Null-eigenvalue localization of quantum walks on real-world complex networks

TL;DR

This paper reveals that real-world complex networks have high-dimensional null eigenspaces in their adjacency matrices, leading to localized quantum states that differ from Anderson localization, with implications for quantum walk dynamics.

Contribution

The study uncovers null-eigenvalue localization in complex networks and demonstrates its distinct nature from Anderson localization through quantum walk analysis.

Findings

Real-world networks have higher null eigenspaces than random networks.

Null eigenstates are strongly localized within network structures.

Localization differs from Anderson localization by eigenvalue position and state decay.

Abstract

First we report that the adjacency matrices of real-world complex networks systematically have null eigenspaces with much higher dimensions than that of random networks. These null eigenvalues are caused by duplication mechanisms leading to structures with local symmetries which should be more present in complex organizations. The associated eigenvectors of these states are strongly localized. We then evaluate these microstructures in the context of quantum mechanics, demonstrating the previously mentioned localization by studying the spread of continuous-time quantum walks. This null-eigenvalue localization is essentially different from the Anderson localization in the following points: first, the eigenvalues do not lie on the edges of the density of states but at its center; second, the eigenstates do not decay exponentially and do not leak out of the symmetric structures. In this sense, it is closer to the bound state in continuum.