Identifying Internet Topology

Polly HuangNTU, EE & INMhttp://cc.ee.ntu.edu.tw/~phuang

Outline

The problemBackgroundSpectral AnalysisConclusion

The Problem

What does the Internet look like?

Routers as verticesCables as edgesInternet topologies as graphs

For example…

The Internet, Circa 1969

A 1999 Internet ISP Map

[data courtesy of Ramesh Govindan and ISI’s SCAN project]

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

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

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

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

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

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

Waxman Example

The Common-sense Era

HierarchyGT-ITM

GT-ITM

Transit Number Connectivity

Stub Number Connectivity

Transit-stub Connectivity

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

Two Important Power-laws

Rank

Node degree

Frequency

Node degree

1 AS with degree ~2000

~2500 ASs with degree 1

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

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

Inet

Fit the node degree power-laws specifically Generate node degrees with power-laws Link degrees preferentially at random

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.

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

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

Number of Links

Date

Matching Cardinality

Date

Summary of Background

Forget about the heuristic oneStructural ones

Miss power-law featuresPower-law ones

Miss other features But what features?

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!!

Spectral Analysis

Danica VukadinovicThomas ErlebachETHZ, TIKPolly HuangNTU, EE INM

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

Outline

The problemBackgroundSpectral AnalysisConclusion

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

No Structural Difference

Eigenvalues are proportionally larger.# of Eigenvalues is proportionally larger.Eigenvalues are proportionally larger.# of Eigenvalues is proportionally larger.

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

Tree vs. Grid

Trees

Eigenvalue Index [0,1]

Eigen-values [0,2]

Grids

Eigenvalue Index [0,1]

Eigen-values [0,2]

AS vs. Inet Graphs

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

Spectral Analysis

Qualitatively useful nls as fingerprint

Quantitatively? Width of horizontal bar at value 1

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|

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)

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

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)

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

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

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

Core Connectivity

Observed Features

Internet graphs have relatively larger edge components

Although ratio of core components decreases, average degree of connectivity increases

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

Our Premise

Encompass both statistical and structural properties

No explicit degree fitting

Not quite there yet, but… do see an end

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

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

Questions?

Polly HuangNTU, EE & INMhttp://cc.ee.ntu.edu.tw/~phuang

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