Abstract
A prominent theme of this book is the spatial analysis of networks and data independent of an embedding in an ambient space. The topology and metric of the network/complex have been sufficient to define the domain upon which we may perform data analysis. However, an intrinsic metric defined on a network may be interpreted as the metric that would have been obtained if the network had been embedded into an ambient space equipped with its own metric. Consequently, it is possible to calculate an embedding map for which the induced metric approximates the intrinsic metric defined on the network. The calculation of such embeddings by manifold learning techniques is one way in which the structure of the network may be examined and visualized. A different method of examining the structure of a network is to calculate an importance ranking for each node. In contrast to the majority of this book, the ranking algorithms are generally used to examine the structure of directed graphs.
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
Note that Chung defined a symmetric conception of the Laplacian operator on a directed graph [79]. See Chap. 2 for more information on this advection process and the corresponding Laplacian matrix used here.
References
Belkin, M., Niyogi, P.: Laplacian eigenmaps and spectral techniques for embedding and clustering. In: Advances in Neural Information Processing Systems, vol. 14, pp. 585–591. MIT Press, Cambridge (2001)
Biggs, N.: Algebraic Graph Theory, 2nd edn. Cambridge University Press, Cambridge (1994)
Brin, S., Page, L.: The anatomy of a large-scale hypertextual web search engine. Computer Networks and ISDN Systems 30(1–7), 107–117 (1998)
Burges, J.C.: Geometric methods for feature extraction and dimensional reduction. In: Data Mining and Knowledge Discovery Handbook, pp. 59–91. Springer, Berlin (2005). Chap. 1
Chan, T.F., Gilbert, J.R., Teng, S.H.: Geometric spectral partitioning. PARC Technical Report CSL-94-15, Xerox (1995)
Chung, F.: Laplacians and the Cheeger inequality for directed graphs. Annals of Combinatorics 9(1), 1–19 (2005)
Constantine, P.G., Gleich, D.F.: Using polynomial chaos to compute the influence of multiple random surfers in the PageRank model. In: Bonato, A., Chung, F.R.K. (eds.) Proc. of WAW. Lecture Notes in Computer Science, vol. 4863, pp. 82–95. Springer, Berlin (2007)
Dale, A.M., Fischl, B., Sereno, M.I.: Cortical surface-based analysis. I. Segmentation and surface reconstruction. NeuroImage 9(2), 179–194 (1999)
Duda, R.O., Hart, P.E., Stork, D.G.: Pattern Classification, 2nd edn. Wiley-Interscience, New York (2000)
Farahat, A., LoFaro, T., Miller, J., Rae, G., Ward, L.: Authority rankings from HITS, Pagerank, and SALSA: Existence, uniqueness, and effect of initialization. SIAM Journal on Scientific Computing 27(4), 1181–1201 (2006)
Fischl, B., Sereno, M.I., Dale, A.M.: Cortical surface-based analysis. II: Inflation, flattening, and a surface-based coordinate system. NeuroImage 9(2), 195–207 (1999)
Fischl, B., Sereno, M.I., Tootell, R.B., Dale, A.M.: High-resolution intersubject averaging and a coordinate system for the cortical surface. Human Brain Mapping 8(4), 272–284 (1999)
Fowler, J., Jeon, S.: The authority of Supreme Court precedent. Social Networks 30(1), 16–30 (2008)
Fowler, J., Johnson, T., Spriggs, J., Jeon, S., Wahlbeck, P.: Network analysis and the law: Measuring the legal importance of precedents at the US Supreme Court. Political Analysis 15(3), 324–346 (2007)
Gleich, D.F.: Models and algorithms for pagerank sensitivity. Ph.D. thesis, Stanford University (2009)
Godsil, C., McKay, B.: Constructing cospectral graphs. Aequationes Mathematicae 25(1), 257–268 (1982)
Gordon, C., Webb, D., Wolpert, S.: You cannot hear the shape of a drum. Bulletin of the American Mathematical Society 27, 134–138 (1992)
He, X., Niyogi, P.: Locality preserving projections. In: Advances in Neural Information Processing Systems, vol. 16, pp. 153–160. MIT Press, Cambridge (2003)
Hoppe, H., DeRose, T., Duchamp, T., McDonald, J., Stuetzle, W.: Surface reconstruction from unorganized points. In: Proc. of SIGGRAPH, vol. 26, pp. 19–26 (1992)
Jiang, X., Lim, L.H., Yao, Y., Ye, Y.: Statistical ranking and combinatorial Hodge theory. Mathematical Programming (to appear)
Kac, M.: Can one hear the shape of a drum? American Mathematical Monthly 73(4), 1–23 (1966)
Katz, L.: A new status index derived from sociometric analysis. Psychometrika 18(1), 39–43 (1953)
Kleinberg, J.: Authoritative sources in a hyperlinked environment. Journal of the ACM 46(5), 604–632 (1999)
Kohlberger, T., Uzunbas, G., Alvino, C., Kadir, T., Slosman, D., Funka-Lea, G.: Organ segmentation with level sets using local shape and appearance priors. In: Yang, G.-Z. et al. (eds.) Int. Conf. on Medical Image Comp. and Comp.-Assisted Intervention (MICCAI ’09). Lecture Notes in Computer Science, vol. 5762, pp. 34–42. Springer, Berlin (2009)
Lévy, B.: Laplace-Beltrami eigenfunctions towards an algorithm that “understands” geometry. In: IEEE International Conference on Shape Modeling and Applications SMI 2006, p. 13 (2006)
Mateus, D., Horaud, R., Knossow, D., Cuzzolin, F., Boyer, E.: Articulated shape matching using Laplacian eigenfunctions and unsupervised point registration. In: IEEE Conference on Computer Vision and Pattern Recognition, 2008. CVPR 2008, pp. 1–8 (2008)
Qiu, H., Hancock, E.: Image segmentation using commute times. In: Proceedings of the 16th British Machine Vision Conference (BMVC 2005), pp. 929–938 (2005)
Qiu, H., Hancock, E.R.: Commute times, discrete Green’s functions and graph matching. In: Proc. of 13th Int. Conf. on Image Analysis and Processing—ICIAP 2005, Cagliari, Italy, September 6–8, 2005, pp. 454–462. Springer, Berlin (2005)
Robinson, S.: The ongoing search for efficient web search algorithms. SIAM News 37(9) (2004)
Saerens, M., Fouss, F., Yen, L., Dupont, P.: The principal components analysis of a graph, and its relationships to spectral clustering. In: Proceedings of the 15th European Conference on Machine Learning (ECML 2004). Lecture Notes in Artificial Intelligence, pp. 371–383. Springer, Berlin (2004)
Schoenberg, I.: Remarks to Maurice Frechet’s article “Sur la définition axiomatique d’une classe d’espace distanciés vectoriellement applicable sur l’espace de Hilbert”. Annals of Mathematics 36, 724–732 (1935)
Shi, J., Malik, J.: Normalized cuts and image segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence 22(8), 888–905 (2000)
Sumner, R., Popovic, J.: Deformation transfer for triangle meshes. Proceedings of SIGGRAPH 23(3), 399–405 (2004)
Tenenbaum, J., de Silva, V., Langford, J.: A global geometric framework for nonlinear dimensionality reduction. Science 290(5500), 2319–2323 (2000)
Van Dam, E., Haemers, W.: Spectral characterizations of some distance-regular graphs. Journal of Algebraic Combinatorics 15(2), 189–202 (2002)
Yen, L., Vanvyve, D., Wouters, F., Fouss, F., Verleysen, M., Saerens, M.: Clustering using a random walk based distance measure. In: Proceedings of the 13th Symposium on Artificial Neural Networks (ESANN 2005), pp. 317–324 (2005)
Zhang, H., van Kaick, O., Dyer, R.: Spectral mesh processing. In: Computer Graphics Forum, pp. 1–29 (2008)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2010 Springer-Verlag London Limited
About this chapter
Cite this chapter
Grady, L.J., Polimeni, J.R. (2010). Manifold Learning and Ranking. In: Discrete Calculus. Springer, London. https://doi.org/10.1007/978-1-84996-290-2_7
Download citation
DOI: https://doi.org/10.1007/978-1-84996-290-2_7
Publisher Name: Springer, London
Print ISBN: 978-1-84996-289-6
Online ISBN: 978-1-84996-290-2
eBook Packages: Computer ScienceComputer Science (R0)