Universality in the spectral and eigenfunction properties of random networks

TL;DR

This study demonstrates that spectral and eigenfunction properties of Erdős-Rényi networks exhibit universal behavior depending on average degree, with Brody distribution effectively describing the transition from isolated vertices to a fully connected network.

Contribution

The paper reveals universality in spectral properties of ER networks across different sizes and degrees, and extends findings to networks with diagonal disorder and small-world networks.

Findings

Universal spectral statistics for fixed average degree

Brody distribution models transition in spectral behavior

Universality persists with diagonal disorder

Abstract

By the use of extensive numerical simulations we show that the nearest-neighbor energy level spacing distribution $P(s)$ and the entropic eigenfunction localization length of the adjacency matrices of Erd\H{o}s-R\'enyi (ER) {\it fully} random networks are universal for fixed average degree $\xi\equiv \alpha N$ ($\alpha$ and $N$ being the average network connectivity and the network size, respectively). We also demonstrate that Brody distribution characterizes well $P(s)$ in the transition from $\alpha=0$, when the vertices in the network are isolated, to $\alpha=1$, when the network is fully connected. Moreover, we explore the validity of our findings when relaxing the randomness of our network model and show that, in contrast to standard ER networks, ER networks with {\it diagonal disorder} also show universality. Finally, we also discuss the spectral and eigenfunction properties of small-world networks.