Abstract
In this paper we make a characteristic polynomial analysis on hypergraphs for the purpose of clustering. Our starting point is the Ihara zeta function [8] which captures the cycle structure for hypergraphs. The Ihara zeta function for a hypergraph can be expressed in a determinant form as the reciprocal of the characteristic polynomial of the adjacency matrix for a transformed graph representation. Our hypergraph characterization is based on the coefficients of the characteristic polynomial, and can be used to construct feature vectors for hypergraphs. In the experimental evaluation, we demonstrate the effectiveness of the proposed characterization for clustering hypergraphs.
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
Agarwal, S., Branson, K., Belongie, S.: Higher-order learning with graphs. In: ICML (2006)
Bretto, A., Cherifi, H., Aboutajdine, D.: Hypergraph imaging: an overview. Pattern Recognition 35(3), 651–658 (2002)
Bunke, H., Dickinson, P., Neuhaus, M., Stettler, M.: Matching of hypergraphs — algorithms, applications, and experiments. Studies in Computational Intelligence 91, 131–154 (2008)
Chung, F.: The laplacian of a hypergraph. AMS DIMACS Series in Discrete Mathematics and Theoretical Computer Science 10, 21–36 (1993)
Li, W., Sole, P.: Spectra of regular graphs and hypergraphs and orthogonal polynomials. European Journal of Combinatorics 17, 461–477 (1996)
Ren, P., Wilson, R.C., Hancock, E.R.: Spectral embedding of feature hypergraphs. In: da Vitoria Lobo, N., Kasparis, T., Roli, F., Kwok, J.T., Georgiopoulos, M., Anagnostopoulos, G.C., Loog, M. (eds.) S+SSPR 2008. LNCS, vol. 5342, pp. 308–317. Springer, Heidelberg (2008)
Rodriguez, J.A.: On the laplacian eigenvalues and metric parameters of hypergraphs. Linear and Multilinear Algebra 51, 285–297 (2003)
Storm, C.K.: The zeta function of a hypergraph. Electronic Journal of Combinatorics 13 (2006)
Wong, A.K.C., Lu, S.W., Rioux, M.: Recognition and shape synthesis of 3d objects based on attributed hypergraphs. IEEE Transactions on Pattern Analysis and Machine Intelligence 11(3), 279–290 (1989)
Zass, R., Shashua, A.: Probabilistic graph and hypergraph matching. In: CVPR (2008)
Zhou, D., Huang, J., Scholkopf, B.: Learning with hypergraphs: Clustering, classification, and embedding. In: NIPS (2007)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Ren, P., Aleksić, T., Wilson, R.C., Hancock, E.R. (2009). Hypergraphs, Characteristic Polynomials and the Ihara Zeta Function. In: Jiang, X., Petkov, N. (eds) Computer Analysis of Images and Patterns. CAIP 2009. Lecture Notes in Computer Science, vol 5702. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03767-2_45
Download citation
DOI: https://doi.org/10.1007/978-3-642-03767-2_45
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-03766-5
Online ISBN: 978-3-642-03767-2
eBook Packages: Computer ScienceComputer Science (R0)