1. The Connectivity and Fault-Tolerance of the Internet Topology
Christopher R. Palmer (CMU)
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 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.
nrdm Christopher R. Palmer – Connectivity and Fault Tolerance of the Internet

3. 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.
nrdm Christopher R. Palmer – Connectivity and Fault Tolerance of the Internet

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?
nrdm Christopher R. Palmer – Connectivity and Fault Tolerance of the Internet

5. 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.
nrdm Christopher R. Palmer – Connectivity and Fault Tolerance of the Internet

6. 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.
nrdm Christopher R. Palmer – Connectivity and Fault Tolerance of the Internet

7. Tool #1: Neighbourhood of a Node
Example Graph
Example Neighbourhood Fn 9 8 7 6 5 4 3 2 1
N(u,h) u h
N(u,h) = # of nodes within h steps of u = |{ v : dist(u,v)  h }|
nrdm Christopher R. Palmer – Connectivity and Fault Tolerance of the Internet

8. 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)
nrdm Christopher R. Palmer – Connectivity and Fault Tolerance of the Internet

9. 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.
nrdm Christopher R. Palmer – Connectivity and Fault Tolerance of the Internet

10. How to Compute them? Approximate Neighbourhood Function Idea:
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
nrdm Christopher R. Palmer – Connectivity and Fault Tolerance of the Internet

11. 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.
nrdm Christopher R. Palmer – Connectivity and Fault Tolerance of the Internet

12. 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,)
nrdm Christopher R. Palmer – Connectivity and Fault Tolerance of the Internet

13. Hop exponent is the slope of the least-squares line we fit to N(u,h).
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).
nrdm Christopher R. Palmer – Connectivity and Fault Tolerance of the Internet

14. 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
nrdm Christopher R. Palmer – Connectivity and Fault Tolerance of the Internet

15. 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.
nrdm Christopher R. Palmer – Connectivity and Fault Tolerance of the Internet

16. 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
nrdm Christopher R. Palmer – Connectivity and Fault Tolerance of the Internet

17. 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
nrdm Christopher R. Palmer – Connectivity and Fault Tolerance of the Internet

18. 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!
nrdm Christopher R. Palmer – Connectivity and Fault Tolerance of the Internet

19. 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.
nrdm Christopher R. Palmer – Connectivity and Fault Tolerance of the Internet

20. “Poor” Nodes ?
Internet
Who and what are these nodes? Data collection error? Poorly connected countries? Other?
nrdm Christopher R. Palmer – Connectivity and Fault Tolerance of the Internet

21. 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.
nrdm Christopher R. Palmer – Connectivity and Fault Tolerance of the Internet

22. 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.
nrdm Christopher R. Palmer – Connectivity and Fault Tolerance of the Internet

23. 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.
nrdm Christopher R. Palmer – Connectivity and Fault Tolerance of the Internet

24. 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
nrdm Christopher R. Palmer – Connectivity and Fault Tolerance of the Internet

25. 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?
nrdm Christopher R. Palmer – Connectivity and Fault Tolerance of the Internet

26. 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 don’t change the “look” of the Internet, the other deletions do.
Disconnection is relatively slow for random failures.
Faster for hop exponent and degree.
nrdm Christopher R. Palmer – Connectivity and Fault Tolerance of the Internet

27. 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!
nrdm Christopher R. Palmer – Connectivity and Fault Tolerance of the Internet