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, researchers have made important steps toward understanding the qualitatively new critical phenomena in complex networks. We review the results, concepts, and methods of this rapidly developing field. Here we mostly consider two closely related classes of these critical phenomena, namely structural phase transitions in the network architectures and transitions in cooperative models on networks as substrates. We also discuss systems where a network and interacting agents on it influence each other. We overview 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. We also discuss strong finite size effects in these systems and highlight open problems and perspectives.Comment: Review article, 79 pages, 43 figures, 1 table, 508 references, extende

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 of non-equilibrium physics, econophysics, evolutionary biology, etc. CONTENTS

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 of non-equilibrium physics, econophysics, evolutionary biology, etc. * Electronic address: sdorogov@fc.up.pt † Electronic address: jfmendes@fc.up.pt

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 q(s,t) of a site s at time t (one site is added per unit of time). At long times, P(q) approximately q(-gamma) at q-->infinity and q(s,t) approximately (s/t)(-beta) at s/t-->0, where the exponent gamma varies from 2 to infinity depending on the initial attractiveness of sites. We show that the relation beta(gamma-1) = 1 between the exponents is universal.

We analytically describe the architecture of randomly damaged uncorrelated networks as a set of successively enclosed substructures--k-cores. The k-core is the largest subgraph where vertices have at least k interconnections. We find the structure of k-cores, their sizes, and their birthpoints--the bootstrap percolation thresholds. We show 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. In contrast, if zeta2 diverges, the networks contain an infinite sequence of k-cores which are ultrarobust against random damage.

We find that scale-free random networks are excellently modeled by simple deterministic graphs. Our graph has a discrete degree distribution (degree is the number of connections of a vertex), which is characterized by a power law with exponent gamma=1+ln 3/ln 2. Properties of this compact structure are surprisingly close to those of growing random scale-free networks with gamma in the most interesting region, between 2 and 3. We succeed to find exactly and numerically with high precision all main characteristics of the graph. In particular, we obtain the exact shortest-path-length distribution. For a large network (ln N>>1) the distribution tends to a Gaussian of width approximately sqrt[ln N] centered at (-)l approximately ln N. We show that the eigenvalue spectrum of the adjacency matrix of the graph has a power-law tail with exponent 2+gamma.

Using the susceptible-infected-susceptible model on unweighted and weighted networks, we consider the disease localization phenomenon. In contrast to the well-recognized point of view that diseases infect a finite fraction of vertices right above the epidemic threshold, we show that diseases can be localized on a finite number of vertices, where hubs and edges with large weights are centers of localization. Our results follow from the analysis of standard models of networks and empirical data for real-world networks.

We find the exact critical temperature Tc of the nearest-neighbor ferromagnetic Ising model on an 'equilibrium' random graph with an arbitrary degree distribution P (k). We observe an anomalous behavior of the magnetization, magnetic susceptibility and specific heat, when P (k) is fat-tailed, or, loosely speaking, when the fourth moment of the distribution diverges in infinite networks. When the second moment becomes divergent, Tc approaches infinity, the phase transition is of infinite order, and size effect is anomalously strong. 87.18.Sn The Ising model is one of the immortal themes in physics. It is a traditional starting point for the study of the effects of cooperative behavior. Networks with complex architecture display a spectrum of unique effects [1][2][3][4][5][6], and so they are an intriguing substrate. The simulation of the Ising model on a growing scale-free network [7] has demonstrated that it is extremely far from that on regular lattices and on 'planar graphs' [8].In this Letter, we report our exact results and results, which, we believe, are asymptotically exact, for the thermodynamic properties of the Ising model on the basic construction for 'equilibrium' random networks. These networks are the undirected graphs, maximally random under the constraint that their degree distribution is a given one, P (k). Here, degree is the number of connections of a vertex. Correlations between degrees of vertices in such graphs are absent. In graph theory, these networks are called 'labelled random graphs with a given degree sequence' or 'the configuration model ' [9]. The earlier interest was mainly in the percolation properties of complex networks and the spread of diseases on them [10][11][12][13][14][15][16][17], and most of the analytical results were obtained just for this basic construction (however,, where the Berezinskii-Kosterlitz-Thouless percolation phase transition was studied in growing networks).Our results demonstrate the strong effect of the fat tail of the degree distribution on the phase transition in the Ising model. The most connected vertices induce strong ferromagnetic correlations in their close neighborhoods at very large temperatures, and so their role is very important. Surprisingly, we observe very strong effects of these vertices, even when they, at first sight, must be insignificant, namely, when the first and the second moments of the degree distribution are still finite but the fourth moment already diverges ( k 4 → ∞). It is convenient to use the power-law degree distribution P (k) ∝ k −γ for parametrization. Then, k 4 diverges for γ ≤ 5, k 3 diverges for γ ≤ 4, and k 2 is divergent for γ ≤ 3. When k 4 < ∞, the phase transition is similar to that in the Ising model on high-dimension regular lattices. However, for γ < 5 its nature is quite different. As γ decreases and the role of the highly connected vertices turns to be more important, T c grows and the phase transition becomes more 'continuous'. Below γ = 4 it is of the higher order than the second in Ehrenfest's termi...