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Transcript of 1 Spectral Analysis of Power-Law Graphs and its Application to Internet Topologies Milena Mihail...
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Spectral Analysis of Power-Law Graphs and its Application to Internet Topologies
Milena MihailGeorgia 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
Sprint
AT&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
GeorgiaTech
CNN
AT&T
Sprint
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The Directed AS Graph
GeorgiaTech
CNN
AT&T
Sprint
Peering Relationships :
Customer – Provider
PeersGao 00, Subramanian et al 01
Five Tier HierarchySubramanian et al 01
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Eigenvalue Power Law
rank
eig
en
valu
e
1 2 3 4 5 10
Faloutsos et al 99
UUNET
SprintC&WUSA
AT&TBBN
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
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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
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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
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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
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Eigenvectors & Eigenvalues
0 1 0 1 1 0 0 1 0 1 1 0 1 0 1 0 0 11 0 1 0 0 10 1 0 1 0 10 1 0 1 0 11 0 0 1 0 11 0 0 1 0 10 1 1 0 1 00 1 1 0 1 0
A =A =
Ax x=
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Matrix as a Linear Transformation
0 1 0 1 1 0 0 1 0 1 1 0 1 0 1 0 0 11 0 1 0 0 10 1 0 1 0 10 1 0 1 0 11 0 0 1 0 11 0 0 1 0 10 1 1 0 1 00 1 1 0 1 0
A =A =
5 2
1
01
1
4 6
3
46
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Matrix as a Linear Transformation
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Stochastic Normalization
0 1/3 0 1/3 1/3 0 1/3 0 1/3 1/3 0 0 1/3 0 1/3 0 0 1/3 0 1/3 0 0 1/31/30 1/3 0 1/3 0 0 1/3 0 1/3 0 1/31/31/3 0 0 1/3 0 1/3 0 0 1/3 0 1/31/30 1/3 1/3 0 1/3 0 1/3 1/3 0 1/3 000 1/3 1/3 0 0 0 1/3 1/3 0 0 1/31/3
A =A =
Ax x=
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The Random WalkEigenvalues between 1 and –1 ( 1 and 0 also easy).
1
0
0
00
0
1/3
1/3
1/3
0
0
0
2/91/9
1/9
1/9
2/9
2/9
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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
1/6
1/6 1/6
1/6
1/6 3/16
2/16
3/163/16
2/16
3/16
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1. Principal Eigenvector ~ Significance
In directed graphs the weights of the principal eigenvector can vary way beyond degrees.
10^-4
2*10^-4 5*10^-4 0.240.39
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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
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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
36
2. Eigenvectors ~ Clustering
1/6
1/6
1/6 1/6
1/6
1/6
1/6
1/6
1/6 1/6
1/6
1/6
-
--
-
-
-~
~ ~
~ ~
~
~ ~
~~
~ ~
Matrix Perturbation Theory
37
Spectral Filtering
1 2 3n 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 )
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.
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Effect of intra-cluster and inter-cluster traffic to most congested link
Internet
Inet Internet Inet
0% 100% 100% 0% 100% 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%
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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
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Which Eigenvectors correspond to Eigenvalue Power Law ?
rank
eig
en
valu
e
1 2 3 4 5 10
Faloutsos et al 99
UUNET
SprintC&WUSA
AT&TBBN
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Large Degrees, or“Stars” of AS Graph
Dominate Spectrum of Adjacency Matrix, Prior to Normalization
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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
Summary0. 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
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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.