The Connectivity and FaultTolerance of the Internet Topology
Christopher R. Palmer (CMU)
crpalmer@cs.cmu.edu
Georgos Siganos (UC Riverside)
Michalis Faloutsos (UC Riverside)
Phillip B. Gibbons (Bell-Labs)
Christos Faloutsos (CMU)
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 don’t 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 can’t deploy your software.
– Must use a simulator to validate your approach.
– Can’t simulate the Internet until we understand it!
• Helps to know where the next problems will arise.
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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
N(u,h)
u
9
8
7
6
5
4
3
2
1
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 router’s view of the world.
– Conjecture: Similar functions => similar routers ?
• Graph’s neighbourhood function:
– Metric that characterizes the overall “look” of a graph.
– Conjecture: Similar functions => 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
Neighbourhood
function for
node 10
Effective
Eccentricity
of 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.
Same graph
Hop exponent is
the slope of the
least-squares line
we fit to N(u,h).
Gives a simple way to compare
two neighbourhood functions.
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Our Data: Scan+Lucent 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.
# Nodes
# Edges
Average
Degree
Max.
Degree
285K
430K
3.15
1,978
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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]
Effective Eccentricity
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We can learn a lot
by looking at the
different parts of
this histogram
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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 <= 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 I’ll 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.
>25K 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 ?
Disconnection is
relatively slow for
random failures.
Faster for
hop exponent
and degree.
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Random deletions
don’t change the
“look” of the Internet,
the other deletions do.
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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!
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