1 Spectral Analysis of Power-Law Graphs and its Application to Internet Topologies Milena Mihail Georgia Tech

2 The Internet Phenomenon Routers WWW P2P Open Decentralized Dynamic Market Competition Security, Privacy Paradigm Shift : Networks as Artifacts that we construct. Networks as Phenomena that we study !

3 Internet Performance Congestion (TCP/IP, ) Stability (Game Theory, ) Scalability (TCP ? Moore’s Law ?) WWW, P2P : Index, Search Van Jacobson 88 Kelly 99 (Kleinberg 97, Google 98)

4 Required Data & Models Routers WWW P2P Connectivity Capacity Traffic / Demand Internet Models, such as GT-ITM, Brite, Inet, for Analytic & Simulation based studies : How do elements organize ?

5 The Internet Phenomenon Routers WWW P2P Open Decentralized Dynamic Market Competition Security, Privacy Paradigm Shift : Networks as Artifacts that we construct. Networks as Phenomena that we study !

6 Level of Autonomous Systems SprintAT&T Georgia Tech CNN Topology data from BGP routing tables, collected by NLANR, looking glass -U. Oregon Decentralized Routing !

7 The AS Graph ~14K nodes in 2002 ( ~2K nodes in 1997) ~30K links in 2002 Georgia Tech CNN AT&T Sprint

8 The Directed AS Graph Georgia Tech CNN AT&T Sprint Peering Relationships : Customer – Provider Peers Gao 00, Subramanian et al 01 Five Tier Hierarchy Subramanian et al 01

9 The Real AS Graph CAIDA

10 Degree-Frequency Power Law Faloutsos et al degree frequency 2100

11 Rank-Degree Power Law rank degree Faloutsos et al 99 UUNET Sprint C&WUSA AT&T BBN

12 Eigenvalue Power Law rank eigenvalue Faloutsos et al 99

13 Eigenvalue Power Law rank eigenvalue Faloutsos et al 99 UUNET Sprint C&WUSA AT&T BBN

14 Heavy Tailed Degree Distribution Departure from standard Internet Models such as Waxman, Transit-Stub Zegura et al 95 Models and techniques must be revisited Degrees not concentrated around mean Highly irregular graphs Departure from Erdos-Renyi Sharp concentration around mean, Exponential Tails

15 Power Law Graphs Which primitives drive their evolution ? Preferential attachment, Barabasi 99, Bollobas et al 00 Multiobjective Optimization, Carlson & Doyle 00, Papadimitriou 02 What are their structural properties ? Hierarchy, Subramanian et al 01, Govindran et al 02 Clustering

16 Spectral Analysis of Matrices Examines eigenvalues and eigenvectors. Useful analogy to signal processing. All eigenvectors form a basis (complete representation). Focus on large eigenvalues and the corresponding eigenvectors. Pervasive in Algebra : Representation Theory Algorithms : Markov chain sampling Complexity : Expanders, Pseudorandomness Datamining, Information Retrieval Highly technical application specific adaptations

17 0. Spectral primitives : eigenvalues and eigenvectors. 1. Eigenvectors ~ Significance, hence HIERARCHY ( capacity / load ) 2. Eigenvectors ~ Clustering CLUSTERING impacts CONGESTION (1.) and (2.) use normalization preprocessing of the data 3. On Eigenvectors of Eigenvalue Power Law 4. Further Directions Outline of Results in this Talk

18 Eigenvectors & Eigenvalues A = Axx=

19 Matrix as a Linear Transformation A = Matrix as a Linear Transformation

Step 0 Step 1 Step 2 Step 3

=

22 Stochastic Normalization 0 1/3 0 1/3 1/3 0 1/3 0 1/ /3 0 1/3 0 1/3 0 1/3 1/ /3 0 1/3 0 1/3 1/3 0 1/ /3 1/ /3 A = Axx=

23 The Random Walk Eigenvalues between 1 and –1 ( 1 and 0 also easy) / /91/9 2/9

24 In undirected graphs the weights of the principal eigenvector are proportional to degrees. 1. Principal Eigenvector ~ Significance Principal Eigenvector is Stationary Distribution, corresponding to to = 1 1/6 3/16 2/16 3/16 2/16 3/16

25 1. Principal Eigenvector ~ Significance In directed graphs the weights of the principal eigenvector can vary way beyond degrees. 10^-42*10^-45*10^

26 1. Hierarchy from Principal Eigenvector of Directed AS Graph Significance by High Degree Significance by Significant Peers and Customers Add 5% prob. Uniform jump to avoid sinks In WWW : Google’s pagerank

27 1. Principal Eigenvector Ranking

28 Eigenvector vs Five Tiers

29 1. Principal Eigenvector Ranking

30 0. Spectral primitives : eigenvalues and eigenvectors. 1. Eigenvectors ~ Significance, hence HIERARCHY ( capacity / load ) 2. Eigenvectors ~ Clustering CLUSTERING impacts CONGESTION (1.) and (2.) use normalization preprocessing of the data 3. On Eigenvectors of Eigenvalue Power Law 4. Further Directions Outline of Results in this Talk

31 2. Eigenvectors ~ Clustering 1/6 = 1

32 2. Eigenvectors ~ Clustering 1/6

33 2. Eigenvectors ~ Clustering 1/

34 2. Eigenvectors ~ Clustering 1/6

35 2. Eigenvectors ~ Clustering 1/6

36 2. Eigenvectors ~ Clustering 1/ ~ ~ ~ ~ ~ ~ ~~ ~~ ~~ Matrix Perturbation Theory

37 Spectral Filtering n K+1 k =1 > > >>> > Find clusters in most positive and most negative ends of eigenvectors associated with large eigenvalues. Heuristic : ( Kleinberg 97, Fiat et al 01 )

38 2. Eigenvectors ~ Clustering Weight of eigenvector k rank n K+1 k =1 > > >>> >

An Example of a Cluster

44 Additional Matrices Similarity Matrix A*A^T, where A is directed AS graph. Complete and Pruned AS topology. In all cases prune leaves of very big degree nodes. Necessary frequency normalization. Clusters consistent and evolving over time. Synthetic Internet topologies have much weaker clustering properties.

45 Clustering and Congestion Assume 1 unit of traffic between each pair of ASes in each direction. Route traffic in the graph (like BGP). Compute # of connections using each link. This is a measure of congestion.

46 Effect of intra-cluster and inter-cluster traffic to most congested link InternetInet Internet Inet 0%100% 0%100% 20%91.5%97.7%20%126%109% 40%83.1%95.3%40%153%117% 60%74.2%92.9%60%172%128% 80%65.7%90.8%80%191%136% 100%57.3%88.5%100%207%142%

47 Outline of Results in this Talk 1. Eigenvectors ~ Significance, hence HIERARCHY ( capacity / load ) 2. Eigenvectors ~ Clustering CLUSTERING impacts CONGESTION (1) and (2) used normalization preprocessing of the data Normalization preprocessing of data is necessary. 3. Eigenvectors of Eigenvalue Power Law LOCALIZED 4. Further Directions

48 Which Eigenvectors correspond to Eigenvalue Power Law ? rank eigenvalue Faloutsos et al 99 UUNET Sprint C&WUSA AT&T BBN

49 Large Degrees, or“Stars” of AS Graph Dominate Spectrum of Adjacency Matrix, Prior to Normalization

50 Principal Eigenvector of a Star d

51 Disjoint Stars

52 “Mostly” Disjoint Stars Proof By Matrix Perturbation Theory, Spectral Graph Theory

53 3. Explanation of Eigenvalue Power Law Theorem : Random graphs whose largest degrees are, in expectation, d_1 > d_2 > … > d_k, d_j ~ j ^ -a have largest eigenvalues sharply concentrated around _j ~ j ^ -b for j = 1,…,k, and corresponding eigenvectors localized on corresponding largest degrees, with very high probability.

54 Summary 0. First Spectral Analysis on Internet Topologies. 1. PRINCIPAL EIGENVECTOR implies HIERARCHY 2. EIGENVECTORS of LARGE EIGENVALUES imply CLUSTERING 3. CLUSTERING impacts CONGESTION 4. Defined Intra-cluster and Inter-cluster Traffic. 5. Introduced Heady Tailed Specific Normalization Preprocessing. 6. Explained Eigenvalue Power Law. (1)Through (5) with C. Gkantsidis and E. Zegura (6) with C. Papadimitriou

55 Further Directions  How does congestion scale in power law graphs ? Other properties, such as resilience. What are the growth primitives of power law graphs ? Optimization tradeoff primitives translate to cost – service Towards efficient and accurate synthetic data. Level of Autonomous Systems : Routing protocol (BGP) stability by game theory.