ITTC © James P.G. Sterbenz Science of Communication Networks · KU EECS SCN – Science of Nets – Graph Spectra . SCN-ST-12 . Graph Spectra . Applications in Computer Science •Expanders

© James P.G. Sterbenz ITTC

11 February 2013 © 2002-2013 James P.G. Sterbenz rev. 13.0

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

[email protected]

http://www.ittc.ku.edu/~jpgs/courses/scinets


11 February 2013 KU EECS SCN – Science of Nets – Graph Spectra SCN-ST-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]



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



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



Analysis of Internet Infrastructure Introduction and Motivation

• [KU-TopView] https://www.ittc.ku.edu/resilinets/maps



Graph Spectra Background and Related Work




Graph Spectra Electromagnetic Spectrum

• Distribution (or range) of em. waves according to wavelength • http://csep10.phys.utk.edu/astr162/lect/light/spectrum.html



Representation of Graphs Matrix Types

• Adjacency matrix A (G ) – if nodes i and j are connected, aij=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 di≠ 0 L (G ) = -1 / √didj , if vi and vj are adjacent 0 , otherwise



Matrix Types Examples

v0

v1

v2

v3

e1

e2 e4

e3

1 0 -0.5 -0.4 0 1 0 -0.6 -0.5 0 1 -0.4

-0.4 -0.6 -0.4 1

v0 v1 v2 v3

v0 v1 v2 v3

0 0 1 1 0 0 0 1 1 0 0 1 1 1 1 0

v0 v1 v2 v3

v0 v1 v2 v3

2 0 0 0 0 1 0 0 0 0 2 0 0 0 0 3

v0 v1 v2 v3

v0 v1 v2 v3

2 0 -1 -1 0 1 0 -1 -1 0 2 -1 -1 -1 -1 3

v0 v1 v2 v3

v0 v1 v2 v3

A (G ) D (G )

L (G ) L (G )

2 0 1 1 0 1 0 1 1 0 2 1 1 1 1 3

v0 v1 v2 v3

v0 v1 v2 v3

Q (G )



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



Graph Spectra Important Characteristics of NLS

• Eigenvalues of normalised Laplacian spectra (nls) – {0 = λ1 ≤ λ2 ≤ … ≤ λn ≤ 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

• λ2(L ) ≤ 1 for non-complete (non-full-mesh) graphs – λ2(L ) ≥ 1/(2e D) ≥ 0

• where e is the graph size (# of links) and D is graph diameter



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]



Graph Spectra Topological Dataset




Topological Dataset US Interstate Highways

• Added 5 highways, 6 interchange nodes, 2 pendants [AASHTO]



Topological Dataset PoP-Level Topologies – AT&T and Sprint

• Included only 48 US contiguous states [Rocketfuel]



Topological Dataset Fibre-Optic Routes – AT&T and Sprint

• Included only 48 US contiguous states [KMI]



Graph Spectra Topology Analysis




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



Topological Analysis via Metrics Baseline Networks with n=10

Topology Star Linear Tree Ring Grid Toroid Mesh

Number of nodes 10 10 10 10 10 10 10

Number of links 9 9 9 10 13 15 45

Maximum degree 9 2 3 2 3 3 9

Average degree 1.8 1.8 1.8 2 2.6 3 9

Degree assortativity -1 -0.13 -0.53 1 0.28 1 1

Node closeness 0.58 0.29 0.37 0.36 0.44 0.53 1

Clustering coefficient 0 0 0 0 0 0 1

Algebraic connectivity 1 0.1 0.18 0.38 0.38 1.38 10

Network diameter 2 9 5 5 5 3 1

Network radius 1 5 3 5 3 3 1

Average hop count 1.8 3.67 2.82 2.78 2.3 1.89 1

Node betweenness 36 20 26 8 11 4 0

Link betweenness 9 25 24 13 12 6 1



Topological Analysis via Metrics Baseline Networks with n=100

Topology Star Linear Tree Ring Grid Toroid Mesh

Number of nodes 100 100 100 100 100 100 100

Number of links 99 99 99 100 180 200 4950

Maximum degree 99 2 3 2 4 4 99

Average degree 1.98 1.98 1.98 2 3.6 4 99

Degree assortativity -1 -0.01 -0.34 1 0.57 1 1

Node closeness 0.51 0.03 0.13 0.04 0.15 0.20 1

Clustering coefficient 0 0 0 0 0 0 1

Algebraic connectivity 1 0.001 0.01 0.004 0.1 0.38 100

Network diameter 2 99 12 50 18 10 1

Network radius 1 50 6 50 10 10 1

Average hop count 1.98 33.7 7.8 25.3 6.67 5.05 1

Node betweenness 4851 2450 3068 1201 616 200 0

Link betweenness 99 2500 2496 1250 341 200 1



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



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)



Graph Spectrum RCF for Baseline Networks with n=100

• λmax = 2 means graph is bipartite



Graph Spectrum Full Mesh Networks

• As n → ∞; λ → 1 for full mesh graphs – higher multiplicity at a point might be crucial for resilience



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



Topological Analysis via Metrics Real Networks

Topology Sprint Physical

Sprint Logical

AT&T Physical

AT&T Logical

US Highways

Number of nodes 263 28 361 107 400

Number of links 311 76 466 140 540

Maximum degree 6 14 7 23 7

Average degree 2.37 5.43 2.58 2.62 2.7

Degree assortativity -0.17 -0.23 -0.16 -0.4 0.11

Node closeness 0.07 0.48 0.08 0.3 0.08

Clustering coefficient 0.03 0.41 0.05 0.09 0.05

Algebraic connectivity 0.0053 0.6844 0.0061 0.1324 0.0059

Network diameter 37 4 37 6 40

Network radius 19 2 19 3 21

Average hop count 14.78 2.19 13.57 3.38 13.34

Node betweenness 11159 100 15970 2168 22798

Link betweenness 9501 27 14270 661 18585



Graph Spectrum RF for Real Networks

• RF floor is noisy to extract useful information



Graph Spectrum RCF for Real Networks

• Spectrum of physical topologies resemble motorways



Spectral Properties of Real Networks cTGD vs. Algebraic Conn. & Spectral Radius Topology cTGD cTGD

Rank a(G) a(G)

Rank ρ(L) ρ(L)

Rank

Level 3 0.4494 1 0.9518 3 1.5033 2

AboveNet 0.4386 2 0.9645 2 1.4978 1

Exodus 0.3617 3 1.0083 1 1.7408 4

EBONE 0.3113 4 0.6477 4 1.7335 3

Tiscali 0.2641 5 0.5255 5 1.7470 5

Sprint 0.2407 6 0.3817 6 1.7853 6

Verio 0.2009 7 0.2448 8 1.8463 7

VSNL 0.1783 8 0.3402 7 1.9053 9

GEANT2 phys. 0.1668 9 0.1515 9 1.8518 8

AT&T 0.1446 10 0.1324 10 1.9127 10

Telstra 0.0941 11 0.0454 11 1.9797 11

AT&T phys. 0.0348 12 0.0061 12 1.9892 13

Sprint phys. 0.0307 13 0.0053 13 1.9839 12



Graph Spectrum RCF for Communication Networks

• Spectrum of physical topologies resemble motorways – Level 3 is richly connected and have small spectral radius



Graph Spectra Conclusions and Future Work




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



Analysis of Internet Infrastructure Review of Graph Spectra

• All but structural graphs have same nodes



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] https://www.ittc.ku.edu/resilinets/maps • [Rocketfuel] http://www.cs.washington.edu/research/networking/rocketfuel



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, 1993




• [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. 247 – 252

• [JWM2006] Almerima Jamaković, Huijuan Wang, and Piet van Mieghem, “Topological Characteristics of the Dutch Road Infrastructure”, in Infrastructure Reliability Seminar , Delft, June 2006




• [CS2011] Dragoš Cvetković and Slobodan Simić, “Graph spectra in Computer Science”, Linear Algebra and its Applications , vol. 434, no. 6, March 2011, pp. 1545 – 1562

• [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. 83 – 95

• [BJ2009] Anirban Banerjee and Jürgen Jost, “Spectral Characterization of Network Structures and Dynamics”, Dynamics On and Of Complex Networks , 2009, pp. 117 – 132



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ITTC © James P.G. Sterbenz Science of Communication Networks · KU EECS SCN – Science of Nets – Graph Spectra . SCN-ST-12 . Graph Spectra . Applications in Computer Science •Expanders

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