter 9). New high-level, nontechnical overview of the goals and tools of cryptography (Chapter 1). New mathematical appendix that summarizes definitions and main results on number theory and algebra (Appendix A). An expanded treatment of stream ciphers, including common design techniques along with coverage of Trivium. Interesting attacks on cryptosystems, including: padding oracle attack correlation attacks and algebraic attacks on stream ciphers attack on the DUAL-EC random bit generator that makes use of a trapdoor. A treatment of the sponge construction for hash functions and its use in the new SHA-3 hash standard. Methods of key distribution in sensor networks. The basics of visual cryptography, allowing a secure method to split a secret visual message into pieces (shares) that can later be combined to reconstruct the secret. The fundamental techniques cryptocurrencies, as used in Bitcoin and blockchain. The basics of the new methods employed in messaging protocols such as Signal, including deniability and Diffie-Hellman key ratcheting.

The main focus of this book is a pair of cooperative control problems: consensus and cooperative output regulation. Its emphasis is on complex multi-agent systems characterized by strong nonlinearity, large uncertainty, heterogeneity, external disturbances and jointly connected switching communication topologies. The cooperative output regulation problem is a generalization of the classical output regulation problem to multi-agent systems and it offers a general framework for handling a variety of cooperative control problems such as consensus, formation, tracking and disturbance rejection. The book strikes a balance between rigorous mathematical proof and engineering practicality. Every design method is systematically presented together with illustrative examples and all the designs are validated by computer simulation. The methods presented are applied to several practical problems, among them the leader-following consensus problem of multiple Euler-Lagrange systems, attitude synchronization of multiple rigid-body systems, and power regulation of microgrids. The book gives a detailed exposition of two approaches to the design of distributed control laws for complex multi-agent systems-the distributed-observer and distributed-internal-model approaches. Mastering both will enhance a reader's ability to deal with a variety of complex real-world problems. Cooperative Control of Multi-agent Systems can be used as a textbook for graduate students in engineering, sciences, and mathematics, and can also serve as a reference book for practitioners and theorists in both industry and academia. Some knowledge of the fundamentals of linear algebra, calculus, and linear systems are needed to gain maximum benefit from this book. Advances in Industrial Control reports and encourages the transfer of technology in control engineering. The rapid development of control technology has an impact on all areas of the control discipline. The series offers an opportunity for researchers to present an extended exposition of new work in all aspects of industrial control.

Focused on the mathematical foundations of social media analysis, Graph-Based Social Media Analysis provides a comprehensive introduction to the use of graph analysis in the study of social and digital media. It addresses an important scientific and technological challenge, namely the confluence of graph analysis and network theory with linear algebra, digital media, machine learning, big data analysis, and signal processing. Supplying an overview of graph-based social media analysis, the book provides readers with a clear understanding of social media structure. It uses graph theory, particularly the algebraic description and analysis of graphs, in social media studies. The book emphasizes the big data aspects of social and digital media. It presents various approaches to storing vast amounts of data online and retrieving that data in real-time. It demystifies complex social media phenomena, such as information diffusion, marketing and recommendation systems in social media, and evolving systems. It also covers emerging trends, such as big data analysis and social media evolution. Describing how to conduct proper analysis of the social and digital media markets, the book provides insights into processing, storing, and visualizing big social media data and social graphs. It includes coverage of graphs in social and digital media, graph and hyper-graph fundamentals, mathematical foundations coming from linear algebra, algebraic graph analysis, graph clustering, community detection, graph matching, web search based on ranking, label propagation and diffusion in social media, graph-based pattern recognition and machine learning, graph-based pattern classification and dimensionality reduction, and much more. This book is an ideal reference for scientists and engineers working in social media and digital media production and distribution. It is also suitable for use as a textbook in undergraduate or graduate courses on digital media, social media, or social networks.

This book gives a comprehensive view of graph theory in informational retrieval (IR) and natural language processing(NLP). This book provides number of graph techniques for IR and NLP applications with examples. It also provides understanding of graph theory basics, graph algorithms and networks using graph. The book is divided into three parts and contains nine chapters. The first part gives graph theory basics and graph networks, and the second part provides basics of IR with graph-based information retrieval. The third part covers IR and NLP recent and emerging applications with case studies using graph theory. This book is unique in its way as it provides a strong foundation to a beginner in applying mathematical structure graph for IR and NLP applications. All technical details that include tools and technologies used for graph algorithms and implementation in Information Retrieval and Natural Language Processing with its future scope are explained in a clear and organized format.

This volume contains the accounts of papers delivered at the Nato Advanced Study Institute on Finite and Infinite Combinatorics in Sets and Logic held at the Banff Centre, Alberta, Canada from April 21 to May 4, 1991. As the title suggests the meeting brought together workers interested in the interplay between finite and infinite combinatorics, set theory, graph theory and logic. It used to be that infinite set theory, finite combinatorics and logic could be viewed as quite separate and independent subjects. But more and more those disciplines grow together and become interdependent of each other with ever more problems and results appearing which concern all of those disciplines. I appreciate the financial support which was provided by the N. A. T. O. Advanced Study Institute programme, the Natural Sciences and Engineering Research Council of Canada and the Department of Mathematics and Statistics of the University of Calgary. 11l'te meeting on Finite and Infinite Combinatorics in Sets and Logic followed two other meetings on discrete mathematics held in Banff, the Symposium on Ordered Sets in 1981 and the Symposium on Graphs and Order in 1984. The growing inter-relation between the different areas in discrete mathematics is maybe best illustrated by the fact that many of the participants who were present at the previous meetings also attended this meeting on Finite and Infinite Combinatorics in Sets and Logic.

The series is aimed specifically at publishing peer reviewed reviews and contributions presented at workshops and conferences. Each volume is associated with a particular conference, symposium or workshop. These events cover various topics within pure and applied mathematics and provide up-to-date coverage of new developments, methods and applications.

Simulating for a crisis is far more than creating a simulation of a crisis situation. In order for a simulation to be useful during a crisis, it should be created within the space of a few days to allow decision makers to use it as quickly as possible. Furthermore, during a crisis the aim is not to optimize just one factor, but to balance various, interdependent aspects of life. In the COVID-19 crisis, decisions had to be made concerning e.g. whether to close schools and restaurants, and the (economic) consequences of a 3 or 4-week lock-down had to be considered. As such, rather than one simulation focusing on a very limited aspect, a framework allowing the simulation of several different scenarios focusing on different aspects of the crisis was required. Moreover, the results of the simulations needed to be easily understandable and explainable: if a simulation indicates that closing schools has no effect, this can only be used if the decision makers can explain why this is the case. This book describes how a simulation framework was created for the COVID-19 crisis, and demonstrates how it was used to simulate a wide range of scenarios that were relevant for decision makers at the time. It also discusses the usefulness of the approach, and explains the decisions that had to be made along the way as well as the trade-offs. Lastly, the book examines the lessons learned and the directions for the further development of social simulation frameworks to make them better suited to crisis situations, and to foster a more resilient society.

In many applications of graph theory, graphs are regarded as geometric objects drawn in the plane or in some other surface. The traditional methods of "abstract" graph theory are often incapable of providing satisfactory answers to questions arising in such applications. In the past couple of decades, many powerful new combinatorial and topological techniques have been developed to tackle these problems. Today geometric graph theory is a burgeoning field with many striking results and appealing open questions. This contributed volume contains thirty original survey and research papers on important recent developments in geometric graph theory. The contributions were thoroughly reviewed and written by excellent researchers in this field.

This textbook thoroughly outlines combinatorial algorithms for generation, enumeration, and search. Topics include backtracking and heuristic search methods applied to various combinatorial structures, such as: Combinations Permutations Graphs Designs Many classical areas are covered as well as new research topics not included in most existing texts, such as: Group algorithms Graph isomorphism Hill-climbing Heuristic search algorithms This work serves as an exceptional textbook for a modern course in combinatorial algorithms, providing a unified and focused collection of recent topics of interest in the area. The authors, synthesizing material that can only be found scattered through many different sources, introduce the most important combinatorial algorithmic techniques - thus creating an accessible, comprehensive text that students of mathematics, electrical engineering, and computer science can understand without needing a prior course on combinatorics.

This book presents interdisciplinary, cutting-edge and creative applications of graph theory and modeling in science, technology, architecture and art. Topics are divided into three parts: the first one examines mechanical problems related to gears, planetary gears and engineering installations; the second one explores graph-based methods applied to medical analyses as well as biological and chemical modeling; and the third part includes various topics e.g. drama analysis, aiding of design activities and network visualisation. The authors represent several countries in Europe and America, and their contributions show how different, useful and fruitful the utilization of graphs in modelling of engineering systems can be. The book has been designed to serve readers interested in the subject of graph modelling and those with expertise in related areas, as well as members of the worldwide community of graph modelers.

This text is an introduction to harmonic analysis on symmetric spaces, focusing on advanced topics such as higher rank spaces, positive definite matrix space and generalizations. It is intended for beginning graduate students in mathematics or researchers in physics or engineering. As with the introductory book entitled "Harmonic Analysis on Symmetric Spaces - Euclidean Space, the Sphere, and the Poincare Upper Half Plane, the style is informal with an emphasis on motivation, concrete examples, history, and applications. The symmetric spaces considered here are quotients X=G/K, where G is a non-compact real Lie group, such as the general linear group GL(n,P) of all n x n non-singular real matrices, and K=O(n), the maximal compact subgroup of orthogonal matrices. Other examples are Siegel's upper half "plane" and the quaternionic upper half "plane". In the case of the general linear group, one can identify X with the space Pn of n x n positive definite symmetric matrices. Many corrections and updates have been incorporated in this new edition. Updates include discussions of random matrix theory and quantum chaos, as well as recent research on modular forms and their corresponding L-functions in higher rank. Many applications have been added, such as the solution of the heat equation on Pn, the central limit theorem of Donald St. P. Richards for Pn, results on densest lattice packing of spheres in Euclidean space, and GL(n)-analogs of the Weyl law for eigenvalues of the Laplacian in plane domains. Topics featured throughout the text include inversion formulas for Fourier transforms, central limit theorems, fundamental domains in X for discrete groups (such as the modular group GL(n,Z) of n x n matrices with integer entries and determinant +/-1), connections with the problem of finding densest lattice packings of spheres in Euclidean space, automorphic forms, Hecke operators, L-functions, and the Selberg trace formula and its applications in spectral theory as well as number theory.

The contributions in this volume cover a broad range of topics including maximum cliques, graph coloring, data mining, brain networks, Steiner forest, logistic and supply chain networks. Network algorithms and their applications to market graphs, manufacturing problems, internet networks and social networks are highlighted. The "Fourth International Conference in Network Analysis," held at the Higher School of Economics, Nizhny Novgorod in May 2014, initiated joint research between scientists, engineers and researchers from academia, industry and government; the major results of conference participants have been reviewed and collected in this Work. Researchers and students in mathematics, economics, statistics, computer science and engineering will find this collection a valuable resource filled with the latest research in network analysis.

Steiner's Problem concerns finding a shortest interconnecting network for a finite set of points in a metric space. A solution must be a tree, which is called a Steiner Minimal Tree (SMT), and may contain vertices different from the points which are to be connected. Steiner's Problem is one of the most famous combinatorial-geometrical problems, but unfortunately it is very difficult in terms of combinatorial structure as well as computational complexity. However, if only a Minimum Spanning Tree (MST) without additional vertices in the interconnecting network is sought, then it is simple to solve. So it is of interest to know what the error is if an MST is constructed instead of an SMT. The worst case for this ratio running over all finite sets is called the Steiner ratio of the space. The book concentrates on investigating the Steiner ratio. The goal is to determine, or at least estimate, the Steiner ratio for many different metric spaces. The author shows that the description of the Steiner ratio contains many questions from geometry, optimization, and graph theory. Audience: Researchers in network design, applied optimization, and design of algorithms.

Computational geometry as an area of research in its own right emerged in the early seventies of this century. Right from the beginning, it was obvious that strong connections of various kinds exist to questions studied in the considerably older field of combinatorial geometry. For example, the combinatorial structure of a geometric problem usually decides which algorithmic method solves the problem most efficiently. Furthermore, the analysis of an algorithm often requires a great deal of combinatorial knowledge. As it turns out, however, the connection between the two research areas commonly referred to as computa tional geometry and combinatorial geometry is not as lop-sided as it appears. Indeed, the interest in computational issues in geometry gives a new and con structive direction to the combinatorial study of geometry. It is the intention of this book to demonstrate that computational and com binatorial investigations in geometry are doomed to profit from each other. To reach this goal, I designed this book to consist of three parts, acorn binatorial part, a computational part, and one that presents applications of the results of the first two parts. The choice of the topics covered in this book was guided by my attempt to describe the most fundamental algorithms in computational geometry that have an interesting combinatorial structure. In this early stage geometric transforms played an important role as they reveal connections between seemingly unrelated problems and thus help to structure the field."

This book gives the complete classification of Moufang polygons, starting from first principles. In particular, it may serve as an introduction to the various important algebraic concepts which arise in this classification including alternative division rings, quadratic Jordan division algebras of degree three, pseudo-quadratic forms, BN-pairs and norm splittings of quadratic forms. This book also contains a new proof of the classification of irreducible spherical buildings of rank at least three based on the observation that all the irreducible rank two residues of such a building are Moufang polygons. In an appendix, the connection between spherical buildings and algebraic groups is recalled and used to describe an alternative existence proof for certain Moufang polygons.

This book develops survey data analysis tools in Python, to create and analyze cross-tab tables and data visuals, weight data, perform hypothesis tests, and handle special survey questions such as Check-all-that-Apply. In addition, the basics of Bayesian data analysis and its Python implementation are presented. Since surveys are widely used as the primary method to collect data, and ultimately information, on attitudes, interests, and opinions of customers and constituents, these tools are vital for private or public sector policy decisions. As a compact volume, this book uses case studies to illustrate methods of analysis essential for those who work with survey data in either sector. It focuses on two overarching objectives: Demonstrate how to extract actionable, insightful, and useful information from survey data; and Introduce Python and Pandas for analyzing survey data.

This book surveys the state-of-the-art in the theory of combinatorial games, that is games not involving chance or hidden information. Enthusiasts will find a wide variety of exciting topics, from a trailblazing presentation of scoring to solutions of three piece ending positions of bidding chess. Theories and techniques in many subfields are covered, such as universality, Wythoff Nim variations, misere play, partizan bidding (a.k.a. Richman games), loopy games, and the algebra of placement games. Also included are an updated list of unsolved problems, extremely efficient algorithms for taking and breaking games, a historical exposition of binary numbers and games by David Singmaster, chromatic Nim variations, renormalization for combinatorial games, and a survey of temperature theory by Elwyn Berlekamp, one of the founders of the field. The volume was initiated at the Combinatorial Game Theory Workshop, January 2011, held at the Banff International Research Station.

The Workshop for Women in Graph Theory and Applications was held at the Institute for Mathematics and Its Applications (University of Minnesota, Minneapolis) on August 19-23, 2019. During this five-day workshop, 42 participants performed collaborative research, in six teams, each focused on open problems in different areas of graph theory and its applications. The research work of each team was led by two experts in the corresponding area, who prior to the workshop, carefully selected relevant and meaningful open problems that would yield high-quality research and results of strong impact. As a result, all six teams have made significant contributions to several open problems in their respective areas. The workshop led to the creation of the Women in Graph Theory and Applications Research Collaboration Network, which provided the framework to continue collaborating and to produce this volume. This book contains six chapters, each of them on one of the different areas of research at the Workshop for Women in Graph Theory and Applications, and written by participants of each team.

This book is intended to provide graduate students and researchers in graph theory with an overview of the elementary methods of graph Ramsey theory. It is especially targeted towards graduate students in extremal graph theory, graph Ramsey theory, and related fields, as the included contents allow the text to be used in seminars. It is structured in thirteen chapters which are application-focused and largely independent, enabling readers to target specific topics and information to focus their study. The first chapter includes a true beginner's overview of elementary examples in graph Ramsey theory mainly using combinatorial methods. The following chapters progress through topics including the probabilistic methods, algebraic construction, regularity method, but that's not all. Many related interesting topics are also included in this book, such as the disproof for a conjecture of Borsuk on geometry, intersecting hypergraphs, Turan numbers and communication channels, etc.