Scaling Invariance in Wave Functions of Quantum Systems on Complex Networks

TL;DR

This paper investigates the wave functions of quantum systems on various complex networks, revealing a universal power-law behavior in their probability distributions that varies with network structure, offering insights into network properties and dynamics.

Contribution

It demonstrates that wave function components follow a power-law with cutoff across different network models, highlighting structure-dependent distribution transitions.

Findings

PDF transitions from power-law to exponential with increasing connectivity in ER networks.

Power-law distributions are observed in WS and GRN networks within specific parameters.

Wave function distributions can serve as structural measurements for complex networks.

Abstract

Structure-induced features of the wave functions for the quantum systems on complex networks are discussed in this paper. For a quantum system on a network, the state corresponding to the eigenvalue close to the center of the spectrum is used as the representative state to display the impacts of the structure on the wave functions. We consider the Erdos-Renyi, the WS small world and the growing randomly network (GRN) models. It is found that the probability distribution functions (PDF) of the representative state's components can be described with a power-law with an exponential cutoff in a unified way. For Erdos-Renyi networks, with the increase of the connectivity probability $p_{ER} $ the PDF turns from power-law-dominated to exponential-dominated functions. For the WS networks in a special region of the rewiring probability $p_r \in (0,0.2)$, where this model can capture the features of real world networks, and the GRN networks, the PDFs obey almost a perfect power-law. These characteristics can be used as the structure measurements of complex networks. They can also provide useful information on dynamical processes on complex networks.