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Identifying Internet Topology. Polly Huang NTU, EE & INM Outline. The problem Background Spectral Analysis Conclusion. The Problem. What does the Internet look like? Routers as vertices Cables as edges Internet topologies as graphs For example….

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Identifying internet topology

Identifying Internet Topology

Polly Huang



  • The problem

  • Background

  • Spectral Analysis

  • Conclusion

The problem
The Problem

  • What does the Internet look like?

  • Routers as vertices

  • Cables as edges

  • Internet topologies as graphs

  • For example…

A 1999 internet isp map
A 1999 Internet ISP Map

[data courtesy of

Ramesh Govindan and ISI’s SCAN project]

Back to the problem
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 graphs

      • To predict the future

      • To design for the future

  • Not a new problem, but

Becoming urgent
Becoming Urgent

  • Internet is an ecosystem

  • Deeply involved in our daily lives

  • Just like the atmosphere

  • We 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
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
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


  • 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
The No-clue Era

  • Heuristic

  • Waxman

    • 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

The common sense era
The Common-sense Era

  • Hierarchy

  • GT-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

~2500 ASs

with degree 1

1 AS with

degree ~2000




Node degree


Two Important Power-laws

The power law era
The Power-law Era

  • Models of the 80s and 90s

    • Fail to capture power-law properties


  • Inet

  • Won’t show examples

    • Too big to make sense


  • Barabasi’s incremental model

    • Create a random core

    • Incrementally add nodes and links

    • Connect new link to existing nodes probabilistically

      • Waxman or preferential

  • Node degrees of these graphs will magically have the power-law properties


  • Fit the node degree power-laws specifically

    • Generate node degrees with power-laws

    • Link degrees preferentially at random

Are they better
Are They Better?

  • Tangmunarunkit et al (Sigcomm 2002)

  • Structural vs. Degree-based

  • Classification

    • Structural: TS (GT-ITM)

    • Degree-based: Inet, BRITE, and etc.

  • Current degree-based generators DO work better than TS.

Which degree based is better
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


  • 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 links

      • Cardinality of matching

Summary of background
Summary of Background

  • Forget about the heuristic one

  • Structural ones

    • Miss power-law features

  • Power-law ones

    • Miss other features

    • But what features?

No idea
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 there

  • Tons of metrics to compare graphs!

  • Will never end this way!!

Spectral analysis

Spectral Analysis

Danica Vukadinovic

Thomas Erlebach


Polly Huang


Our rationale
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


  • The problem

  • Background

  • Spectral Analysis

  • Conclusion

Research statement
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 eigenvalues

      • Condensed info about the matrix

No structural difference
No Structural Difference

Eigenvalues are proportionally larger.

# of Eigenvalues is proportionally larger.


  • 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 spectrum

Thus referred to as spectral analysis

Tree vs grid







Eigenvalue Index [0,1]

Eigenvalue Index [0,1]

Tree vs. Grid

Nls as graph fingerprint
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 analysis1
Spectral Analysis

  • Qualitatively useful

    • nls as fingerprint

  • Quantitatively?

    • Width of horizontal bar at value 1

Width of horizontal bar at 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
For a Graph G

  • P: subgraph containing pendant nodes

  • Q: subgraph containing quasi-pendant nodes

  • Inner: G - P - Q

  • I: isolated nodes in Inner

  • R: Inner - I (R for the rest)

Physical interpretation
Physical Interpretation

  • Q: high-connectivity domains, core

  • R: regional alliances, partial core

  • I: multi-homed leaf domains, edge

  • P: single-homed leaf domains, edge

  • Core vs. edge classification

    • A bit fuzzy

    • For the sake of simplicity

Validation by examples
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
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
Evolution of Edge

Ratio of nodes in P

Ratio of nodes in I

The edge components are indeed

large and growing

Strong growth of I component ~ increasing

number of multi-homed domains

Evolution of core
Evolution of Core

Ratio of nodes in Q

Ratio of links in Q

The core components get more links than nodes.

Observed features
Observed Features

  • Internet graphs have relatively largeredge components

  • Although ratio of core components decreases, average degree of connectivityincreases

Towards a hybrid model





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

Our premise
Our Premise

  • Encompass both statistical and structural properties

  • No explicit degree fitting

  • Not quite there yet, but… do see an end


  • 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
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



Polly Huang



  • 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.

  • M. Faloutsos, P. Faloutsos and C. Faloutsos. On power-law relationships of the Internet opology. Proceedings of Sigcomm 1999.

  • H. Tangmunarunkit, R. Govindan, S. Jamin, S. Shenker, W. Willinger. Network Topology Generators: Degree-Based vs. Structural. Proceedings of Sigcomm 2002.

  • D. Vukadinovic, P. Huang, T. Erlebach. On the Spectrum and Structure of Internet Topology Graphs. To appear in the proceedings of I2CS 2002.