Abstract
We show that the largest eigenvalues of graphs whose highest degrees are Zipf-like distributed with slope a are distributed according to a power law with slope α/2. This follows as a direct and almost certain corollary of the degree power law. Our result has implications for the singular value decomposition method in information retrieval.
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
Achlioptas, D., Fiat, A., Karlin, A. and McSherry, F., “Web Search via Hub Synthesis”, Proceedings of the 42nd Annual IEEE Symposium on Foundations of Computer Science, (FOCS 2001), pp 500–509.
Aiello, W., Chung, F.R.K. and Lu, L., “A random graph model for power law graphs”, Proceedings of the Thirtysecond Annual ACM Symposium on Theory of Computing (STOC 2000), pp 171–180.
Aiello, W., Chung, F.R.K. and Lu, L., “Random Evolution in Massive Graphs”, Proceedings of the Fourty-Second Annual IEEE Symposium on Foundations of Computer Science, (FOCS 2001), pp. 510–519.
Azar, Y., Fiat, A., Karlin, A., McSherry, F. and J. Saia, “Spectral Analysis for Data Mining”, Proceedings of the Thirty-Third Annual ACM Symposium on Theory of Computing, (STOC 2001), pp 619–626.
Barabási, A.-L. and Albert, R., “Emergence of scaling in random graphs”, Science 286 (1999), pp 509–512.
Bollobás, B., Riordan, O., Spencer, J. and Tusnády, G., “The degree sequence of a scale-free random graph process”, Random Structures and Algorithms,Volume 18, Issue 3, 2001, pp 279–290.
Broder, A., Kumar, R., Maghoul, F., Raghavan, P., Rajagopalan, S., Stata, R., Tomikns, A. and Wiener, J., “Graph structure in the Web”, Proc. 9th International World Wide Web Conference (WWW9)/Computer Networks, 33(1-6), 2000, pp. 309–320.
Chung, F.R.K. and Lu, L., “Connected components in random graphs with given degree sequences”, Available at http://www.math.ucsd.edu/fan.
Cooper, C. and Frieze, A., A general model for web graphs, Proceedings of ESA, 2001, pp 500–511.
Dill, S., Kumar, R., McCurley, K., Rajagopalan, S., Sivakumar, D. and Tomkins, A., “Self-similarity in the Web”, In Proceedings of International Conference on Very Large Data Bases, Rome, 2001, pp. 69–78.
Dorogovtsev, S.N. and Mendes, J.F.F., “Evolution of Networks”, Advances in Physics, to appear (2002). Available at http://www.fc.up.pt/fis/sdorogov.
Erdös. P. and Rényi, A., “On the Evolution of Random Graphs”, Publications of the Mathematical Institute of the Hungarian Academy of Science 5, (1960), pp17–61.
Fabrikant, A., Koutsoupias, E. and Papadimitriou, C.H. “Heuristically Optimized Tradeoffs”, Available at http://www.cs.berkeley.edu/christos.
Faloutsos, M., Faloutsos, P. and Faloutsos, C., “On Power-law Relationships of the Internet Topology”, In Proceedings Sigcomm 1999, pp 251–262.
Farkas, I.J., Derényi, I., Barabási, A.L. and Vicsek, T., “Spectra of Real-World Graphs: Beyond the Semi-Circle Law”, e-print cond-mat/0102335.
Goh, K.I., Kahng, B. and Kim, D., “Spectra and eigenvectors of scale-free networks”, Physical Review E., Vol 64, 2001.
Husbands, P., Simon, H. and Ding, C., “On the use of the Singular Value Decomposition for Text Retrieval”, 1st SIAM Computational Information Retrieval Workshop, October 2000, Raleigh, NC.
Jin, C., Chen, Q. and Jamin, S., “Inet: Internet Topology Generator”, University of Michigan technical Report, CSE-TR-433-00. Available at http://irl.eecs.umich.edu/jamin.
Kleinberg, J., “Authoritative sources in a hyperlinked environment”, Proc. 9th ACM-SIAM Symposium on Discrete Algorithms, 1998. Extended version in Journal of the ACM 46 (1999).
Kumar, R., Rajagopalan, S., Sivakumar, D. and Tomkins, A., “Trawling the web for emerging cyber-communities”, WWW8/Computer Networks, Vol. 31, No 11-16, 1999, pp. 1481–1493.
Kumar, R., Raghavan, P., Rajagopalan, S., Sivakumar, D., Tomkins, A. and Upfal, E., “Stochastic models for the Web graph”, Proceedings of the 41st IEEE Symposium on Foundations of Computer Science, (FOCS 2000), pp 57–65.
Lovász, L., Combinatorial Problems and Exercises, North-Holland Publishing Co., Amsterdam-New York, 1979.
Medina, A., Lakhina, A., Matta, I. and Byers, J., BRITE: Universal Topology Generation from a User’s Perspective. Technical Report BUCS-TR2001-003, Boston University, 2001. Available at http://www.cs.bu.edu/brite/publications.
Medina, A., Matta, I. and Byers, J., “On the origin of power laws in Internet topologies”, ACM Computer Communication Review, vol. 30, no. 2, pp. 18–28, Apr. 2000.
Gkantsidis, C., Mihail, M. and Zegura, E., “Spectral Analysis of Internet Topologies”, Georgia Institute of Technology, Technical Report GIT-CC-0710.
Palmer, C. and Steffan, J., “Generating network topologies that obey power laws”, In Proceedings of Globecom 2000.
Papadimitriou, C.H., Raghavan, P., Tamaki, H. and Vempala, S., “Latent Semantic Indexing: A Probabilistic Analysis”, Journal of Computer and System Sciences, 61, 2000, pp. 217–235.
Spencer, J., Ten Lecture Notes on the Probabilistic Method, SIAM Lecture Notes, Philadelphia, 1987.
Stewart, G. W. and Sun, J., Matrix Perturbation Theory, Academic Press, 1990.
Wilkinson, J. H., The Algebraic Eigenvalue Problem, Numerical Mathematics and Scientific Computation, Oxford University Press, 1965.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2002 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Mihail, M., Papadimitriou, C. (2002). On the Eigenvalue Power Law. In: Rolim, J.D.P., Vadhan, S. (eds) Randomization and Approximation Techniques in Computer Science. RANDOM 2002. Lecture Notes in Computer Science, vol 2483. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45726-7_20
Download citation
DOI: https://doi.org/10.1007/3-540-45726-7_20
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-44147-2
Online ISBN: 978-3-540-45726-8
eBook Packages: Springer Book Archive