ITTC Science of Communication Networks The University of Kansas EECS SCN Graph Spectra

Transcription

1 Science of Communication Networks The University of Kansas EECS SCN Graph Spectra Egemen K. Çetinkaya and James P.G. Sterbenz Department of Electrical Engineering & Computer Science Information Technology & Telecommunications Research Center The University of Kansas 11 February 2013 rev James P.G. Sterbenz

2 Graph Spectra Outline ST.1 Introduction and motivation ST.2 Background and related work ST.3 Topological dataset ST.4 Topology analysis ST.5 Conclusions and future work Primary reference: [CARS2012] 11 February 2013 KU EECS SCN Science of Nets Graph Spectra SCN-ST-2

3 Graph Spectra Introduction and Motivation ST.1 Introduction and motivation ST.2 Background and related work ST.3 Topological dataset ST.4 Topology analysis ST.5 Conclusions and future work 11 February 2013 KU EECS SCN Science of Nets Graph Spectra SCN-ST-3

4 Analysis of Internet Infrastructure Introduction and Motivation Modelling the Internet complexity of the Internet is non-trivial structural levels and protocols layers unrealistic synthetic topology models Steps towards understanding evolution of Internet physical topology modelling security and competitiveness hinders physical graphs we have the US fibre-optic map [KMI] how accurate is physical topology data? Utilisation of graph spectra for network analysis graph metrics exist; however, not suitable for different graph 11 February 2013 KU EECS SCN Science of Nets Graph Spectra SCN-ST-4

5 Analysis of Internet Infrastructure Introduction and Motivation [KU-TopView] 11 February 2013 KU EECS SCN Science of Nets Graph Spectra SCN-ST-5

6 Graph Spectra Background and Related Work ST.1 Introduction and motivation ST.2 Background and related work ST.3 Topological dataset ST.4 Topology analysis ST.5 Conclusions and future work 11 February 2013 KU EECS SCN Science of Nets Graph Spectra SCN-ST-6

7 Graph Spectra Electromagnetic Spectrum Distribution (or range) of em. waves according to wavelength 11 February 2013 KU EECS SCN Science of Nets Graph Spectra SCN-ST-7

8 Representation of Graphs Matrix Types Adjacency matrix A (G ) if nodes i and j are connected, a ij =1 else 0 Laplacian matrix L (G ) = D (G ) A (G ) D (G ) is diagonal matrix of node degrees L + (G ) = Q (G ) is signless Laplacian Normalised Laplacian matrix 1, if i = j and d i 0 L (G ) = -1 / d i d j, if v i and v j are adjacent 0, otherwise 11 February 2013 KU EECS SCN Science of Nets Graph Spectra SCN-ST-8

9 v 1 e 3 v 0 v 3 e 1 e 2 e 4 v 2 Matrix Types v 0 v 1 v 2 v 3 Examples v 0 v 1 v 2 v v 0 v 1 v 2 v 3 v 0 v 1 v 2 v A (G ) D (G ) v 0 v 1 v 2 v 3 v 0 v 1 v 2 v v 0 v 1 v 2 v 3 v 0 v 1 v 2 v v 0 v 1 v 2 v 3 v 0 v 1 v 2 v L (G) Q (G) L (G) 11 February 2013 KU EECS SCN Science of Nets Graph Spectra SCN-ST-9

10 Graph Spectra Eigenvalues and Eigenvectors Given a matrix M, eigenvalues λ, and eigenvectors x Eigenvalues and eigenvectors satisfy, M x = λ x Eigenvalues are roots of characteristic polynomial det ( M λ I ) for x 0 Spectrum of M is its eigenvalues and multiplicities multiplicity is the number of occurrences of an eigenvalue 11 February 2013 KU EECS SCN Science of Nets Graph Spectra SCN-ST-10

11 Graph Spectra Important Characteristics of NLS Eigenvalues of normalised Laplacian spectra (nls) {0 = λ 1 λ 2 λ n 2} number of 0s represent number of connected components Quasi-symmetric about 1 Spectral radius ρ(l ) : largest eigenvalue if ρ(l ) = 2, then the graph is bipartite closer to 2 means nearly bipartite λ 2 (L ) 1 for non-complete (non-full-mesh) graphs λ 2 (L ) 1/(2e D) 0 where e is the graph size (# of links) and D is graph diameter 11 February 2013 KU EECS SCN Science of Nets Graph Spectra SCN-ST-11

12 Graph Spectra Applications in Computer Science Expanders and combinatorial optimization Complex networks and the Internet Data mining Computer vision and pattern recognition Web search Statistical databases and social networks Quantum computing [CS2011] 11 February 2013 KU EECS SCN Science of Nets Graph Spectra SCN-ST-12

13 Graph Spectra Topological Dataset ST.1 Introduction and motivation ST.2 Background and related work ST.3 Topological dataset ST.4 Topology analysis ST.5 Conclusions and future work 11 February 2013 KU EECS SCN Science of Nets Graph Spectra SCN-ST-13

14 Topological Dataset US Interstate Highways Added 5 highways, 6 interchange nodes, 2 pendants [AASHTO] 11 February 2013 KU EECS SCN Science of Nets Graph Spectra SCN-ST-14

15 Topological Dataset PoP-Level Topologies AT&T and Sprint Included only 48 US contiguous states [Rocketfuel] 11 February 2013 KU EECS SCN Science of Nets Graph Spectra SCN-ST-15

16 Topological Dataset Fibre-Optic Routes AT&T and Sprint Included only 48 US contiguous states [KMI] 11 February 2013 KU EECS SCN Science of Nets Graph Spectra SCN-ST-16

17 Graph Spectra Topology Analysis ST.1 Introduction and motivation ST.2 Background and related work ST.3 Topological dataset ST.4 Topology analysis ST.5 Conclusions and future work 11 February 2013 KU EECS SCN Science of Nets Graph Spectra SCN-ST-17

18 Topology Analysis of Graphs Metrics We started our analysis with baseline networks star, linear, tree, ring, grid, toroid, mesh order (number of nodes) of the baseline graphs: 10 and 100 Average degree is same for star, linear, tree do these graphs share same structural properties? Clustering coefficient is to coarse for baseline graphs except for mesh which the value is 1, rest is 0 Algebraic connectivity a (G ): second smallest eigenvalue of the Laplacian matrix a (G ) = 0.1 for n=10 (linear) and n=100 (grid) not very useful for comparing different order graphs 11 February 2013 KU EECS SCN Science of Nets Graph Spectra SCN-ST-18

19 Topological Analysis via Metrics Baseline Networks with n=10 Topology Star Linear Tree Ring Grid Toroid Mesh Number of nodes Number of links Maximum degree Average degree Degree assortativity Node closeness Clustering coefficient Algebraic connectivity Network diameter Network radius Average hop count Node betweenness Link betweenness February 2013 KU EECS SCN Science of Nets Graph Spectra SCN-ST-19

20 Topological Analysis via Metrics Baseline Networks with n=100 Topology Star Linear Tree Ring Grid Toroid Mesh Number of nodes Number of links Maximum degree Average degree Degree assortativity Node closeness Clustering coefficient Algebraic connectivity Network diameter Network radius Average hop count Node betweenness Link betweenness February 2013 KU EECS SCN Science of Nets Graph Spectra SCN-ST-20

21 Topology Analysis of Graphs Spectra Metrics are useful, but have limitations We investigate spectra of graphs in particular normalised Laplacian spectrum since it is normalised, the eigenvalues range [0,2] We calculate: RF: relative frequency [BJ2009] RCF: relative cumulative frequency [VHE2002] Eigenvalue of 2 indicates how bipartite a graph is Eigenvalue 1 multiplicity indicates node duplications nodes having similar neighbours Spectrum is symmetric around 1 11 February 2013 KU EECS SCN Science of Nets Graph Spectra SCN-ST-21

22 Graph Spectrum RF for Baseline Networks with n=100 RF of eigenvalue multiplicities is noisy mesh and star graphs look similar (except λ = 2 eigenvalue) 11 February 2013 KU EECS SCN Science of Nets Graph Spectra SCN-ST-22

23 Graph Spectrum RCF for Baseline Networks with n=100 λ max = 2 means graph is bipartite 11 February 2013 KU EECS SCN Science of Nets Graph Spectra SCN-ST-23

24 Graph Spectrum Full Mesh Networks As n ; λ 1 for full mesh graphs higher multiplicity at a point might be crucial for resilience 11 February 2013 KU EECS SCN Science of Nets Graph Spectra SCN-ST-24

25 Topology Analysis of Real Networks Metrics and Spectra Communication and transportation networks studied Metrics indicate: physical topologies are closer to transportation network difference between physical and logical topologies higher number of nodes in physical topologies rich connectivity in logical topologies Normalised Laplacian spectrum indicates: similar conclusions, visually helpful in reliable network design Spectral properties can help resilience evaluation 11 February 2013 KU EECS SCN Science of Nets Graph Spectra SCN-ST-25

26 Topological Analysis via Metrics Real Networks Topology Sprint Physical Sprint Logical AT&T Physical AT&T Logical US Highways Number of nodes Number of links Maximum degree Average degree Degree assortativity Node closeness Clustering coefficient Algebraic connectivity Network diameter Network radius Average hop count Node betweenness Link betweenness February 2013 KU EECS SCN Science of Nets Graph Spectra SCN-ST-26

27 Graph Spectrum RF for Real Networks RF floor is noisy to extract useful information 11 February 2013 KU EECS SCN Science of Nets Graph Spectra SCN-ST-27

28 Graph Spectrum RCF for Real Networks Spectrum of physical topologies resemble motorways 11 February 2013 KU EECS SCN Science of Nets Graph Spectra SCN-ST-28

29 Spectral Properties of Real Networks ctgd vs. Algebraic Conn. & Spectral Radius Topology ctgd ctgd Rank a(g) a(g) Rank Level AboveNet Exodus EBONE Tiscali Sprint Verio VSNL GEANT2 phys February 2013 KU EECS SCN Science of Nets Graph Spectra SCN-ST-29 ρ(l) ρ(l) Rank AT&T Telstra AT&T phys Sprint phys

30 Graph Spectrum RCF for Communication Networks Spectrum of physical topologies resemble motorways Level 3 is richly connected and have small spectral radius 11 February 2013 KU EECS SCN Science of Nets Graph Spectra SCN-ST-30

31 Graph Spectra Conclusions and Future Work ST.1 Introduction and motivation ST.2 Background and related work ST.3 Topological dataset ST.4 Topology analysis ST.5 Conclusions and future work 11 February 2013 KU EECS SCN Science of Nets Graph Spectra SCN-ST-31

32 Analysis of Internet Infrastructure Conclusions and Future Work Physical topologies resemble motorways known: but not rigorously studied grid-like structures The normalised Laplacian spectrum is powerful spectral radius indicates bipartiteness λ = 1 multiplicity indicates duplicates (i.e. star-like structures) Future work: study other physical critical infrastructures railways, power grid, pipelines investigate metrics and relationship to resiliency/connectivity 11 February 2013 KU EECS SCN Science of Nets Graph Spectra SCN-ST-32

33 Analysis of Internet Infrastructure Review of Graph Spectra All but structural graphs have same nodes 11 February 2013 KU EECS SCN Science of Nets Graph Spectra SCN-ST-33

34 Graph Theory References and Further Reading [CARS2012] Egemen K. Çetinkaya, Mohammed J.F. Alenazi, Justin P. Rohrer, and James P.G. Sterbenz, Topology Connectivity Analysis of Internet Infrastructure Using Graph Spectra, in Proc. of the IEEE/IFIP RNDM, St. Petersburg, October 2012 [AASHTO] American Association of State Highway and Transportation Officials, Guidelines for the Selection of Supplemental Guide Signs for Traffic Generators Adjacent to Freeways, Washington, D.C., 2001 [KMI] KMI Corporation, North American Fiberoptic Long-haul Routes Planned and in Place, 1999 [KU-TopView] [Rocketfuel] 11 February 2013 KU EECS SCN Science of Nets Graph Spectra SCN-ST-34

35 Graph Spectra References and Further Reading [C1994] Fan R. K. Chung, Spectral Graph Theory, American Mathematical Society, 1994 [M2011] Piet van Mieghem, Graph Spectra for Complex Networks, Cambridge University Press, 2011 [BH2012] Andries E. Brouwer and Willem H. Haemers, Spectra of Graphs, Springer, 2012 [CRS2010] Dragoš Cvetković, Peter Rowlinson, and Slobodan Simić, An Introduction to the Theory of Graph Spectra, London Mathematical Society, 2010 [B1993] Norman Biggs, Algebraic Graph Theory, Cambridge University Press, February 2013 KU EECS SCN Science of Nets Graph Spectra SCN-ST-35

36 Graph Spectra References and Further Reading [GN2006] M. T. Gastner and M. E. Newman, The spatial structure of networks,the European Physical Journal B, vol. 49, no. 2, January 2006, pp [JWM2006] Almerima Jamaković, Huijuan Wang, and Piet van Mieghem, Topological Characteristics of the Dutch Road Infrastructure, in Infrastructure Reliability Seminar, Delft, June February 2013 KU EECS SCN Science of Nets Graph Spectra SCN-ST-36

37 Graph Spectra References and Further Reading [CS2011] Dragoš Cvetković and Slobodan Simić, Graph spectra in Computer Science, Linear Algebra and its Applications, vol. 434, no. 6, March 2011, pp [VHE2002] Danica Vukadinović, Polly Huang, and Thomas Erlebach, On the Spectrum and Structure of Internet Topology Graphs, in Proc. of the IICS, June 2002, pp [BJ2009] Anirban Banerjee and Jürgen Jost, Spectral Characterization of Network Structures and Dynamics, Dynamics On and Of Complex Networks, 2009, pp February 2013 KU EECS SCN Science of Nets Graph Spectra SCN-ST-37

38 End of Foils 11 February 2013 KU EECS SCN Science of Nets Graph Spectra SCN-ST-38