Discrete-time quantum walk on complex networks for community detection

TL;DR

This paper introduces a quantum walk-based method for community detection in complex networks, demonstrating its effectiveness on synthetic and real-world networks by analyzing localization behaviors of different quantum coins.

Contribution

It presents a novel quantum walk approach with Fourier and Grover coins for identifying community structures in complex networks, outperforming classical methods in clarity.

Findings

Fourier-coin quantum walk localizes within communities

Grover-coin quantum walk localizes around initial nodes

Classical random walk converges to uniform distribution

Abstract

We define the discrete-time quantum walk on complex networks and utilize it for community detection. We numerically show that the quantum walk with the Fourier coin is localized in a community to which the initial node belongs. Meanwhile, the quantum walk with the Grover coin tends to be localized around the initial node, not over a community. The probability of the classical random walk on the same network converges to the uniform distribution with a relaxation time generally a priori. We thus claim that the time average of the probability of the Fourier-coin quantum walk on complex networks reveals the community structure more explicitly than that of the Grover-coin quantum walk and a snapshot of the classical random walk. We first demonstrate our method of community detection for a prototypical three-community network, producing the correct grouping. We then apply our method to two real-world networks, namely Zachary's karate club and the US Airport network. We successfully reveals the community structure, the two communities of the instructor and the administrator in the former and major airline companies in the latter.