J. F. F. Mendes

We review the recent fast progress in statistical physics of evolving networks. Interest has focused mainly on the structural properties of random complex networks in communications, biology, social sciences and economics. A number of giant artificial networks of such a kind came into existence recently. This opens a wide field for the study of their topology, evolution, and complex processes occurring in them. Such networks possess a rich set of scaling properties. A number of them are scale-free and show striking resilience against random breakdowns. In spite of large sizes of these networks, the distances between most their vertices are short — a feature known as the “smallworld” effect. We discuss how growing networks self-organize into scale-free structures and the role of the mechanism of preferential linking. We consider the topological and structural properties of evolving networks, and percolation in these networks. We present a number of models demonstrating the main features of evolving networks and discuss current approaches for their simulation and analytical study. Applications of the general results to particular networks in Nature are discussed. We demonstrate the generic connections of the network growth processes with the general problems

The combination of the compactness of networks, featuring small diameters, and their complex architectures results in a variety of critical effects dramatically different from those in cooperative systems on lattices. In the last few years, important steps have been made toward understanding the qualitatively new critical phenomena in complex networks. The results, concepts, and methods of this rapidly developing field are reviewed. Two closely related classes of these critical phenomena are considered, namely, structural phase transitions in the network architectures and transitions in cooperative models on networks as substrates. Systems where a network and interacting agents on it influence each other are also discussed. A wide range of critical phenomena in equilibrium and growing networks including the birth of the giant connected component, percolation, $k$-core percolation, phenomena near epidemic thresholds, condensation transitions, critical phenomena in spin models placed on networks, synchronization, and self-organized criticality effects in interacting systems on networks are mentioned. Strong finite-size effects in these systems and open problems and perspectives are also discussed.

Only recently did mankind realise that it resides in a world of networks. The Internet and World Wide Web are changing our life. Our physical existence is based on various biological networks. We have recently learned that the term "network" turns out to be a central notion in our time, and the consequent explosion of interest in networks is a social and cultural phenomenon. The principles of the complex organization and evolution of networks, natural and artificial, are the topic of this book, which is written by physicists and is addressed to all involved researchers and students. The aim of the text is to understand networks and the basic principles of their structural organization and evolution. The ideas are presented in a clear and a pedagogical way, with minimal mathematics, so even students without a deep knowledge of mathematics and statistical physics will be able to rely on this as a reference. Special attention is given to real networks, both natural and artifical. Collected empirical data and numerous real applications of existing theories are discussed in detail, as well as the topical problems of communication networks. Available in OSO: http://www.oxfordscholarship.com/oso/public/content/physics/9780198515906/toc.html

Abstract The aim of this book is to understand networks and the basic principles of their structural organization and evolution. The ideas are presented in a clear and a pedagogical way. Special attention is given to real networks, both natural and artificial, including the Internet and the World Wide Web. Collected empirical data and numerous real applications of existing theories are discussed in detail, as well as the topical problems of communication and other networks.

The model of growing networks with the preferential attachment of new links is generalized to include initial attractiveness of sites. We find the exact form of the stationary distribution of the number of incoming links of sites in the limit of long times, $P(q)$, and the long-time limit of the average connectivity $\overline{q}(s,t)$ of a site $s$ at time $t$ (one site is added per unit of time). At long times, $P(q)\ensuremath{\sim}{q}^{\ensuremath{-}\ensuremath{\gamma}}$ at $q\ensuremath{\rightarrow}\ensuremath{\infty}$ and $\overline{q}(s,t)\ensuremath{\sim}(s/t{)}^{\ensuremath{-}\ensuremath{\beta}}$ at $s/t\ensuremath{\rightarrow}0$, where the exponent $\ensuremath{\gamma}$ varies from $2$ to $\ensuremath{\infty}$ depending on the initial attractiveness of sites. We show that the relation $\ensuremath{\beta}(\ensuremath{\gamma}\ensuremath{-}1)\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1$ between the exponents is universal.