Localization and non-ergodicity in clustered random networks

Abstract

Abstract 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.