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Random Networks for Communication: From Statistical Physics to Information Systems (Cambridge Series in Statistical and Probabilistic Mathematics) ReviewThis book is about random network models and how local connectivity properties give rise to large scale properties that emerge as the network grows in size. The study of emergent properties of evolving, random structures, with most prominent that of random graphs, has been the focus of many researchers coming from widely diverse disciplines that include physics, mathematics, computer science as well as social sciences. The core idea behind all these studies is that a simple local connectivity rule that defines how two elements of the structure interact with each other can give rise to more complex connectivity properties that hold globally on the structure and manifest themselves (emerge) as the structure's size increases. Moreover, it appears that there is some critical point, or threshold value, for the local connectivity rule such that the properties emerge suddenly from non-existent to existent, when the rule crosses this point. These emergent properties are, then, aptly called threshold properties. The book is focused on the study of two elementary, but rich in properties and modelling power, combinatorial structures: the random tree and the random grid. In the random tree model, we have a tree composed of an infinite number of vertices each having k children, with k > 0. Also, a probability value p is fixed and then each edge of the tree appears in the tree, independently of the others, with probability p. In the random grid model, the nodes are positioned on the points of the two-dimensional integer grid. These models are in contrast with the classical pioneering Erdos-Renyi random graph models in which that in these models adjacency between two vertices is defined by physical proximity while in the latter adjacency can be potentially appear between any pair of vertices.In addition to the theoretical exposition, each chapter is aptly complemented by exercises that, most often, encourage the reader to finish sketched or incomplete proofs given in the text. The exercises are carefully designed so as to be tractable, with some effort, and to increase, at the same time, the intuition and understanding of the reader of the similarities and differences between the various random graph models. Also, in the end of the book, the authors provide an Appendix with some useful background material on basic probability theory.
In summary, this book is a clear, readable and highly intuitive introduction to the properties and applications of random network models that, also, provides all the rigorous details or invites the reader to fill them in, in the exercises section. The models tackled by the authors are characterized by the important property that the geometry of the nodes has a pivotal role in the formation of the network connections, as opposed to classical Erdos-Renyi random graph models in which there is no notion of geometry and edges can be inserted (with some probability) between any pair of nodes. The balance between intuition and rigor is ideal, in my opinion, and reading the book is an enjoyable and highly rewarding endeavour. I believe this book will be useful to physicists, mathematicians, and computer scientists alike that look at random graph models where point locations affects the shape and properties of the resulting network: physicists will acquaint themselves with complex networks having rich modelling capabilities (e.g. models for random interaction particle systems such as spin glasses), mathematicians may discover connections of the networks with formal systems (much like the connection of the classical Erdos-Renyi random graph properties with first and second order logic), and computer scientists will greatly appreciate the applicability of the theory given in the book to
the study of realistic, ad-hoc mobile networks in which network node connections change rapidly and unpredictably as a function of the geometry of the current node positions.
Random Networks for Communication: From Statistical Physics to Information Systems (Cambridge Series in Statistical and Probabilistic Mathematics) Overview
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