Measuring the Mixing Time of Social Graphs
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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
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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
211/3/10 IMC'10
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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]
311/3/10 IMC'10
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The Idea
4
Intersection node
Verifier
Attackedge
Limited # of attack edges10-15 per million nodes
11/3/10 IMC'10
Suspect
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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.
511/3/10 IMC'10
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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
611/3/10 IMC'10
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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.
711/3/10 IMC'10
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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
811/3/10 IMC'10
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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
911/3/10 IMC'10
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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
1011/3/10 IMC'10
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Main Results
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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
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13
Big difference between the measurements using the two methods
Difference across datasets is related to the social network model
11/3/10 IMC'10
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Physics 3
Physics 2
Physics 1
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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
11/3/10 IMC'10
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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.
11/3/10 IMC'10
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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
1711/3/10 IMC'10
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Implication
• What’s the amount of the mixing time we need indeed for these designs to work?
1811/3/10 IMC'10
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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)
1911/3/10 IMC'10
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Future work
2011/3/10 IMC'10
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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