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Hitting times for second-order random walks

Part of: Stochastic processes Markov processes

Published online by Cambridge University Press:  04 July 2022

DARIO FASINO Open the ORCID record for DARIO FASINO [Opens in a new window] ,
ARIANNA TONETTO  and
FRANCESCO TUDISCO Open the ORCID record for FRANCESCO TUDISCO [Opens in a new window]
DARIO FASINO
Affiliation:
University of Udine, 33100 Udine, Italy emails: dario.fasino@uniud.it; arianna.tonetto@spes.uniud.it
ARIANNA TONETTO
Affiliation:
University of Udine, 33100 Udine, Italy emails: dario.fasino@uniud.it; arianna.tonetto@spes.uniud.it
FRANCESCO TUDISCO
Affiliation:
Gran Sasso Science Institute, 67100, L’Aquila, Italy email: francesco.tudisco@gssi.it
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Abstract

A second-order random walk on a graph or network is a random walk where transition probabilities depend not only on the present node but also on the previous one. A notable example is the non-backtracking random walk, where the walker is not allowed to revisit a node in one step. Second-order random walks can model physical diffusion phenomena in a more realistic way than traditional random walks and have been very successfully used in various network mining and machine learning settings. However, numerous questions are still open for this type of stochastic processes. In this work, we extend well-known results concerning mean hitting and return times of standard random walks to the second-order case. In particular, we provide simple formulas that allow us to compute these numbers by solving suitable systems of linear equations. Moreover, by introducing the ‘pullback’ first-order stochastic process of a second-order random walk, we provide second-order versions of the renowned Kac’s and Random Target Lemmas.

Keywords

Second-order random walksnon-backtracking walkshitting timesreturn times

MSC classification

Primary: 60G12: General second-order processes
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)
Type
Papers
Information
European Journal of Applied Mathematics , Volume 34 , Issue 4 , August 2023 , pp. 642 - 666
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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References

Alon, N., Benjamini, I., Lubetzky, E. & Sodin, S. (2007) Non-backtracking random walks mix faster. Commun. Contemp. Math. 9, 585–603.CrossRefGoogle Scholar
Arrigo, F., Grindrod, P., Higham, D. J. & Noferini, V. (2018) Non-backtracking walk centrality for directed networks. J. Complex Netw. 6, 54–78.CrossRefGoogle Scholar
Arrigo, F., Higham, D. J. & Noferini, V. (2021) A theory for backtrack-downweighted walks. SIAM J. Matrix Anal. Appl. 42, 1229–1247.CrossRefGoogle Scholar
Arrigo, F., Higham, D. J. & Tudisco, F. (2020) A framework for second-order eigenvector centralities and clustering coefficients. Proc. Royal Soc. A 476(2236), 20190724.CrossRefGoogle Scholar
Benson, A., Gleich, D. F. & Lim, L.-H. (2017) The spacey random walk: A stochastic process for higher-order data. SIAM Rev. 59, 321–345.CrossRefGoogle Scholar
Benson, A. R., Abebe, R., Schaub, M. T., Jadbabaie, A. & Kleinberg, J. (2018) Simplicial closure and higher-order link prediction. Proc. Nat. Acad. Sci. 115(48), E11221–E11230.CrossRefGoogle Scholar
Benson, A. R., Gleich, D. F. & Lim, L.-H. (2017) The spacey random walk: A stochastic process for higher-order data. SIAM Rev. 59(2), 321–345.CrossRefGoogle Scholar
Breen, J., Faught, N., Glover, C., Kempton, M., Knudson, A. & Oveson, A. (2022) Kemeny’s constant for non-backtracking random walks. arXiv preprint arXiv:2203.12049. Google Scholar
Cioabă, S.M & Xu, P. (2015) Mixing rates of random walks with little backtracking. In: SCHOLARa Scientific Celebration Highlighting Open Lines of Arithmetic Research, volume 655. Contemp. Math. Amer. Math. Soc., Providence, RI, pp. 27–58.CrossRefGoogle Scholar
Cipolla, S., Durastante, F. & Tudisco, F. (2021) Nonlocal PageRank. ESAIM Math. Model. Numer. Anal. 55(1), 77–97.CrossRefGoogle Scholar
Croft, D. P., Krause, J. & James, R. (2004) Social networks in the guppy (Poecilia reticulata). Proc. Royal Soc. B Biolog. Sci. 271, S516–S519.CrossRefGoogle Scholar
Cucuringu, M., Rombach, P., Lee, S. & Porter, M. (2016) Detection of core-periphery structure in networks using spectral methods and geodesic paths. Eur. J. Appl. Math. 27, 846–887.CrossRefGoogle Scholar
Della Rossa, F., Dercole, F. & Piccardi, C. (2013) Profiling core-periphery network structure by random walkers. Sci. Rep. 3, 1–8.Google Scholar
Estrada, E., Delvenne, J.-C., Hatano, N., Mateos, J. L., Metzler, R., Riascos, A. P. & Schaub, M. T. (2017) Random multi-hopper model: super-fast random walks on graphs. J. Compl. Netw. 6(3), 382–403.CrossRefGoogle Scholar
Fasino, D. and Tudisco, F. (2020) Ergodicity coefficients for higher-order stochastic processes. SIAM J. Math. Data Sci. 2, 740–769.CrossRefGoogle Scholar
Fitzner, R. & Hofstad, R. (2012) Non-backtracking random walk. J. Stat. Phys. 150, 264–284.CrossRefGoogle Scholar
Grindrod, P., Higham, D. J. & Noferini, V. (2018) The deformed graph Laplacian and its applications to network centrality analysis. J. Stat. Phys. 39, 310–341.Google Scholar
Grover, A. & Leskovec, J. (2016) Node2vec: Scalable feature learning for networks. In: Proceedings of the ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 855–864.CrossRefGoogle Scholar
Kac, M. (1947) On the notion of recurrence in discrete stochastic processes. Bull. Amer. Math. Soc. 53(10), 1002–1010.CrossRefGoogle Scholar
Kawamoto, T. (2016) Localized eigenvectors of the non-backtracking matrix. J. Stat. Mech. Theory Exp. 2016(2), 023404.CrossRefGoogle Scholar
Kemeny, J. G. & Snell, J. L. (1976) Finite Markov Chains . Undergraduate Texts in Mathematics. Springer-Verlag, New York-Heidelberg.Google Scholar
Kempton, M. (2016) Non-backtracking random walks and a weighted Ihara’s theorem. Open J. Disc. Math. 6, 207–226.10.4236/ojdm.2016.64018CrossRefGoogle Scholar
Krzakala, F., Moore, C., Mossel, E., Neeman, J., Sly, A., Zdeborová, L. & Zhang, P. (2013) Spectral redemption in clustering sparse networks. Proc. Nat. Acad. Sci. 110(52), 20935–20940.CrossRefGoogle Scholar
Kumar, R., Raghu, M., Sarlós, T. & Tomkins, A. (2017) Linear additive Markov processes. In: Proceedings of the 26th International Conference on World Wide Web, pp. 411–419.CrossRefGoogle Scholar
Levin, D. A., Peres, Y. & Wilmer, E. L. (2009) Markov Chains and Mixing Times. American Mathematical Society, Providence, RI.Google Scholar
Liben-Nowell, D. & Kleinberg, J. (2007) The link-prediction problem for social networks. J. Amer. Soc. Inform. Sci. Technol. 58(7), 1019–1031.CrossRefGoogle Scholar
Lusseau, D., Schneider, K., Boisseau, O. J., Haase, P., Slooten, E. & Dawson, S. M. (2003) The bottlenose dolphin community of Doubtful Sound features a large proportion of long-lasting associations. Behav. Ecol. Sociobiol. 54(4), 396–405.CrossRefGoogle Scholar
Martin, T., Zhang, X. & Newman, M. E. (2014) Localization and centrality in networks. Phys. Rev. E 90(5), 052808.CrossRefGoogle ScholarPubMed
Meyer, C. D. (2000) Matrix Analysis and Applied Linear Algebra, vol. 71. SIAM.CrossRefGoogle Scholar
Moler, C. (2013) Lake Arrowhead Coauthor Graph. http://blogs.mathworks.com/cleve/2013/06/10/lake-arrowhead-coauthor-graph/, Retrieved May 1, 2022.Google Scholar
Nadler, B., Srebro, N. & Zhou, X. (2009) Semi-supervised learning with the graph Laplacian: The limit of infinite unlabelled data. Adv. Neural Inform. Process. Syst. 22, 1330–1338.Google Scholar
Newman, M. (2010) Networks: An Introduction. Oxford University Press.CrossRefGoogle Scholar
Newman, M. E. (2005) A measure of betweenness centrality based on random walks. Social Netw. 27(1), 39–54.CrossRefGoogle Scholar
Noh, J. D. & Rieger, H. (2004) Random walks on complex networks. Phys. Rev. Lett. 92(11), 118701.CrossRefGoogle ScholarPubMed
Norris, J. R. (1998) Markov Chains. Number 2 in Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, 1998.Google Scholar
Raftery, A. E. (1985) A model for high-order Markov chains. J. Roy. Statist. Soc. Ser. B 47(3), 528–539.Google Scholar
Rombach, P., Porter, M. A., Fowler, J. H. & Mucha, P. J. (2017) Core-periphery structure in networks (revisited). SIAM Rev. 59(3), 619–646.CrossRefGoogle Scholar
Rubino, G. & Sericola, B. (1991) A finite characterization of weak lumpable Markov processes. I. The discrete time case. Stochastic Process. Appl. 38(2), 195–204.CrossRefGoogle Scholar
Torres, L., Chan, K. S., Tong, H. & Eliassi-Rad, T. (2021) Nonbacktracking eigenvalues under node removal: X-centrality and targeted immunization. SIAM J. Math. Data Sci. 3(2), 656–675.CrossRefGoogle Scholar
Tudisco, F., Benson, A. R. & Prokopchik, K. (2021) Nonlinear higher-order label spreading. In: Proceedings of the Web Conference 2021, pp. 2402–2413.CrossRefGoogle Scholar
Tudisco, F. & Higham, D. J. (2019) A nonlinear spectral method for core-periphery detection in networks. SIAM J. Math. Data Sci. 1, 269–292.CrossRefGoogle Scholar
Wu, S.-J. & Chu, M. T. (2017) Markov chains with memory, tensor formulation, and the dynamics of power iteration. Appl. Math. Comput., 303:226–239, 2017.Google Scholar
Wu, T. & Gleich, D. F. (2017) Retrospective higher-order Markov processes for user trails. In: Proceedings of the 23rd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 1185–1194.CrossRefGoogle Scholar
Wu, Y., Bian, Y. & Zhang, X. (2016) Remember where you came from: On the second-order random walk based proximity measures. Proc. VLDB Endow. 10(1), 13–24.CrossRefGoogle Scholar