Название: Random Graphs and Networks: A First Course
Автор: Alan Frieze, Michal Karonski
Издательство: Cambridge University Press
Год: 2023
Страниц: 233
Язык: английский
Формат: pdf (true)
Размер: 12.6 MB

Networks surround us, from social networks to protein–protein interaction networks within the cells of our bodies. The theory of random graphs provides a necessary framework for understanding their structure and development. This text provides an accessible introduction to this rapidly expanding subject. It covers all the basic features of random graphs – component structure, matchings and Hamilton cycles, connectivity and chromatic number – before discussing models of real-world networks, including intersection graphs, preferential attachment graphs and small-world models. Based on the authors' own teaching experience, it can be used as a textbook for a one-semester course on random graphs and networks at advanced undergraduate or graduate level. The text includes numerous exercises, with a particular focus on developing students' skills in asymptotic analysis. More challenging problems are accompanied by hints or suggestions for further reading.

In the past 30 years, random graphs, and more generally, random discrete structures, have become the focus of research of large groups of mathematicians, computer scientists, physicists, and social scientists. All these groups contribute to this area differently: mathematicians try to develop models and study their properties in a formal way while physicists and computer scientists apply those models to study real-life networks and systems and, through simulations, to develop interesting and fruitful intuitions as to how to bring mathematical models closer to reality. The abrupt development of study in the theory and applications of random graphs and networks is in a large part due to the Internet and WWW revolution, which exploded at the end of the twentieth century and, in consequence, the worldwide popularity of different social media such as Facebook or Twitter, just to name the most influential ones. Our textbook aims to give a gentle introduction to the mathematical foundations of random graphs and to build a platform to understand the nature of real-life networks.

Random graphs, at the beginning of the twenty-first century is recognized as a young but quickly maturing area of mathematics, with strong connections to Computer Science and physics. Computer Science exploits various ways of applying probabilistic concepts in the analysis of algorithms and in the construction of randomized algorithms. A common ground of random graphs and physics is particularly visible in the analysis of phase transition phenomena and in percolation. In the past 20 years, one can observe a veritable tsunami of publications dealing with various models of random graphs introduced to analyze very large real-world networks: WWW linkage, social, neural, communication, information, and transportation networks, as well as a wide range of large-scale systems.

Nowadays, research in random graphs and networks is thriving, and the subject is included in the curriculum of many mathematics and Computer Science departments across the world. Our textbook should help readers not only gain mathematical knowledge about the basic results of the theory of random graphs but also allow them to better understand how to model and explore real-world networks.