Localization and non-ergodicity in clustered random networks

We consider clustering in rewired Erdos-Renyi 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 μcr of closed triad of bonds, the evolving networks decay into the maximally possible number of dense subgraphs. The adjacency matrix acquires above μcr 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 einstantly f ad hoc prepared networks with in- A nd cross-cluster probabilities exactly as at the final stage of the evolutionary designed network. For such einstant f networks the eigenvalues are delocalized in both zones. We speculate about the difference in eigenvalue statistics between eevolutionary f and einstant f networks from the perspective of a possible phase transition between ergodic and non-ergodic network patterns with a strong ememory dependence f, thus advocating possible existence of non-ergodic delocalized states in the clustered random networks at least at finite network sizes. © 2019 The authors. Published by Oxford University Press. All rights reserved.