Exploring universality of the $\beta \text{−}\text{Gaussian}$ 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 $\beta \text{−}\mathrm{Gaussian}$ 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 $\beta \text{−}\mathrm{Gaussian}$ ensemble in complex networks. In this work, we study the eigenvalue statistics of various network models, such as small-world, Erdős-Rényi random, and scale-free networks, as well as in comparing the intermediate level statistics of the model networks with that of a $\beta \text{−}\mathrm{Gaussian}$ ensemble. It is found that the nearest-neighbor eigenvalue statistics of all the model networks are in excellent agreement with the $\beta \text{−}\mathrm{Gaussian}$ ensemble. However, the $\beta \text{−}\mathrm{Gaussian}$ 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 $\beta \text{−}\mathrm{Gaussian}$ ensemble is in excellent agreement with the intermediate higher order eigenvalue statistics of model networks.