Random Graph Dynamics

Random Graph Dynamics
Author: Rick Durrett
Publisher: Cambridge University Press
Total Pages: 203
Release: 2010-05-31
Genre: Mathematics
ISBN: 1139460889

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.

Dynamical Systems, Graphs, and Algorithms

Dynamical Systems, Graphs, and Algorithms
Author: George Osipenko
Publisher: Springer
Total Pages: 286
Release: 2006-10-28
Genre: Mathematics
ISBN: 3540355952

This book describes a family of algorithms for studying the global structure of systems. By a finite covering of the phase space we construct a directed graph with vertices corresponding to cells of the covering and edges corresponding to admissible transitions. The method is used, among other things, to locate the periodic orbits and the chain recurrent set, to construct the attractors and their basins, to estimate the entropy, and more.

Graph Dynamics

Graph Dynamics
Author: Erich Prisner
Publisher: CRC Press
Total Pages: 252
Release: 1995-11-22
Genre: Mathematics
ISBN: 9780582286962

Study of "graph operators" or "graph-valued functions" such as the line graph, the clique graph, the complement, and powers, raises several immediate questions: Which graphs are fixed under the operator? Which graphs appear as images of graphs? What happens if the operator is iterated? Over the last 30 years these questions have been answered and methods developed for particular operators in literally hundreds of papers on the subject. Nowhere, however, could one find a comprehensive treatment-a unification of terminology, questions, and methods. Graph Dynamics provides that comprehensive treatment. Its purpose is threefold: it serves as an introductory textbook on the topic, offers an encyclopedic survey of the literature, and reports recent research-both new tools and results on concrete operators. Part I explicitly presents graph dynamics general theory, stating general principles illustrated by application to graph operators. Part 2 addresses the operators themselves. It lists all known graph operators grouped together in families and recounts, with complete references, all that is known about the dynamical behavior of these concrete operators. Graph Dynamics is the book you need if you are looking for information on a particular operator, need a text for advanced students, or want to review collected research results presented with a common terminology. It is clearly an essential resource for anyone working in or studying algebra, combinatorics, or graph theory.

System Dynamics and Control with Bond Graph Modeling

System Dynamics and Control with Bond Graph Modeling
Author: Javier Kypuros
Publisher: CRC Press
Total Pages: 517
Release: 2013-04-25
Genre: Technology & Engineering
ISBN: 1466560762

Written by a professor with extensive teaching experience, System Dynamics and Control with Bond Graph Modeling treats system dynamics from a bond graph perspective. Using an approach that combines bond graph concepts and traditional approaches, the author presents an integrated approach to system dynamics and automatic controls. The textbook guide

Dynamics on Graphs

Dynamics on Graphs
Author: Rick Durrett
Publisher: Cambridge University Press
Total Pages: 0
Release: 2024-10-31
Genre: Mathematics
ISBN: 9781009521451

This extensive revision of the 2007 book 'Random Graph Dynamics,' covering the current state of mathematical research in the field, is ideal for researchers and graduate students. It considers a small number of types of graphs, primarily the configuration model and inhomogeneous random graphs. However, it investigates a wide variety of dynamics. The author describes results for the convergence to equilibrium for random walks on random graphs as well as topics that have emerged as mature research areas since the publication of the first edition, such as epidemics, the contact process, voter models, and coalescing random walk. Chapter 8 discusses a new challenging and largely uncharted direction: systems in which the graph and the states of their vertices coevolve.

Engineering System Dynamics

Engineering System Dynamics
Author: Forbes T. Brown
Publisher: CRC Press
Total Pages: 1088
Release: 2006-08-15
Genre: Science
ISBN: 9780849396489

For today's students, learning to model the dynamics of complex systems is increasingly important across nearly all engineering disciplines. First published in 2001, Forbes T. Brown's Engineering System Dynamics: A Unified Graph-Centered Approach introduced students to a unique and highly successful approach to modeling system dynamics using bond graphs. Updated with nearly one-third new material, this second edition expands this approach to an even broader range of topics. What's New in the Second Edition? In addition to new material, this edition was restructured to build students' competence in traditional linear mathematical methods before they have gone too far into the modeling that still plays a pivotal role. New topics include magnetic circuits and motors including simulation with magnetic hysteresis; extensive new material on the modeling, analysis, and simulation of distributed-parameter systems; kinetic energy in thermodynamic systems; and Lagrangian and Hamiltonian methods. MATLABĀ® figures prominently in this edition as well, with code available for download from the Internet. This code includes simulations for problems that appear in the later chapters as well as code for selected thermodynamic substances. Using a step-by-step pedagogy accompanied by abundant examples, graphs, illustrations, case studies, guided exercises, and homework problems, Engineering System Dynamics: A Unified Graph-Centered Approach, Second Edition is a text that students will embrace and continue to use well into their careers. While the first half of the book is ideal for junior-level undergraduates, the entire contents are suited for more advanced students.

Applying Graph Theory in Ecological Research

Applying Graph Theory in Ecological Research
Author: Mark R.T. Dale
Publisher: Cambridge University Press
Total Pages: 355
Release: 2017-11-09
Genre: Mathematics
ISBN: 110708931X

This book clearly describes the many applications of graph theory to ecological questions, providing instruction and encouragement to researchers.

Random Graph Dynamics

Random Graph Dynamics
Author: Rick Durrett
Publisher: Cambridge University Press
Total Pages: 222
Release: 2006-10-23
Genre: Mathematics
ISBN: 9780521866569

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 about the same 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. While this literature is extensive, many of the papers are based on simulations and nonrigorous arguments. 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 of this book is the interest in the dynamics of process taking place on the graph in addition to their geometric properties, such as connectedness and diameter.

Modularity and Dynamics on Complex Networks

Modularity and Dynamics on Complex Networks
Author: Renaud Lambiotte
Publisher: Cambridge University Press
Total Pages: 102
Release: 2022-02-03
Genre: Science
ISBN: 1108808654

Complex networks are typically not homogeneous, as they tend to display an array of structures at different scales. A feature that has attracted a lot of research is their modular organisation, i.e., networks may often be considered as being composed of certain building blocks, or modules. In this Element, the authors discuss a number of ways in which this idea of modularity can be conceptualised, focusing specifically on the interplay between modular network structure and dynamics taking place on a network. They discuss, in particular, how modular structure and symmetries may impact on network dynamics and, vice versa, how observations of such dynamics may be used to infer the modular structure. They also revisit several other notions of modularity that have been proposed for complex networks and show how these can be related to and interpreted from the point of view of dynamical processes on networks.

Evolutionary Games and Population Dynamics

Evolutionary Games and Population Dynamics
Author: Josef Hofbauer
Publisher: Cambridge University Press
Total Pages: 356
Release: 1998-05-28
Genre: Mathematics
ISBN: 9780521625708

Every form of behaviour is shaped by trial and error. Such stepwise adaptation can occur through individual learning or through natural selection, the basis of evolution. Since the work of Maynard Smith and others, it has been realised how game theory can model this process. Evolutionary game theory replaces the static solutions of classical game theory by a dynamical approach centred not on the concept of rational players but on the population dynamics of behavioural programmes. In this book the authors investigate the nonlinear dynamics of the self-regulation of social and economic behaviour, and of the closely related interactions between species in ecological communities. Replicator equations describe how successful strategies spread and thereby create new conditions which can alter the basis of their success, i.e. to enable us to understand the strategic and genetic foundations of the endless chronicle of invasions and extinctions which punctuate evolution. In short, evolutionary game theory describes when to escalate a conflict, how to elicit cooperation, why to expect a balance of the sexes, and how to understand natural selection in mathematical terms.