Exploring universality of the -Gaussian ensemble in complex networks via intermediate eigenvalue statistics

The eigenvalue statistics are an important tool to capture localization to delocalization transition in physical systems. Recently, a beta-Gaussian ensemble is being proposed as a single parameter to describe the intermediate eigenvalue statistics of many physical systems. It is critical to explore the universality of a beta-Gaussian ensemble in complex networks. In this work, we study the eigenvalue statistics of various network models, such as small-world, Erdos-Renyi random, and scale-free networks, as well as in comparing the intermediate level statistics of the model networks with that of a beta-Gaussian ensemble. It is found that the nearest-neighbor eigenvalue statistics of all the model networks are in excellent agreement with the beta-Gaussian ensemble. However, the beta-Gaussian ensemble fails to describe the intermediate level statistics of higher order eigenvalue statistics, though there is qualitative agreement till n < 4. Additionally, we show that the nearest-neighbor eigenvalue statistics of the beta-Gaussian ensemble is in excellent agreement with the intermediate higher order eigenvalue statistics of model networks.