Abstract
In this paper, we explore analytically and experimentally the commute time of the continuous-time quantum walk. For the classical random walk, the commute 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 commute time of the continuous-time quantum walk can be determined via integrals of the Laplacian spectrum, calculated using Gauss-Laguerre quadrature. We analyse the quantum commute times with reference to their classical counterpart. Experimentally, we show that the quantum commute times can be used to emphasise cluster-structure.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Borgwardt, K., Schönauer, S., Vishwanathan, S., Smola, A., Kriegel, H.: Protein function prediction via graph kernels. Bioinformatics 21, (June 2005)
Childs, A., Farhi, E., Gutmann, S.: An example of the difference between quantum and classical random walks. Quantum Information Processing 1(1-2), 35–43 (2002)
DePiero, F., Carlin, J.: Structural matching via optimal basis graphs. In: ICPR 2006, pp. 449–452 (2006)
Emms, D., Severini, S., Wilson, R.C., Hancock, E.: Coined quantum walks lift the cospectrality of graphs and trees. In: EMMCVPR, pp. 332–345 (2005)
Kempe, J.: Quantum random walks – an introductory overview. Contemporary Physics 44(4), 307–327 (2003)
Lovász, L.: Combinatorics, Paul Erdös is Eighty, chapter Random Walks on Graphs: A Survey, János Bolyai Mathematical Society, vol. 2, pp. 353–398., Budapest (1996)
Luo, B., Wilson, R.C., Hancock, E.: Spectral embedding of graphs. Pattern Recognition 36(10), 2213–2230 (2003)
Meila, M., Shi, J.: A random walks view of spectral segmentation. In: AI and STATISTICS (AISTATS) 2001 (2001)
Nayak, A., Vishwanath, A.: Quantum walk on the line (2000)
Neuhaus, M., Bunke, H.: A random walk kernel derived from graph edit distance. In: SSPR, pp. 191–199 (2006)
Qiu, H., Hancock, E.: Robust multi-body motion tracking using commute time clustering. In: Leonardis, A., Bischof, H., Pinz, A. (eds.) ECCV 2006. LNCS, vol. 3954, pp. 160–173. Springer, Heidelberg (2006)
Robles-Kelly, A., Hancock, E.: String edit distance, random walks and graph matching. International Journal of Pattern Recognition and Artificial Intelligence 18(3), 315–327 (2004)
Zhu, X., Ghahramani, Z., Lafferty, J.: Semi-supervised learning using gaussian fields and harmonic functions. In: ICML, pp. 1561–1566 (2003)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Emms, D., Wilson, R.C., Hancock, E. (2007). Graph Embedding Using Quantum Commute Times. In: Escolano, F., Vento, M. (eds) Graph-Based Representations in Pattern Recognition. GbRPR 2007. Lecture Notes in Computer Science, vol 4538. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-72903-7_34
Download citation
DOI: https://doi.org/10.1007/978-3-540-72903-7_34
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-72902-0
Online ISBN: 978-3-540-72903-7
eBook Packages: Computer ScienceComputer Science (R0)