Computer Science

Sampling Biases in IP Topology Measurements

John Byers

with Anukool Lakhina, Mark Crovella and Peng XieDepartment of Computer ScienceBoston University

Discovering the Internet topology

Goal: Discover the Internet Router Graph• Vertices represent routers,• Edges connect routers that are one IP hop apart

Measurement Primitive: traceroute Reports the IP path from A to B i.e., how IP paths are overlaid on the router graph

source destination

212.12.5.77

212.12.58.3 163.55.221.98

163.55.1.41

163.55.1.10

• k sources: Few active sources, strategically located.

• m destinations: Many passive destinations, globally dispersed.

• Union of many traceroute paths.

(k,m)-traceroute study

Traceroute studies today

Destinations

Sources

DegreeF

req

uen

cy

Dataset from [PG98]

Heavy tails in Topology Measurements

A surprising finding: [FFF99]

Let be a given node degree.Let be frequency of degree vertices in a graph

Power-law relationship:

dfd

cd df

d

Subsequent measurements show that the degree distribution is a heavy tail,[GT00, BC01, …]

log

(Pr[

X>

x])

log( )

cdxX ]Pr[

We’re skeptical

We will argue that the evidence for power laws is at best insufficient.

Insufficient does not mean noisy or incomplete. (which these datasets certainly are!)

For us, insufficient means that measurements are statistically biased.

We will show that (k,m)-traceroute studies exhibit significant sampling bias.

A thought experiment

Idea: Simulate topology measurements on a random graph.

1. Generate a sparse Erdös-Rényi random graph, G=(V,E). Each edge present independently with probability pAssign weights: w(e) = 1 + , where in

2. Pick k unique source nodes, uniformly at random

3. Pick m unique destination nodes, uniformly at random

4. Simulate traceroute from k sources to m destinations, i.e. learn shortest paths between k sources and m destinations.

5. Let Ĝ be union of shortest paths.

Ask: How does Ĝ compare with G ?

||

1,||

1

VV

Ĝ is a biased sample of G that looks heavy-tailedAre heavy tails a measurement artifact?

MeasuredGraph, Ĝ

Underlying Random Graph, G

Underlying Graph: N=100000, p=0.00015Measured Graph: k=3, m=1000

log(Degree)

log

(Pr[

X>

x])

Outline

Motivation and Thought Experiments

Understanding Bias on Simulated TopologiesWhere and Why

Detecting and Defining BiasStatistical hypotheses to infer presence

of bias

Examining Internet Maps

Understanding Bias

(k,m)-traceroute sampling of graphs is biased

An intuitive explanation: When traces are run from few sources to large

destinations, some portions of underlying graph are explored more than others.

We now investigate the causes behind bias.

Are nodes sampled unevenly?

• Conjecture: Shortest path routing favors higher degree nodes nodes sampled unevenly

• Validation:Examine true degrees of nodes in measured graph, Ĝ.

Expect true degrees of nodes in Ĝ to be higher than degrees of nodes in G, on average.

True Degrees of nodes in Ĝ

Degrees of all nodes in G

Measured Graph: k=5,m=1000

• Conclusion: Difference between true degrees of Ĝ and degrees of G is insignificant; dismiss conjecture.

Are edges sampled unevenly?

• Conjecture:Edges selected incident to a node in Ĝ not proportional to true degree.

• Validation:For each node in Ĝ, plot true degree vs. measured degree.

If unbiased, ratio of true to measured degree should be constant. Points clustered around y=cx line (c<1).

• Conclusion: Edges incident to a node are sampled disproportionately; supports conjecture.

Ob

serv

ed D

egre

e

True Degree

Why: Analyzing Bias

• Question: Given some vertex in Ĝ that is h hops from the source, what fraction of its true edges are contained in Ĝ?

• Messages:

• As h increases, number of edges discovered falls off sharply.*

* We can prove exponential fall-off analytically, in a simplified model.

Distance from source

Fra

cti

on

of

no

de

ed

ges

dis

cove

red

1000dst

100dst

600dst

Result of adding more destinations: most new nodes and edges closer to the source.

What does this suggest?

Summary:

Edges are sampled unevenly by (k,m)-traceroute methods.

Edges close to the source are sampled more often than edges further away.

Intuitive Picture:

Neighborhood near sources is well explored but, visibility of edges declines sharply with hop distance from sources.

Hop1lo

g(P

r[X

>x]

)

log(Degree)

Hop2

Hop3

Underlying Graph

Measured Graph

Hop4

Outline

Motivation and Thought Experiments

Understanding Bias in Simulated TopologiesWhere and Why

Detecting and Defining BiasStatistical hypotheses to infer presence

of bias

Examining Internet Maps

Inferring Bias

Goal:Given a measured Ĝ, does it appear to be biased?

Why this is difficult: Don’t have underlying graph. Don’t have formal criteria for checking bias.

General Approach: Examine statistical properties as a function of distance from nearest source. Unbiased sample No change Change Bias

Detecting Bias

Examine Pr[D=d|H=h], the conditional probability that a node has degree d, given that it is at distance h from the source.

Two observations:1. Highest degree nodes are near the source.2. Degree distribution of nodes near the source different from those far away

log(Degree)

Ĝ degrees| H=3

log

(Pr[

X>

x])

Underlying Graph

Ĝ degrees| H=2

A Statistical Test for C1

2

)1(2)1(

)1(]Pr[

v

ek

Cut vertex set in half: N (near) and F (far), by distance from nearest source.Let v : (0.01) |V|

k : fraction of v that lies in N

Can bound likelihood k deviates from 1/2 using Chernoff-bounds:

H0C1

Reject hypothesis with confidence 1- if:

2

)1()1(

v

e

C1: Are the highest-degree nodes near the source? If so, then consistent with bias.

The 1% highest degree nodes occur at random with distance to nearest source.

A Statistical Test for C2

Partition vertices across median distance: N (near) and F (far)

Compare degree distribution of nodes in N and F, using the Chi-Square Test:

l

iiii EEO

1

22 /)(

where O and E are observed and expected degree frequencies and l is histogram bin size.

Reject hypothesis with confidence 1- if:

H0C2

2]1,[

2 l

C2: Is the degree distribution of nodes near the source different from those further away? If so, consistent with bias.

Chi Square Test succeeds on degree distribution for nodes near the source and far from the source.

Our Definition of Bias

• Bias (Definition): Failure of a sampled graph to meet statistical tests for randomness associated with C1 and C2.

• Disclaimers:Tests are not conclusive.Tests are binary and don’t tell us how

biased datasets are.

• But dataset that fails both tests is a poor choice to make generalizations of underlying graph.

Introducing datasets

Pansiot-Grad

log(Degree)

Mercator Skitter

log

(Pr[

X>

x])

Dataset Name Date # Nodes # Links # Srcs # Dsts

Pansiot-Grad 1995 3,888 4,857 12 1270

Mercator 1999 228,263 320,149 1 NA

Skitter 2000 7,202 11,575 8 1277

Testing C1

H0C1 The 1% highest degree nodes occur at random with distance

to source.

Pansiot-Grad: 93% of the highest degree nodes are in NMercator: 90% of the highest degree nodes are in NSkitter: 84% of the highest degree nodes are in N

Testing C2

H0C2

Pansiot-Grad Mercator Skitter

log

(Pr[

X>

x])

log(Degree)

Near

Far

All

Near

Far

All

Near

Far

All

Summary of Statistical Tests

All datasets pass both statistical tests for evidence of bias.

Likely that true degree distribution of the routers is different than that of these datasets.

Final Remarks

• Using (k,m)-traceroute methods to discover Internet topology yields biased samples.

• Rocketfuel [SMW:02] is limited-scale but may avoid some pitfalls of (k,m)-traceroute studies.

• One open question: How to sample the degree of a router at random?

• Node degree distributions are especially sensitive to biased sampling may not be a sufficiently robust metric for characterizing or comparing graphs.

Sampling Power-Law Graphs

Even though distributional shape similar, different exponents matter for topology modeling. Again, Ĝ is a biased sample of G

MeasuredGraph

Underlying, Power-Law Graph

Underlying PLRG: N=100000Measured Graph: k=3, m=1000

log

(Pr[

X>

x])

log(Degree)