Sergey N. Dorogovtsev

CRITICAL PHENOMENA IN COMPLEX NETWORKS

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,

Lectures on Complex Networks

k

Size-dependent degree distribution of a scale-free growing network.

-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.

Scaling behaviour of developing and decaying networks

This text is a concise modern introduction to the science of complex networks, and is based on lectures for university students and non-specialists. The author aims to introduce a reader without serious background in mathematics or physics to the world of networks. The lectures fill the existing gap between popular science books and comprehensive reference volumes on complex networks, and provide the shortest path to the world of networks, discussing the main directions of modern research in this active field, as well as the history of network studies. The text describes the current state of the art in complex networks and includes recent results, and will be useful for both teaching and self-study.

Giant strongly connected component of directed networks.

We show that the connectivity distributions

Spectra of complex networks

P(k,t)

Bootstrap percolation on complex networks.

of scale-free growing networks (

Exactly solvable small-world network

t

Multiple percolation transitions in a configuration model of a network of networks.

is the network size) have the generic scale -- the cut-off at

Principles of statistical mechanics of uncorrelated random networks

k_{cut} \sim t^\beta