Hostname: page-component-8448b6f56d-qsmjn Total loading time: 0 Render date: 2024-04-24T05:57:36.579Z Has data issue: false hasContentIssue false
  • English
  • Français

A continuum limit for the PageRank algorithm

Part of: Representations of solutions Elliptic equations and systems

Published online by Cambridge University Press:  27 April 2021

A. YUAN Open the ORCID record for A. YUAN [Opens in a new window] ,
J. CALDER Open the ORCID record for J. CALDER [Opens in a new window]  and
B. OSTING Open the ORCID record for B. OSTING [Opens in a new window]
A. YUAN
Affiliation:
Department of Mathematics, University of Minnesota, Minneapolis, MN55455, USA emails: yuanx290@umn.edu; jwcalder@umn.edu
J. CALDER
Affiliation:
Department of Mathematics, University of Minnesota, Minneapolis, MN55455, USA emails: yuanx290@umn.edu; jwcalder@umn.edu
B. OSTING
Affiliation:
Department of Mathematics, University of Utah, Salt Lake City, UT84112, USA email: osting@math.utah.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Semi-supervised and unsupervised machine learning methods often rely on graphs to model data, prompting research on how theoretical properties of operators on graphs are leveraged in learning problems. While most of the existing literature focuses on undirected graphs, directed graphs are very important in practice, giving models for physical, biological or transportation networks, among many other applications. In this paper, we propose a new framework for rigorously studying continuum limits of learning algorithms on directed graphs. We use the new framework to study the PageRank algorithm and show how it can be interpreted as a numerical scheme on a directed graph involving a type of normalised graph Laplacian. We show that the corresponding continuum limit problem, which is taken as the number of webpages grows to infinity, is a second-order, possibly degenerate, elliptic equation that contains reaction, diffusion and advection terms. We prove that the numerical scheme is consistent and stable and compute explicit rates of convergence of the discrete solution to the solution of the continuum limit partial differential equation. We give applications to proving stability and asymptotic regularity of the PageRank vector. Finally, we illustrate our results with numerical experiments and explore an application to data depth.

Keywords

Partial differential equations on graphs and networkssecond-order elliptic equationsviscosity solutions

MSC classification

Primary: 35J15: Second-order elliptic equations
Secondary: 35D40: Viscosity solutions
Type
Papers
Information
European Journal of Applied Mathematics , Volume 33 , Issue 3 , June 2022 , pp. 472 - 504
Check for updates
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

References

Ando, R. K. & Zhang, T. (2007) Learning on graph with Laplacian regularization. In: Advances in Neural Information Processing Systems, Vol. 19, p. 25.Google Scholar
Belkin, M. & Niyogi, P. (2002) Laplacian eigenmaps and spectral techniques for embedding and clustering. In: Advances in Neural Information Processing Systems, pp. 585–591.Google Scholar
Belkin, M. & Niyogi, P. (2003) Using manifold structure for partially labeled classification. In: Advances in Neural Information Processing Systems, pp. 953–960.Google Scholar
Belkin, M. & Niyogi, P. (2005) Towards a theoretical foundation for Laplacian-based manifold methods. In: International Conference on Computational Learning Theory, Springer, pp. 486–500.Google Scholar
Belkin, M. & Niyogi, P. (2007) Convergence of Laplacian eigenmaps. In: Advances in Neural Information Processing Systems, pp. 129–136.Google Scholar
Bertozzi, A. L. & Flenner, A. (2012) Diffuse interface models on graphs for classification of high dimensional data. Multiscale Model. Simul. 10(3), 1090–1118.CrossRefGoogle Scholar
Bousquet, O., Chapelle, O. & Hein, M. (2004) Measure based regularization. In: Advances in Neural Information Processing Systems, pp. 1221–1228.Google Scholar
Burago, D., Ivanov, S. & Kurylev, Y. (2014) A graph discretization of the Laplace-Beltrami operator. J. Spectr. Theory 4(4), 675–714.CrossRefGoogle Scholar
Calder, J. (2018) The game theoretic p-Laplacian and semi-supervised learning with few labels. Nonlinearity 32(1), 301.Google Scholar
Calder, J. (2018) Lecture notes on viscosity solutions. Online Lecture Notes: http://www-users.math.umn.edu/jwcalder/viscosity_solutions.pdf.Google Scholar
Calder, J. (2019) Consistency of Lipschitz learning with infinite unlabeled data and finite labeled data. SIAM J. Math. Data Sci. 1(4), 780–812.CrossRefGoogle Scholar
Calder, J., Esedoglu, S. & Hero, A. O. A Hamilton–Jacobi equation for the continuum limit of nondominated sorting. SIAM J. Math. Anal. 46(1), 603–638 (2014).CrossRefGoogle Scholar
Calder, J. & García Trillos, N. (2019) Improved spectral convergence rates for graph Laplacians on ε-graphs and k-NN graphs. arXiv preprint.Google Scholar
Calder, J. & Slepčev, D. (2019) Properly-weighted graph Laplacian for semi-supervised learning. Appl. Math. Optim. Spec. Issue Optim. Data Sci. 82, 1111–1159.CrossRefGoogle Scholar
Calder, J., Slepčev, D. & Thorpe, M. (2020) Rates of convergence for Laplacian semi-supervised learning with low labeling rates. arXiv:2006.02765.Google Scholar
Calder, J. & Smart, C. K. (2018) The limit shape of convex hull peeling. arXiv preprint arXiv:1805.08278.Google Scholar
Coifman, R. R. & Lafon, S. (2006) Diffusion maps. Appl. Comput. Harmon. Anal. 21(1), 5–30.CrossRefGoogle Scholar
Crandall, M. G., Ishii, H. & Lions, P.-L. (1992) User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. 27(1), 1–67.Google Scholar
Dunson, D. B., Wu, H.-T. & Wu, N. (2019) Diffusion based Gaussian process regression via heat kernel reconstruction. arXiv preprint arXiv:1912.05680.Google Scholar
El Alaoui, A., Cheng, X., Ramdas, A., Wainwright, M. J. & Jordan, M. I. (2016) Asymptotic behavior of lp-based Laplacian regularization in semi-supervised learning. In: Conference on Learning Theory, pp. 879–906.Google Scholar
Evans, L. (2010) Partial Differential Equations . Graduate Studies in Mathematics, American Mathematical Society.CrossRefGoogle Scholar
Flores, M., Calder, J. & Lerman, G. (2018) Algorithms for lp-based semi-supervised learning on graphs. arXiv preprint.Google Scholar
Garcia-Cardona, C., Merkurjev, E., Bertozzi, A. L., Flenner, A. & Percus, A. G. (2014) Multiclass data segmentation using diffuse interface methods on graphs. IEEE Trans. Pattern Anal. Mach. Intell. 36(8), 1600–1613.CrossRefGoogle ScholarPubMed
García Trillos, N. (2019) Variational limits of k-NN graph-based functionals on data clouds. SIAM J. Math. Data Sci. 1(1), 93–120.CrossRefGoogle Scholar
García Trillos, N., Gerlach, M., Hein, M. & Slepčev, D. (2020) Error estimates for spectral convergence of the graph Laplacian on random geometric graphs toward the Laplace–Beltrami operator. Found. Comput. Math. 20(4), 827–887.CrossRefGoogle Scholar
García Trillos, N. & Murray, R. W. (2020) A maximum principle argument for the uniform convergence of graph Laplacian regressors. SIAM J. Math. Data Sci. 2(3), 705–739.CrossRefGoogle Scholar
García Trillos, N. & Slepčev, D. (2018) A variational approach to the consistency of spectral clustering. Appl. Comput. Harm. Anal. 45(2), 239–281.CrossRefGoogle Scholar
García Trillos, N., Slepčev, D., Von Brecht, J., Laurent, T. & Bresson, X. (2016) Consistency of Cheeger and ratio graph cuts. J. Mach. Learn. Res. 17(1), 6268–6313.Google Scholar
Gilbarg, D. & Trudinger, N. (2001) Elliptic Partial Differential Equations of Second Order . Classics in Mathematics, U.S. Government Printing Office.CrossRefGoogle Scholar
Gleich, D. F. (2015) PageRank beyond the web. SIAM Rev. 57(3), 321–363.CrossRefGoogle Scholar
Haveliwala, T. & Kamvar, S. (2003) The Second Eigenvalue of the Google Matrix. Technical report, Stanford.Google Scholar
He, J., Li, M., Zhang, H.-J., Tong, H. & Zhang, C. (2004) Manifold-ranking based image retrieval. In: Proceedings of the 12th Annual ACM International Conference on Multimedia, ACM, pp. 9–16.CrossRefGoogle Scholar
He, J., Li, M., Zhang, H.-J., Tong, H. & Zhang, C. (2006) Generalized manifold-ranking-based image retrieval. IEEE Trans. Image Process. 15(10), 3170–3177.CrossRefGoogle ScholarPubMed
Hein, M., Audibert, J.-Y. & Von Luxburg, U. (2007) Graph Laplacians and their convergence on random neighborhood graphs. J. Mach. Learn. Res. 8, 1325–1368.Google Scholar
Hein, M., Audibert, J.-Y. & Von Luxburg, U. (2005) From graphs to manifolds–weak and strong pointwise consistency of graph Laplacians. In: International Conference on Computational Learning Theory, Springer, pp. 470–485.Google Scholar
Hoffmann, F., Hosseini, B., Oberai, A. A. & Stuart, A. M. (2019) Spectral analysis of weighted Laplacians arising in data clustering. arXiv preprint arXiv:1909.06389.Google Scholar
Lafon, S. S. (2004) Diffusion Maps and Geometric Harmonics. PhD thesis, Yale University PhD dissertation.Google Scholar
Langville, A. N. & Meyer, C. D. (2004) Deeper inside PageRank. Internet Math. 1(3), 335–380.CrossRefGoogle Scholar
LeCun, Y., Bottou, L., Bengio, Y. & Haffner, P. (1998) Gradient-based learning applied to document recognition. Proc. IEEE 86(11), 2278–2324.CrossRefGoogle Scholar
Ng, A. Y., Jordan, M. I. & Weiss, Y. (2002) On spectral clustering: analysis and an algorithm. In: Advances in Neural Information Processing Systems, pp. 849–856.Google Scholar
Osting, B. & Reeb, T. H. (2017) Consistency of Dirichlet partitions. SIAM J. Math. Anal. 49(5), 4251–4274.CrossRefGoogle Scholar
Shi, J. & Malik, J. (2000) Normalized cuts and image segmentation. Departmental Papers (CIS), p. 107.Google Scholar
Shi, Z. (2015) Convergence of Laplacian spectra from random samples. arXiv preprint arXiv:1507.00151.Google Scholar
Shi, Z., Wang, B. & Osher, S. J. (2018) Error estimation of weighted nonlocal Laplacian on random point cloud. arXiv preprint arXiv:1809.08622.Google Scholar
Shnitzer, T., Ben-Chen, M., Guibas, L., Talmon, R. & Wu, H.-T. (2019) Recovering hidden components in multimodal data with composite diffusion operators. SIAM J. Math. Data Sci. 1(3), 588–616.CrossRefGoogle Scholar
Singer, A. (2006) From graph to manifold Laplacian: the convergence rate. Appl. Comput. Harmon. Anal. 21(1), 128–134.CrossRefGoogle Scholar
Slepčev, D. & Thorpe, M. (2019) Analysis of p-Laplacian regularization in semi-supervised learning. SIAM J. Math. Anal. 51(3), 2085–2120.CrossRefGoogle Scholar
Szummer, M. & Jaakkola, T. (2002) Partially labeled classification with Markov random walks. In: Advances in Neural Information Processing Systems, pp. 945–952.Google Scholar
Ting, D., Huang, L. & Jordan, M. (2010) An analysis of the convergence of graph Laplacians. In: International Conference on Machine Learning (ICML).Google Scholar
Trillos, N. G. & Slepčev, D. (2016) Continuum limit of total variation on point clouds. Arch. Ration. Mech. Anal. 220(1), 193–241.Google Scholar
Von Luxburg, U., Belkin, M. & Bousquet, O. (2008) Consistency of spectral clustering. Ann. Stat. 36(2), 555–586.CrossRefGoogle Scholar
Wang, Y., Cheema, M. A., Lin, X. & Zhang, Q. (2013) Multi-manifold ranking: using multiple features for better image retrieval. In: Pacific-Asia Conference on Knowledge Discovery and Data Mining, Springer, pp. 449–460.CrossRefGoogle Scholar
Xiao, H., Rasul, K. & Vollgraf, R. (2017) Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv:1708.07747.Google Scholar
Xu, B., Bu, J., Chen, C., Cai, D., He, X., Liu, W. & Luo, J. (2011) Efficient manifold ranking for image retrieval. In: Proceedings of the 34th International ACM SIGIR Conference on Research and Development in Information Retrieval, ACM, pp. 525–534.CrossRefGoogle Scholar
Yang, C., Zhang, L., Lu, H., Ruan, X. & Yang, M.-H. (2013) Saliency detection via graph-based manifold ranking. In: Proceedings of the IEEE conference on Computer Vision and Pattern Recognition, pp. 3166–3173.CrossRefGoogle Scholar
Zhou, D., Bousquet, O., Lal, T. N., Weston, J. & Schölkopf, B. (2004) Learning with local and global consistency. Adv. Neural Inf. Process. Syst. 16(16), 321–328.Google Scholar
Zhou, D., Hofmann, T. & Schölkopf, B. (2005) Semi-supervised learning on directed graphs. In: Advances in Neural Information Processing Systems, pp. 1633–1640.Google Scholar
Zhou, D., Huang, J. & Schölkopf, B. (2005) Learning from labeled and unlabeled data on a directed graph. In: Proceedings of the 22nd International Conference on Machine Learning, ACM, pp. 1036–1043.CrossRefGoogle Scholar
Zhou, D., Weston, J., Gretton, A., Bousquet, O. & Schölkopf, B. (2004) Ranking on data manifolds. In: Advances in Neural Information Processing Systems, Vol. 16, pp. 169–176.Google Scholar
Zhou, X., Belkin, M. & Srebro, N. (2011) An iterated graph Laplacian approach for ranking on manifolds. In: Proceedings of the 17th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, ACM, pp. 877–885.CrossRefGoogle Scholar
Zhu, X., Ghahramani, Z. & Lafferty, J. (2003) Semi-supervised learning using Gaussian fields and harmonic functions. In: International Conference on Machine Learning, Vol. 3, pp. 912–919.Google Scholar
Zosso, D. & Osting, B. (2016) A minimal surface criterion for graph partitioning. Inverse Probl. Imaging 10(4), 1149–1180.Google Scholar