Measuring the Mixing Time of Social Graphs

Abedelaziz Mohaisen, Aaram Yun, and Yongdae Kim

Computer Science and Engineering DepartmentUniversity of Minnesota Twin Cities

ACM Internet Measurement ConferenceNov. 1—3, 2010, Mel, Australia

Background

• Sybil Attack: nodes with multiple fake identities– P2P, Sensor/Ad hoc networks, Reputation System

• “The Sybil Attack” by John DouceurImpossibility of defending against the attack without a trusted centralized authority

• A new directionUse of social networks to defend against Sybil attack without centralized authority

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The new direction

• SybilGuard [SIGCOMM’06]• SybilLimit [Oakland’08]• GateKeeper [PODC’10]• SybilInfer [NDSS’09]• SumUp [NSDI’09]• MobID [INFOCOM’10]• Whanau [NSDI’10]• … [SIGCOMM’10]

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The Idea

4

Intersection node

Verifier

Attackedge

Limited # of attack edges10-15 per million nodes

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Suspect

Main findings

• Good news– Some Sybil defenses do

not need ``fast mixing’’ graphs in order to work for ``good nodes’’.

• Bad news– Social graphs are not fast

mixing.– Some theoretical arguments

in Sybil defenses are inaccurate.

– The applicability of social network-based Sybil defenses is infeasible for some social graphs.

– Negative correlation between trust and mixing.

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The rest of this talk

• Assumptions and Preliminaries

• How to measure the mixing time

• Results and Implications

• Conclusion and Future Work

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The assumptions reloaded

• Trust in social network– Face-to-face network, not OSN– However…

• Small mixing time– The cost and effectiveness of designs.– Number of accepted Sybils per attack edge.

• Small number of attack edges– Justify the sparse-cut hypothesis.

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Preliminaries

• Social networks – Undirected graph, edges = interdependencies– A is the adjacency matrix– P is transition matrix, π is stationary distribution

• Mixing time– The time to reach the stationary distribution

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Computing the mixing time• Bounded by the second largest eigenvalue (µ)

• Computed directly from the definition.

• Methodology– Compute the lower bound as an indicator– Compute the mixing time of 1000 random sources selected

uniformly at random in the social graph

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DatasetsDataset Nodes Edges μ

Wiki-vote 7066 100736 0.899418

Slashdot 1 77360 546487 0.987531

Facebook A 1000000 20353743 0.982477

Facebook B 1000000 15807563 0.992020

Youtube 113490 2987624 0.997494

Enron 33696 180811 0.996879

Physics 1 4158 13428 0.998133

Physics 2 11204 117649 0.998221

Physics 3 8638 8638 0.996879

Livejournal A 1000000 26151771 0.999387

Livejournal B 1000000 27562349 0.999695

DBLP 614981 1155148 0.997972

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Main Results

12

Faster mixing

Slower mixing

Slower mixing

Big difference between the measurements using the two methods

Difference across datasets is related to the social network model11/3/10 IMC'10

13

Big difference between the measurements using the two methods

Difference across datasets is related to the social network model

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Physics 3

Physics 2

Physics 1

14

1. A few slow-mixing sources are enough to slow down the overall mixing of the network.

2. The use of the mixing time, as the maximal time, for reasoning about Sybil defenses is inaccurate

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Physics 1

Physics 3

Physics 2

15

Such slow mixing nodes represent a large percent of nodes in the social graph.

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Other measurements

• The impact of trimming low degree nodes– Using the same method as in SIGCOMM’06– Graph size reduced to only 20% of the original

graph after trimming up to 5 degrees– The total variation distance moves from 0.2 to

0.03 for walk length of 100 (SLEM technique)– From 0.015 to 0.003 (sampling technique)– From 0.6 to 0.2 at walk length of 15

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Implication

• What’s the amount of the mixing time we need indeed for these designs to work?

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Conclusion• Measured the mixing time of several social networks

using sampling• Showed that 2nd largest Eigenvalue is not accurate for

representing the mixing of the whole graph.

• Findings• Social graphs are slower mixing than anticipated and used• Negative correlation between trust and mixing time• Smaller walk length is sufficient to accept most honest nodes.

• Still larger than theoretically assumed• Relaxed mixing assumption (larger statistical distance)

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Future work

2011/3/10 IMC'10

Measuring the Mixing Time of Social Graphs

Abedelaziz Mohaisen, Aaram Yun, and Yongdae Kim

Computer Science and Engineering DepartmentUniversity of Minnesota Twin Cities

ACM Internet Measurement ConferenceNov. 1—3, 2010, Mel, Australia