Christopher R' Palmer CMU

1
The Connectivity and Fault-Tolerance of the
Internet Topology

• Christopher R. Palmer (CMU)
• crpalmer_at_cs.cmu.edu
• Georgos Siganos (UC Riverside)
• Michalis Faloutsos (UC Riverside)
• Phillip B. Gibbons (Bell-Labs)
• Christos Faloutsos (CMU)

2
Understanding the Internet

• The Internet is very important in daily life!
• How long has it been since you sent bits into the
  Internet?
• But we dont really know much about it. Why?
• The Internet is huge.
• Detailed data only recently available for study.
• Hard to process using existing tools.

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Who Cares if we Understand it?

• It helps for designing new algorithms!
• E.g. How can you design a new routing algorithm?
• Once we have new algorithms we need to test them
• Typically cant deploy your software.
• Must use a simulator to validate your approach.
• Cant simulate the Internet until we understand
  it!
• Helps to know where the next problems will arise.

4
Our Approach

• Treat the Internet (at a Router level) as a large
  graph.
• Unweighted undirected graph.
• 285K nodes (routers) and 430K edges (links).
• Look at the properties of the nodes of this
  graph
• In the past, looked at degree (avg / max /
  power-laws).
• Now we are going to try to start to classify
  them.
• Use properties of the graph to look at fault
  tolerance
• What if a communication channel fails?
• What if a Router fails?

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Our Contributions

• Add to our understanding of the topology
• Get a better idea of what makes up the core.
• Get a better idea of the robustness of the
  Internet.
• Introduce some tools to help people do more!
• At least as important as our new understanding.
• Gives others tools to explore their ideas.

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Roadmap

• Introduce and motivate our data-mining tools and
  data
• Neighbourhood function of a node (router).
• Neighbourhood function of a graph (network).
• Effective eccentricity.
• Hop plot exponent.
• Router level Internet data that we will study.
• Use our tools to identify interesting routers.
• Use our tools to examine fault tolerance.
• Conclusions.

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Tool 1 Neighbourhood of a Node
Example Graph
Example Neighbourhood Fn
9 8 7 6 5 4 3 2 1
N(u,h)
u
1 2 3 4 5
h

• N(u,h) of nodes within h steps of u v
  dist(u,v) ? h

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Tool 2 Neighbourhood Function
N(u,h) of nodes within h steps of u v
dist(u,v) ? h N(h) of pairs of nodes
with h steps of each other ?u N(u,h)
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Why use the Neighbourhood?

• Individual neighbourhood function
• Metric that characterizes a routers view of the
  world.
• Conjecture Similar functions gt similar routers
  ?
• Graphs neighbourhood function
• Metric that characterizes the overall look of a
  graph.
• Conjecture Similar functions gt similar graphs?
• Now we need ways of computing and comparing them.

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How to Compute them?

• Approximate Neighbourhood Function
• Developed as a tool for Data Mining large graphs
• Going to use it here to analyze network graphs
• Very fast approximation with good error bounds.

• Idea
• approximate the set operations in the previous
  algorithm

u
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Properties of our Approximation

• Very fast
• More than 400 times faster on an Internet graph!
• Very accurate
• About a 5 relative error.
• Works for very large graphs
• We have a version that uses secondary storage
  efficiently.
• See the paper for more details and references.

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Tool 3 Effective Eccentricity
90 of the reachable
Effective Eccentricity of 10
Neighbourhood function for node 10

• Effective eccentricity is the first distance, h,
  at which you can reach 90 of the nodes in your
  connected component.
• EffEcc(u) min h N(u,h) ? .9 ? N(u,?)

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Tool 4 Hop Exponent

• Faloutsos, Faloutsos and Faloutsos
• Internet follows a hop plot exponent power law?
  N(h) ? hH

• Hop exponent, H
• Slope of l.s. line.
• Characterizes growth of N(u,h) or N(h).
• Succinct description.
• Gives a simple way to compare two neighbourhood
  functions.

Same graph Hop exponent is the slope of the
least-squares line we fit to N(u,h).
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Our Data ScanLucent Data Set

• Two projects used traceroute like probes
• SCAN Multiple robots collect linkage
  information.
• Lucent Single source probes network over time.
• Carefully merged to form best picture of
  Internet.
• Data was current as of late 1999.

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Roadmap

• Introduced our data-mining tools and data.
• Use our tools to classify routers
• Effective Eccentricity vs. Hop Exponent ?
• Find pathologies in the data.
• Find core or important routers.
• Use our tools to examine fault tolerance.
• Conclusions.

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Hop Exponent vs. Eff. Eccentricity

• Strongly correlated may use either metric
• Use hop exponent for a continuous value.
• Use effective eccentricity for binned values.

Hop Exponent
Effective eccentricity
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Effective Eccentricity

• Compute effective eccentricities for each node in
  graph
• View this data as a histogram (number of nodes is
  log scale)

of nodes with this eccentricity log scale
We can learn a lot by looking at the different
parts of this histogram
Effective Eccentricity
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Identify Outliers / Data Errors
Actual Subgraph of these nodes
Eff. Ecc. of 1 or 2

• Maximum degree of a node is lt 2K
• Effective eccentricity of 1 implies can reach at
  most 2K/.9 nodes
• That is, those nodes cannot reach entire 285K
  node graph!

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Identify Important Nodes

• Topologically important nodes very well
  connected.
• Conjecture These are core routers in the
  Internet.
• Will try to show that this is the case later in
  this talk.

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Poor Nodes ?
Internet
Who and what are these nodes? Data collection
error? Poorly connected countries? Other?
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Classifying Routers

• Effective Eccentricity is a new metric that
  allows us to
• Identify data irregularities.
• Found errors in the collected data.
• Found routers that were surprising and should be
  investigated.
• Find core routers ?
• We found topologically important nodes.
• In a few slides Ill add some evidence to suggest
  that they are really core routers.

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Roadmap

• Introduced our data-mining tools and data.
• Used our tools to classify routers.
• Use our tools to examine fault tolerance
• What if communication links fail?
• What if routers fail?
• Are our core routers actually important?
• Conclusions.

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Fault Tolerance

• Want to understand inherent fault tolerance
• Not concerned about protocol errors.
• Instead, focus on the communication that is
  possible.
• Types of faults simulated
• Link failures e.g. backhoe digs into a network
  cable.
• Router failures e.g. fire at the data center.
• Measure
• Impact on possible communication.
• Impact on the Internet structure.

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Link Failures

• Experiment Pick an edge at random, delete it and
  measure network disruption.

gt25K deletions for big change
150K deletions, it still looks like the Internet
Internet very resilient to link failures
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Node Failures

• We will model three different events.
• Random router failures
• Pick a node at random and delete it (and all
  incident edges).
• Hop exponent rank failures
• Delete nodes in decreasing order of hop
  exponent.
• Test our claim of finding core routers.
• Degree rank failures
• Delete nodes in decreasing order of node degree.
• Most aggressive way of attacking the Internet?

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Effect of node deletions

• Robust to random failures, focussed failures are
  a problem
• Core routers are
• different from high degree routers and
• identified by the individual hop exponents ?

Random deletions dont change the look of the
Internet, the other deletions do.
Disconnection is relatively slow for random
failures.
Faster for hop exponent and degree.
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Conclusions

• Neighbourhood function a good metric of
  importance
• Found core routers in the Internet.
• Found data errors / outliers.
• Found interesting fault tolerance results
• Internet is not sensitive to link failures.
• Internet is not sensitive to random router
  failures.
• Internet is sensitive to targeted attacks.
• Our data-mining tools provide a promising step
  forward in understanding the Internet topology!