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.

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 $z_2$ of the second-nearest neighbors, the emergence of a $k$-core is a hybrid phase transition. In contrast, if $z_2$ 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 γ 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 $\left(\ln N\gg 1\right)$ the distribution tends to a Gaussian of width $\sim \sqrt{\ln N}$ centered at $\mathcal{l}¯\sim \ln N.\mathrm{}$ We show that the eigenvalue spectrum of the adjacency matrix of the graph has a power-law tail with exponent $2+\gamma .$

We reveal the nature of the avalanche collapse of the giant viable component in multiplex networks under perturbations such as random damage. Specifically, we identify latent critical clusters associated with the avalanches of random damage. Divergence of their mean size signals the approach to the hybrid phase transition from one side, while there are no critical precursors on the other side. We find that this discontinuous transition occurs in scale-free multiplex networks whenever the mean degree of at least one of the interdependent networks does not diverge.

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.