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.
Highlights
We show that the state of the Fourier walk on complex networks is localized in a community of the initial node and, thereby, reveals the community structure
We define the discrete-time quantum walk on complex networks and utilize it for community detection
We numerically show that the Fourier walk is localized in a community to which the initial node belongs
Summary
The quantum walk has been studied in various areas of physics. The quantum walk is divided into two types: the discrete-time quantum walk [1] and the continuous-time quantum walk [2]. Quantum walks with different inner states and different coin operators behave differently. The probability distribution of the two-state quantum walk in one dimension, on the other hand, has only two peaks, which spread linearly to the left and right, without any peak that localizes [5]. We focus on the twostate walk, using the Fourier coin and the Grover coin [6,7]. The quantum walk on networks occupies an important role in search problems. It takes classical algorithms O(N ) steps to identify the target record from an unsorted database of N√records, while it takes quantum mechanical systems only O( N ) steps [8]
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