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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.