Abstract
In this paper, we show how to quantify graph complexity in terms of the normalized entropies of convex Birkhoff combinations. We commence by demonstrating how the heat kernel of a graph can be decomposed in terms of Birkhoff polytopes. Drawing on the work of Birkhoff and von Neuman, we next show how to characterise the complexity of the heat kernel. Finally, we provide connections with the permanent of a matrix, and in particular those that are doubly stochastic. We also include graph embedding experiments based on polytopal complexity, mainly in the context of Bioinformatics (like the clustering of protein-protein interaction networks) and image-based planar graphs.
Chapter PDF
Similar content being viewed by others
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.
References
Robles-Kelly, A., Hancock, E.R.: A riemannian approach to graph embedding. Pattern RecognitionĀ (40), 1042ā1056 (2007)
Luo, B., Wilson, R.C., Hancock, E.: Spectral embedding of graphs. Pattern RecognitionĀ (36), 2213ā2223 (2003)
Shokoufandeh, A., Dickinson, S., Siddiqi, K., Zucker, S.: Indexing using a spectral encoding of topological structure. In: IEEE ICPR, pp. 491ā497
Torsello, A., Hancock, E.: Learning shape-classes using a mixture of tree-unions. IEEE Trans. on PAMIĀ 28(6), 954ā967 (2006)
Lozano, M., Escolano, F.: Protein classification by matching and clustering surface graphs. Pattern RecognitionĀ 39(4), 539ā551 (2006)
Kƶrner, J.: Coding of an information source having ambiguous alphabet and the entropy of graphs. In: Trans. of the 6th Prague Conference on Information Theory, pp. 411ā425 (1973)
Escolano, F., Hancock, E., Lozano, M.: Birkhoff polytopes, heat kernels, and graph embedding. In: ICPR (2008)
Birkhoff, G.D.: Tres observaciones sobre el algebra lineal. Universidad Nacional de Tucuman Revista, Serie AĀ 5, 147ā151 (1946)
Chang, C., Chen, W., Huang, H.: On service guarangees for input buffered crossbar switches: A capacity decomposition approach by birkhoff and von neumann. In: IEEE IWQoS, pp. 79ā86 (1998)
Kondor, R.I., Lafferty, J.: Diffusion kernels on graphs and other discrete structures. In: Proc. ICML (2002)
Mirsky, L.: Proofs of two theorems on doubly stochastic matrices. Proc. Amer. Math. Soc.Ā 9, 371ā374 (1958)
Jerrum, M., Sinclair, A., Vigoda, E.: A polynomial-time approximation algorithm for the permanent of a matrix with nonnegative entries. Journal of the ACMĀ 51(4), 671ā697 (2004)
Qiu, H., Hancock, E.: Graph simplification and matching using conmute times. Pattern Recognition (40), 2874ā2889 (2007)
von Mering, C., Huynen, M., Jaeggi, D., Schmidt, S., Bork, P., Snell, B.: String: a database of predicted functional associations. Nuc. Acid Res.Ā 31, 258ā261 (2003)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
Ā© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Escolano, F., Hancock, E.R., Lozano, M.A. (2008). Polytopal Graph Complexity, Matrix Permanents, and Embedding. In: da Vitoria Lobo, N., et al. Structural, Syntactic, and Statistical Pattern Recognition. SSPR /SPR 2008. Lecture Notes in Computer Science, vol 5342. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-89689-0_28
Download citation
DOI: https://doi.org/10.1007/978-3-540-89689-0_28
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-89688-3
Online ISBN: 978-3-540-89689-0
eBook Packages: Computer ScienceComputer Science (R0)