What is this book about, and who is it written for? To start with the first question, this book
introduces a subject placed at the interface between mathematics, physics, and information
theory of systems. In doing so, it is not intended to be a comprehensive monograph and
collect all the mathematical results available in the literature, but rather pursues the more
ambitious goal of laying the foundations. We have tried to give emphasis to the relevant
mathematical techniques that are the essential ingredients for anybody interested in the
field of random networks. Dynamic coupling, renormalisation, ergodicity and deviations
from the mean, correlation inequalities, Poisson approximation, as well as some other
tricks and constructions that often arise in the proofs are not only applied, but also
discussed with the objective of clarifying the philosophy behind their arguments. We
have also tried to make available to a larger community the main mathematical results
on random networks, and to place them into a new communication theory framework,
trying not to sacrifice mathematical rigour. As a result, the choice of the topics was
influenced by personal taste, by the willingness to keep the flow consistent, and by
the desire to present a modern, communication-theoretic view of a topic that originated
some fifty years ago and that has had an incredible impact in mathematics and statistical
physics since then. Sometimes this has come at the price of sacrificing the presentation
of results that either did not fit well in what we thought was the ideal flow of the
book, or that could be obtained using the same basic ideas, but at the expense of
highly technical complications. One important topic that the reader will find missing,
for example, is a complete treatment of the classic Erdös–Rényi model of random graphs
and of its more recent extensions, including preferential attachment models used to
describe properties of the Internet. Indeed, we felt that these models, lacking a geometric
component, did not fit well in our framework and the reader is referred to the recent
account of Durrett (2007) for a rigorous treatment of preferential attachment models. Other
omissions are certainly present, and hopefully similarly justified. We also refer to the
monographs by Bollobás (2001), Bollobás and Riordan (2006), Grimmett (1999), Meester
and Roy (1996), and Penrose (2003), for a compendium of additional mathematical results.