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On the bias of Breadth First Search (BFS)

and of other graph sampling techniques

Maciej Kurant (EPFL / UCI)

Joint work with:Athina Markopoulou (UCI),

Patrick Thiran (EPFL).

08 Sep 2010, ITC’22, Amsterdam, Netherlands

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Breadth First Search (BFS)

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Not feasible for huge online graphs! E.g., a full BFS of the friendship graph of would require 200TB of html traffic.

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BFS sample of a large graph

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sampling budget

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Why sample with BFS?

• BFS is a well known textbook technique

• BFS sample is a nice looking graph– E.g.., BFS of a lattice is a lattice– We can study its topological characteristics, which is not possible with random walks

• It is used in practice:– Y. Ahn, S. Han, H. Kwak, S. Moon, and H. Jeong, “Analysis of Topological Characteristics of Huge

Online Social Networking Services,” in Proc. of WWW, 2007.– A. Mislove, M. Marcon, K. P. Gummadi, P. Druschel, and S. Bhattacharjee, “Measurement and

Analysis of Online Social Networks,” in Proc. of IMC, 2007.– C. Wilson, B. Boe, A. Sala, K. P. Puttaswamy, and B. Y. Zhao, “User interactions in social networks

and their implications,” in Proc. of EuroSys, 2009.

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Why?

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node degree distribution

pk - real node

degree distribution

Our BFS samples of .

BFS sample size: 100K nodes Facebook size: 500M nodes

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- real average node degree

- real average squared node degree.

Our Goal

Graph traversals on RG(pk):

?BFS

qk ( f ) = ?

This bias has been empirically observed in the past, but never formally analyzed.

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Graph model RG(pk)

• Random graph RG(pk) with a given node

degree distribution pk

• Can be generated by configuration model:Example: |V| = 4 and pk: p1= p2= p3= p4 = 0.25

‘stubs’

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Approach 1: Brute force

Remedy: “The Principle of Deferred Decisions”

So we can generate the graph ‘on the fly’, while exploring it!

Generate all possible graphs, and ... No way!!

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Approach 2: The Principle of Deferred Decisions

This does not scale! (because of dependencies between stubs)

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* we assumed that the generated graph is connected

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Originally proposed in:J. H. Kim, “Poisson cloning model for random graphs,” International Congress of Mathematicians (ICM), 2006 (preprint in 2004).

Developped in:D. Achlioptas, A. Clauset, D. Kempe, and C. Moore, “On the bias of traceroute sampling: or, power-law degree distributions in regular graphs,” in STOC, 2005.

(both in a different context)

Approach 3: Breaking the stub dependencies

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Originally proposed in:J. H. Kim, “Poisson cloning model for random graphs,” International Congress of Mathematicians (ICM), 2006 (preprint in 2004).

Developped in:D. Achlioptas, A. Clauset, D. Kempe, and C. Moore, “On the bias of traceroute sampling: or, power-law degree distributions in regular graphs,” in STOC, 2005.

(both in a different context)

Approach 3: Breaking the stub dependencies

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Approach 3: Breaking the stub dependencies

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The analysis is exactly the same for other graph traversal techniques:

• BFS• DFS• Forest Fire• Snowball Sampling• Node sampling weighted by degrees• …

Approach 3: Breaking the stub dependencies

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Simulations on a power law random graph with 10K nodes

Theory vs

Simulations

degree distribution

corrected!

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What if the graph is not random?

Random graph RG(pk):Purely random, given the degree distribution pk.

Assortative RG(pk):Nodes of similar degree are more likely to connect.

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BFS deg av k

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- real average node degree

- real average squared node degree.

Summary

Graph traversals on RG(pk):

MHRW, RWRW

Random Walk

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- real average node degree

- real average squared node degree.

Summary

Graph traversals on RG(pk):

MHRW, RWRW

Random Walk

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- real average node degree

- real average squared node degree.

Graph traversals on RG(pk):

For small sample size (for f→0),BFS has the same bias as RW.(also in our Facebook measurements)

This bias monotonically decreases with f. We found analytically the shape of this curve.

Thank you!

MHRW, RWRW

For large sample size (for f→1), BFS becomes unbiased.

Summary

Random Walk