Random graphs, also known as complex networks, model the structure of a population. The spread of opinions and diseases on these structures is much different from the usual ODE models. This book will be of interest to mathematicians, computer scientists, epidemiologists, social scientists, and physicists who study these systems.
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.
Recenzijas
'This fully revised book showcases the enormous recent progress made in the field of random graphs and dynamics on them. The chosen topics, including small-world properties, random walks, and interacting particle systems, are carefully picked and well aligned. The author gives a high-level explanation of the proofs of their main results. Omitting the full proofs gives the reader insight in a wide range of topics and the tools of the trade for them, while keeping the book relatively short. Thus, it is an excellent starting point to this exciting field, with dozens of pointers to the literature for more details.' Remco van der Hofstad, Eindhoven University of Technology 'Random graph theory (preferential attachment graphs, small-world networks, configuration model, etc.) and interacting particle systems (contact process, voter model, etc.) are currently two of the most important branches of probability theory in terms of mathematics and their applications in epidemiology and sociology. Durrett's book Dynamics on Graphs covers both topics in parallel before merging them in an elegant way along with a rigorous mathematical treatment. The book concludes with the very hot topic of adaptive networks that combine dynamics on the graph and dynamics of the graph, opening the door to future challenging research. Great job!' Nicolas Lanchier, Arizona State University
Papildus informācija
The book concentrates on dynamics on graphs important for applications to random walks, epidemics, the contact process, and voter models.
Preface; Notation;
1. ErdsRenyi random graphs;
2. General degree
distributions;
3. Inhomogeneous random graphs;
4. Epidemics;
5. Contact
process;
6. Random walks, mixing times;
7. Voter models, coalescing RWs;
8.
Coevolving systems; Appendix. Large deviations; Books and Long Surveys; Index.
Rick Durrett is James B. Duke Emeritus Professor of Mathematics at Duke University. He received his Ph.D. in Operations Research from Stanford University in 1976. After nine years at the University of California, Los Angeles and twenty-five at Cornell University, he moved to Duke University in 2010, where he worked until he retired in 2023. Durrett is the author of eight books and more than 240 journal articles on a wide variety of topics in probability theory and its applications to ecology, genetics, cancer, and epidemiology. He has supervised fifty-one Ph.D. students. He has been a member of the National Academy of Sciences since 2007.
