Multifractal analysis of eigenvectors of small-world networks. (March 2021) - Mishra, Ankit; Bandyopadhyay, Jayendra N.; Jalan, Sarika
Highlights: Rewiring regular network will affect two major structural properties; Clustering and shortest path length. Higher clustering drives localization whereas shorter path length known to support delocalization. Central regime of the eigenvalue spectrum of smallworld networks depicts strong multifractal behavior. Based on effect of rewiring on localization properties of eigenvectors, rewiring probability can be divided into two domains. Decreasing the average degree increases the strength of the multifractality of eigenvectors. Abstract: Many real-world complex systems have small-world topology characterized by the high clustering of nodes and short path lengths. It is well-known that higher clustering drives localization while shorter path length supports delocalization of the eigenvectors of networks. Using multifractals technique, we investigate localization properties of the eigenvectors of the adjacency matrices of small-world networks constructed using Watts-Strogatz algorithm. We find that the central part of the eigenvalue spectrum is characterized by strong multifractality whereas the tail part of the spectrum have D q → 1. Before the onset of the small-world transition, an increase in the random connections leads to an enhancement in the eigenvectors localization, whereas just after the onset, the eigenvectors show a gradual decrease in the localization. We have verified an existence of sharp change in the correlation dimension at the localization-delocalization transition.