Graph embedding using a quasi-quantum analogue of the hitting times of continuous time quantum walks

In this paper, we explore analytically and experimentally a quasi-quantum analogue of the hitting time of the continuous-time quantum walk on a graph. For the classical random walk, the hitting time has been shown to be robust to errors in edge weight structure and to lead to spectral clustering algorithms with improved performance. Our analysis shows that the quasi-quantum analogue of the hitting time of the continuoustime quantum walk can be determined via integrals of the Laplacian spectrum, calculated using Gauss-Laguerre quadrature. We analyse the quantum hitting times with reference to their classical counterpart. Specifically, we explore the graph embeddings that preserve hitting time. Experimentally, we show that the quantum hitting times can be used to emphasise cluster-structure.