Null-eigenvalue localization of quantum walks on complex networks

Ruben Bueno,Naomichi Hatano

doi:10.1103/physrevresearch.2.033185

Ruben Bueno, Naomichi Hatano

Open Access

https://doi.org/10.1103/physrevresearch.2.033185

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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.

Highlights

Complex networks define relations between entities such as atoms, proteins, or even humans [1,2,3], as in acquaintance networks [4] and the World Wide Web [5]
We look into the null-eigenvalue degeneracy in terms of network structures, its localized properties, and its consequences in quantum mechanics
We can see that the main part of χ jk comes from the null-eigenvalue contribution χ j(k0). This means that the quantum particle stays on the encircled nodes in Fig. 4, namely, the local structures in Fig. 2(b), and this localization is due to the null eigenspace. This is another instance of the null-eigenvalue localization, which concept we introduced in the previous section

Summary

INTRODUCTION

Complex networks define relations between entities such as atoms, proteins, or even humans [1,2,3], as in acquaintance networks [4] and the World Wide Web [5]. Mathematical models, either of regular lattices or of random graphs, appeared to be insufficient [6] to explain the complex characteristics that real-world networks exhibit, including the small-world phenomenon [7] and scale-free property [1]. The degeneracy is not a cause of localization, but a consequence of the fact that there are a large number of localized states of null eigenvalues all over the real-world network. We stress that the adjacency matrix of real-world complex networks typically has an eigenvalue spectrum that is very distinct from the one of the random matrix theory, in that the former have high degeneracy of null eigenvalues. We claim that locally symmetric structures of typical complex networks produce geometrical constriction of wave functions, caging the null eigenvectors.

HIGH NULL-EIGENVALUE DEGENERACY

QUANTUM MECHANICAL INTERPRETATION OF THE NULL EIGENVALUES

CONCLUSIONS