The recent rapid progress in the statistical physics of evolving networks is reviewed, and how growing networks self-organize into scale-free structures is discussed, and the role of the mechanism of preferential linking is investigated.Expand

The aim of the text is to understand networks and the basic principles of their structural organization and evolution, so even students without a deep knowledge of mathematics and statistical physics will be able to rely on this as a reference.Expand

A wide range of critical phenomena in equilibrium and growing networks including the birth of the giant connected component, percolation, $k$-core percolations, 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.Expand

The model of growing networks with the preferential attachment of new links is generalized to include initial attractiveness of sites and it is shown that the relation beta(gamma-1) = 1 between the exponents is universal.Expand

It is found that scale-free random networks are excellently modeled by simple deterministic graphs and exactly and numerically with high precision all main characteristics of the graph are found.Expand

Using the susceptible-infected-susceptible model on unweighted and weighted networks, the disease localization phenomenon is considered and it is shown that diseases can be localized on a finite number of vertices.Expand

It follows from the theory of the evolution of language that the size of the core part of language, the ‘kernel lexicon’, does not vary as language evolves, and the two regimes in the distribution naturally emerge from the evolutionary dynamics of the word web.Expand

It is shown that spectra of locally treelike random graphs may serve as a starting point in the analysis of spectral properties of real-world networks, e.g., of the Internet.Expand

It is shown that in networks with a finite mean number zeta2 of the second-nearest neighbors, the emergence of a k-core is a hybrid phase transition, and in contrast, ifZeta2 diverges, the networks contain an infinite sequence of k-cores which are ultrarobust against random damage.Expand

It is found that the accelerating growth of networks establishes their structure, and two scenarios of self-organization of nonlinearly growing networks into free-scale structures are considered.Expand