Analysis of the Internet Topology

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Analysis of the Internet Topology

• Michalis Faloutsos, U.C. Riverside (PI)
• Christos Faloutsos, CMU (sub-contract, co-PI)
• DARPA NMS, no. 00-1-8936

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Goal Find Order in Chaos
Power-law Frequency of degree vs. degree
A real Internet instance, Oct 1995

• Analyze and model the Internet topology
• preferably with a few simple numbers

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Motivation

• Intuitively
• You cant resolve the traffic jam problem of
  a city without looking at the street layout.
• The Internet community has little knowledge of
  the topology

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What is the novelty?

• Power-laws to describe skewed distributions,
• i.e. y a xp.
• Spectral-analysis to capture the finger-print
  of the graph
• Multi-fractals to identify 80-20 distributions
• i.e. 20 nodes have 80 of the edges
• Use the above to define and identify a hierarchy

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Deliverables

• New graph metrics
• A model for the Internet topology
• A software tool to analyze graphs

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Road Map

• Current Research
• Four Power-laws
• Future Directions
• Graph metrics
• Applications
• Conclusion

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Our Power-laws

• I. Outdegree of nodes vs. rank
• II. Frequency of outdegree
• III. Eigenvalues of adj. matrix
• IV. Pairs of nodes within h hops
• Accuracy correlation coeff. gt 0.97

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I. Power-law rank R
outdegree
Exponent slope R -0.74
R
Dec98
Rank nodes in decreasing outdegree order

• The plot is a line in log-log scale

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II.Power-law outdegree O
Frequency
Exponent slope
O -2.15
Nov97
Outdegree

• The plot is linear in log-log scale

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III. Eigenvalues

• Let A be the adjacency matrix of graph
• The eigenvalue ? is
• A v ? v, where v some vector
• Eigenvalues are related to topological properties

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III.Power-law eigen exponent E
Eigenvalue
Exponent slope
E -0.48
Dec98
Rank of decreasing eigenvalue

• Find the eigenvalues of the adjacency matrix
• Eigenvalues in decreasing order (first 20)

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IV. Power-law hopplot H

• Pairs of reachable nodes as a function of hops
• Simpler neighborhood size vs hops

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Road Map

• Current Research
• Four Power-laws
• Future Directions
• Graph metrics
• Applications
• Conclusion

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Looking For More Patterns

• Analyze the dynamic nature of the Internet
• Study the eigenvalues of the adjacency matrix
• Identify multi-fractal relationships
• Identify a hierarchy

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Practical Applications and our Patterns

• Provide realistic models for simulations
• Faster simulations
• Improve the design of protocols
• Estimate useful network properties
• Facilitate traffic engineering
• Predict network evolution, what-if scenarios
• Estimate fault-tolerance of the topology

Applications will inspire and guide the search
for an Internet model and its metrics
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Conclusions

• The Internet has more structure than we thought!
• Our tools look very promising they characterize
  concisely the topology
• Modeling the Internet will have significant
  practical impact

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Analysis of the Internet Topology
Chaos
NEW IDEAS
Finding Order in
Use power-laws Use the eigenvalue analysis of
the adjacency matrix Use multi-fractal
analysis Describe the topology concisely i.e.
with a few simple numbers
Frequency distribution of node degree
Internet Topology, 1995
IMPACT
SCHEDULE
YEAR 1
YEAR 2
YEAR 3
A model for the Internet topology will Improve
the design of routing protocols Help explain the
behavior of traffic Improve the validity of
network simulations Estimate the vulnerability of
topology to malicious users
1. Set of topological metrics 2.
Comprehensive model of the Internet
topology 3. Tool for analyzing the topology
of graphs
U.C.Riverside Michalis Faloutsos
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I. Estimating with the rank exponent R

• Lemma
• Given the nodes N, and an estimate for the
  rank exponent R, we predict the edges E

See paper for details SIGCOMM99
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IV. The Average Neighborhood Size
With hopplot
Avg. Neighborhood
Real
With avg. degree
Hops

• Avg. Neighborhood size versus h hops
• Before with avg. degree d dh
• Now with hopplot hH
• Qualitative sphere in H-dimensions, H 4.86

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Time Evolution rank R
Domain level

• The rank exponent has not changed!