We consider the canonical ensemble of N -vertex Erdős-Rényi (ER) random topological graphs with quenched vertex degree, and with fugacity µ for each closed triple of bonds. We claim complete defragmentation of large-N graphs into the collection of [p −1 ] almost full subgraphs (cliques) above critical fugacity, µc, where p is the ER bond formation probability. Evolution of the spectral density, ρ(λ), of the adjacency matrix with increasing µ leads to the formation of two-zonal support for µ > µc. Eigenvalue tunnelling from one (central) zone to the other means formation of a new clique in the defragmentation process. The adjacency matrix of the ground state of a network has the block-diagonal form where number of vertices in blocks fluctuate around the mean value N p. The spectral density of the whole network in this regime has triangular shape. We interpret the phenomena from the viewpoint of the conventional random matrix model and speculate about possible physical applications.Investigation of critical and collective effects in graphs and networks has becoming a new rapidly developing interdisciplinary area, with diverse applications and variety of questions to be asked, see [1] for review. Ensembles of random Erdős-Rényi topological graphs (networks) provide an efficient laboratory for testing collective phenomena in statistical physics of complex systems, being also tightly linked to conventional random matrix theory. Besides investigating typical statistical properties of networks, like vertex degree distribution, clustering coefficients, "small world" structure etc, last two decades have been marked by rapidly growing interest in more refined graph characteristics, such as distribution of small subgraphs involving triads of vertices.Triadic interactions, being the simplest interactions beyond the free-field theory, play crucial role in the network statistics. Presence of such interactions is responsible for emergence of phase transitions in complex distributed systems. First example of a phase transition in random networks, known as Strauss clustering [2], has been treated by the Random Matrix Theory (RMT) in [3]. It was argued that, when the increasing fugacity, µ, the system develops two phases with essentially different triad concentrations. At large µ the system falls into the Strauss phase with the single clique of nodes. The condensation of triads is a non-perturbative phenomenon identified in [4] with the 1st order phase transition in the framework of mean-field cavity-like approach.Similar critical behavior was found in [5] for the vertexdegree-conserved ER graphs. It was demonstrated in the framework of the mean-field approach that the phase transition takes place in this case as well. The hysteresis for dependence of the triad concentration on the fugacity, µ also has been observed in [5]. For bi-color networks with conserved vertex degree a new phenomena of a wide plateau formation in concentration of black-white bonds as a function of the fugacity of unicolor triples of bonds has been found in...

We study the asymptotic behavior of the number of paths of length N on several classes of infinite graphs with a single special vertex. This vertex can work as an entropic trap for the path, i.e. under certain conditions the dominant part of long paths become localized in the vicinity of the special point instead of spreading to infinity. We study the conditions for such localization on decorated star graphs, regular trees and regular hyperbolic graphs as a function of the functionality of the special vertex. In all cases the localization occurs for large enough functionality. The particular value of transition point depends on the large-scale topology of the graph. The emergence of localization is supported by the analysis of the spectra of the adjacency matrices of corresponding finite graphs.

We consider clustering in rewired Erdős–Rényi networks with conserved vertex degree and in random regular graphs from the localization perspective. It has been found in Avetisov et al. (2016, Phys. Rev. E, 94, 062313) that at some critical value of chemical potential $\mu_{\rm cr}$ of closed triad of bonds, the evolving networks decay into the maximally possible number of dense subgraphs. The adjacency matrix acquires above $\mu_{\rm cr}$ the two-zonal support with the triangle-shaped main (perturbative) zone separated by a wide gap from the side (non-perturbative) zone. Studying the distribution of gaps between neighbouring eigenvalues (the level spacing), we demonstrate that in the main zone the level spacing matches the Wigner–Dyson law and is delocalized, however it shares the Poisson statistics in the side zone, which is the signature of localization. In parallel with the evolutionary designed networks, we consider ‘instantly’ ad hoc prepared networks with in- and cross-cluster probabilities exactly as at the final stage of the evolutionary designed network. For such ‘instant’ networks the eigenvalues are delocalized in both zones. We speculate about the difference in eigenvalue statistics between ‘evolutionary’ and ‘instant’ networks from the perspective of a possible phase transition between ergodic and non-ergodic network patterns with a strong ‘memory dependence’, thus advocating possible existence of non-ergodic delocalized states in the clustered random networks at least at finite network sizes.

We propose two models of social segregation inspired by the Schelling model. Agents in our models are nodes of evolving social networks. The total number of social connections of each node remains constant in time, though may vary from one node to the other. The first model describes a "polychromatic" society, in which colors designate different social categories of agents. The parameter µ favors/disfavors connected "monochromatic triads", i.e. connected groups of three individuals within the same social category, while the parameter ν controls the preference of interactions between two individuals from different social categories. The polychromatic model has several distinct regimes in (µ, ν)-parameter space. In ν-dominated region, the phase diagram is characterized by the plateau in the number of the inter-color connections, where the network is bipartite, while in µ-dominated region, the network looks as two weakly connected unicolor clusters. At µ > µcrit and ν > νcrit two phases are separated by a critical line, while at small values of µ and ν, a gradual crossover between the two phases occurs. The second "colorless" model describes a society in which the advantage/disadvantage of forming small fully connected communities (short cycles or cliques in a graph) is controlled by a parameter γ. We analyze the topological structure of a social network in this model and demonstrate that above a critical threshold, γ + > 0, the entire network splits into a set of weakly connected clusters, while below another threshold, γ − < 0, the network acquires a bipartite graph structure. Our results propose mechanisms of formation of self-organized communities in international communication between countries, as well as in crime clans and prehistoric societies.