Identifying Internet Topology
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Transcript of Identifying Internet Topology
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Identifying Internet Topology
Polly HuangNTU, EE & INMhttp://cc.ee.ntu.edu.tw/~phuang
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Outline
The problemBackgroundSpectral AnalysisConclusion
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The Problem
What does the Internet look like?
Routers as verticesCables as edgesInternet topologies as graphs
For example…
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The Internet, Circa 1969
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A 1999 Internet ISP Map
[data courtesy of Ramesh Govindan and ISI’s SCAN project]
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Back To The Problem
What does the Internet look like?
Equivalent of Can we describe the graphs
String, mesh, tree?Or something in the middle?
Can we generate similar graphsTo predict the future To design for the future
Not a new problem, but
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Becoming Urgent
Internet is an ecosystemDeeply involved in our daily lives
Just like the atmosphereWe have now daily weather reports to
determine whether To carry umbrella To fly as scheduled To fish as usual To fill holes at the construction site
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Internet Weather
We need in the future continuous Internet weather reports to determine When and how much to bit on eBay Whether to exchange stock online today Whether to do free KTV over Skype at
7pm Friday night Whether to stay with HiNet ADSL or
switch to So-Net
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Nature of the Research
Need to analyze dig into the details of Internet
topologies hopefully to find invariants
Need to model formulate the understanding hopefully in a compact way
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Background
As said, the problem is not new!Three generations of network topology
analysis and modeling already: 80s - No clue, not Internet specific 90s - Common sense 00s - Some analysis on BGP Tables
To describe: basic idea and example
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The No-clue Era
HeuristicWaxman
Define a plane; e.g., [0,100] X [0,100] Place points uniformly at random Connect two points probabilistically
p(u, v) ~ 1 / e d ; d: distance between u, v
The farther apart the two nodes are, the less likely they will be connected
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Waxman Example
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The Common-sense Era
HierarchyGT-ITM
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GT-ITM
Transit Number Connectivity
Stub Number Connectivity
Transit-stub Connectivity
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Break-through
Faloutsos et al (SIGCOMM 99) Analyze routing tables Autonomous System level graphs (AS graphs)
A domain is a vertex
Find power-law properties in AS graphs
Power-law by definition Linear relationship in log-log plot
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Two Important Power-laws
Rank
Node degree
Frequency
Node degree
1 AS with degree ~2000
~2500 ASs with degree 1
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The Power-law Era
Models of the 80s and 90s Fail to capture power-law properties
BRITEInet
Won’t show examples Too big to make sense
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BRITE
Barabasi’s incremental model Create a random core Incrementally add nodes and links Connect new link to existing nodes
probabilisticallyWaxman or preferential
Node degrees of these graphs will magically have the power-law properties
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Inet
Fit the node degree power-laws specifically Generate node degrees with power-laws Link degrees preferentially at random
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Are They Better?
Tangmunarunkit et al (Sigcomm 2002)
Structural vs. Degree-basedClassification
Structural: TS (GT-ITM) Degree-based: Inet, BRITE, and etc.
Current degree-based generators DO work better than TS.
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Which Degree-based is better?
Compare AS, Inet, and BRITE graphs
Take the AS graph history From NLANR 1 AS graph per 3-month period 1998, January - 2001, March
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Methodology
For each AS graph Find number of nodes, average degree Generate an Inet graph with the same
number of nodes and average degree Generate a BRITE graph with the same
number of nodes and average degree Compare with addition metrics
Number of linksCardinality of matching
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Number of Links
Date
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Matching Cardinality
Date
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Summary of Background
Forget about the heuristic oneStructural ones
Miss power-law featuresPower-law ones
Miss other features But what features?
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No Idea!
Try to look into individual metrics Doesn’t help much
What do you expect from a number that describes a whole graph
A bit information here, a bit thereTons of metrics to compare graphs!Will never end this way!!
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Spectral Analysis
Danica VukadinovicThomas ErlebachETHZ, TIKPolly HuangNTU, EE INM
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Our Rationale
So power-laws on node degree Good But not enough
Take a step back Need to know more Try the extreme Full details of the inter-connectivity Adjacency matrix
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Outline
The problemBackgroundSpectral AnalysisConclusion
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Research Statement
Objective Identify missing features
Approach Analysis on the adjacency matrix
can re-produce the complete graph from it
To begin with, look at its eigenvaluesCondensed info about the matrix
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No Structural Difference
Eigenvalues are proportionally larger.# of Eigenvalues is proportionally larger.Eigenvalues are proportionally larger.# of Eigenvalues is proportionally larger.
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Normalization
Normalized adjacency matrix Normalized Laplacian Eigenvalues always in [0,2]
Normalized eigenvalue index Eigenvalue index always in [0,1]
Sorted in an increasing order
Normalized Laplacian Spectrum (nls)
Looking at a whole spectrumThus referred to as spectral analysisLooking at a whole spectrumThus referred to as spectral analysis
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Tree vs. Grid
Trees
Eigenvalue Index [0,1]
Eigen-values [0,2]
Grids
Eigenvalue Index [0,1]
Eigen-values [0,2]
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AS vs. Inet Graphs
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nls as Graph Fingerprint
Unique for an entire class of graphs Same structure ~ same nls
Distinctive among different classes of graphs Different structure ~ different nls
Do have exception but rare
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Spectral Analysis
Qualitatively useful nls as fingerprint
Quantitatively? Width of horizontal bar at value 1
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Width of horizontal bar at 1
Different in quantity for types of graphs AS, Inet, tree, grid Wider to narrower
The width Defined as Multiplicity 1 Denoted as mG(1)
An extended theorem mG(1) |P| + |I| - |Q|
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For a Graph G
P: subgraph containing pendant nodesQ: subgraph containing quasi-pendant
nodes
Inner: G - P - QI: isolated nodes in InnerR: Inner - I (R for the rest)
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Physical Interpretation
Q: high-connectivity domains, coreR: regional alliances, partial coreI: multi-homed leaf domains, edgeP: single-homed leaf domains, edge
Core vs. edge classification A bit fuzzy For the sake of simplicity
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Validation by Examples
Q UUNET, Sprint, Cable & Wireless, AT&T Backbone ISPs
R RIPE, SWITCH, Qwest Sweden National networks
I DEC, Cisco, HP, Nortel Big companies
P (trivial)
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Revisit the Theory
mG(1) |P| + |I| - |Q|Correlation
Ratio of the edge components -> Width of horizontal bar at value 1
Grid, tree, Inet, AS graphs Increasingly larger mG(1) Likely proportionally larger edge
components
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Evolution of Edge
The edge components are indeed large and growingThe edge components are indeed large and growingStrong growth of I component ~ increasing
number of multi-homed domainsStrong growth of I component ~ increasing number of multi-homed domains
Ratio of nodes in P Ratio of nodes in I
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Evolution of Core
The core components get more links than nodes.The core components get more links than nodes.
Ratio of nodes in Q Ratio of links in Q
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Core Connectivity
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Observed Features
Internet graphs have relatively larger edge components
Although ratio of core components decreases, average degree of connectivity increases
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Towards a Hybrid Model
Form Q, R, I, P components Average degree -> nodes, links Radio of nodes, links in Q, R, I, P
Connect nodes within Q, R Preferential to node degree
Inter-connect R, I, P to the Q component Preferential to degree of nodes in Q
Q
R I P
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Our Premise
Encompass both statistical and structural properties
No explicit degree fitting
Not quite there yet, but… do see an end
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Conclusion
Firm theoretical ground nls as graph fingerprint Ratio of graph edge -> multiplicity 1
Plausible physical interpretation Validation by actual AS names Validation by AS graph analysis
Framework for a hybrid model
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On-going Work
Towards a hybrid model Fast computation of the graph theoretical
metrics Implementation of the graph generator
Topology scaler Shrink the topology to its smaller equivalent To make it possible to simulate at all
Nature of the Internet graph structure The mechanism that gives rise to the power-laws
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Questions?
Polly HuangNTU, EE & INMhttp://cc.ee.ntu.edu.tw/~phuang
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References
E. W. Zegura, K. Calvert and M. J. Donahoo. A Quantitative Comparison of Graph-based Models for Internet Topology. IEEE/ACM Transactions on Networking, December 1997. http://www.cc.gatech.edu/projects/gtitm/papers/ton-model.ps.gz
M. Faloutsos, P. Faloutsos and C. Faloutsos. On power-law relationships of the Internet opology. Proceedings of Sigcomm 1999. http://www.acm.org/sigcomm/sigcomm99/papers/session7-2.html
H. Tangmunarunkit, R. Govindan, S. Jamin, S. Shenker, W. Willinger. Network Topology Generators: Degree-Based vs. Structural. Proceedings of Sigcomm 2002. http://www.isi.edu/~hongsuda/publication/USCTech02_draft.ps.gz
D. Vukadinovic, P. Huang, T. Erlebach. On the Spectrum and Structure of Internet Topology Graphs. To appear in the proceedings of I2CS 2002. http://www.tik.ee.ethz.ch/~vukadin/pubs/topologynpages.pdf