Remco van der Hofstad - Böcker

This rigorous introduction to network science presents random graphs as models for real-world networks. Such networks have distinctive empirical properties and a wealth of new models have emerged to capture them. Classroom tested for over ten years, this text places recent advances in a unified framework to enable systematic study. Designed for a master's-level course, where students may only have a basic background in probability, the text covers such important preliminaries as convergence of random variables, probabilistic bounds, coupling, martingales, and branching processes. Building on this base - and motivated by many examples of real-world networks, including the Internet, collaboration networks, and the World Wide Web - it focuses on several important models for complex networks and investigates key properties, such as the connectivity of nodes. Numerous exercises allow students to develop intuition and experience in working with the models.

Complex networks are key to describing the connected nature of the society that we live in. This book, the second of two volumes, describes the local structure of random graph models for real-world networks and determines when these models have a giant component and when they are small-, and ultra-small, worlds. This is the first book to cover the theory and implications of local convergence, a crucial technique in the analysis of sparse random graphs. Suitable as a resource for researchers and PhD-level courses, it uses examples of real-world networks, such as the Internet and citation networks, as motivation for the models that are discussed, and includes exercises at the end of each chapter to develop intuition. The book closes with an extensive discussion of related models and problems that demonstratemodern approaches to network theory, such as community structure and directed models.

The text may also be useful in advanced courses and seminars, as well as for reference and individual study. Part I, consisting of 3 chapters, presents a general introduction to percolation, stating the main results, defining the central objects, and proving its main properties.