Weighted spectral distribution

Comparison of graph structures is a frequently encountered problem across a number of problem domains. Comparing graphs requires a metric to discriminate which features of the graphs are considered important. The spectrum of a graph is often claimed to contain all the information within a graph, but the raw spectrum contains too much information to be directly used as a useful metric. In this paper we introduce a metric, the weighted spectral distribution, that improves on the raw spectrum by discounting those eigenvalues believed to be unimportant and emphasizing the contribution of those believed to be important. We use this metric to optimize the selection of parameter values for generating Internet topologies. Our metric leads to parameter choices that appear sensible given prior knowledge of the problem domain: the resulting choices are close to the default values of the topology generators and, in the case of the AB generator, fall within the expected region. This metric provides a means for meaningfully optimizing parameter selection when generating topologies intended to share structure with, but not match exactly, measured graphs.

[1]  Fan Chung,et al.  Spectral Graph Theory , 1996 .

[2]  Albert,et al.  Topology of evolving networks: local events and universality , 2000, Physical review letters.

[3]  A. Seary,et al.  Spectral methods for analyzing and visualizing networks : an introduction , 2000 .

[4]  Edwin R. Hancock,et al.  Structural Graph Matching Using the EM Algorithm and Singular Value Decomposition , 2001, IEEE Trans. Pattern Anal. Mach. Intell..

[5]  Michael I. Jordan,et al.  On Spectral Clustering: Analysis and an algorithm , 2001, NIPS.

[6]  Sugih Jamin,et al.  Inet-3.0: Internet Topology Generator , 2002 .

[7]  Thomas Erlebach,et al.  On the Spectrum and Structure of Internet Topology Graphs , 2002, IICS.

[8]  Christos Gkantsidis,et al.  Spectral analysis of Internet topologies , 2003, IEEE INFOCOM 2003. Twenty-second Annual Joint Conference of the IEEE Computer and Communications Societies (IEEE Cat. No.03CH37428).

[9]  Richard Mortier,et al.  Using Magpie for Request Extraction and Workload Modelling , 2004, OSDI.

[10]  Ronald R. Coifman,et al.  Diffusion Maps, Spectral Clustering and Eigenfunctions of Fokker-Planck Operators , 2005, NIPS.

[11]  B. Sudakov,et al.  Pseudo-random Graphs , 2005, math/0503745.

[12]  Stéphane Lafon,et al.  Diffusion maps , 2006 .

[13]  Dmitri Loguinov,et al.  Wealth-Based Evolution Model for the Internet AS-Level Topology , 2006, Proceedings IEEE INFOCOM 2006. 25TH IEEE International Conference on Computer Communications.

[14]  A. Jamakovic,et al.  On the relationship between the algebraic connectivity and graph's robustness to node and link failures , 2007, 2007 Next Generation Internet Networks.

[15]  S. Hanna.  Representation and generation of plans using graph spectra , 2007 .

[16]  Miguel Rio,et al.  Network topologies: inference, modeling, and generation , 2008, IEEE Communications Surveys & Tutorials.