Turning Privacy Leaks into Floods: Surreptitious Discovery of Social Network Friendships

Michael T. GoodrichUniv. of California, Irvine

joint w/ Arthur U. Asuncion

Problem Definition• Discover the friendships

Problem Definition• Discover the friendships

Leveraging Information Leaks• Leak: Friendship list can be

viewed by friends-of-friends. This allows:– Given two people, X and Y, we can tell

whether X and Y have a friend in common.

• Leverage: We use this to discover the friends list for members of the network

Abstracting the Problem• Viewed abstractly, we are trying to

learn binary attribute vectors.

Group Testing• Input: n items, numbered 0,1, …, n-1, at most d

of which are defective.• Output: the indices of the defective items.• Items can be grouped into subsets, each of which

can be tested to see it contains a defective item or not.

• Goal: minimize the total number of tests• Original problem: Testing blood samples.

Testing Schemes

• Non-adaptive: All tests must be done in parallel

• Adaptive: Tests can be done sequentially

• Adaptive is easier, but our framework requires a non-adaptive approach

Facebook Application• Each member has a “vector” of friendships• For any member M, the system returns a bit

for whether M has a friend in common with the attacker, even if M restricts this information to friends-of-friends

• We can use non-adaptive scheme to learn friendship relationships in any sub-community in Facebook.

DNA Application• DNA sequences are stored in a database,

D.• For any sequence Q, the database returns a

score for how close Q is to each sequence in D

• We form a binary vector w.r.t. places where mutations happen relative to a reference string R

• We can use non-adaptive scheme to learn DNA strings in D.

Netflix Application• Movie ratings vectors are stored in a

database, D.• For any vector V, the database returns a

score for how close V is to each vector in the database

• We can form a binary attribute vector for movies

• We can use non-adaptive scheme to learn ratings vectors in D.

Matrix View of Testing• A non-adaptive testing regimen can be

viewed as a t x n binary matrix M:

– M[i,j] = 1 if and only if test i includes item j

• M is d-disjunct if the Boolean sum of any d columns does not contain any other column.– An item is defective iff all its tests are positive

• M is d-separable if the Boolean sums of each set of at most d columns are distinct (harder analysis algorithm)

t

n

M

Randomized Approach• Use a randomized approach motivated

by Bloom filtering.• Construct a matrix M, but relax

requirements• Given a set D of d columns in M and a

column j, say j is distinguishable from D if there is a row i such that M[i,j]=1 but M[i,j’]=0 for each j’ in D.

• M is D-distinguishable if, for a particular collection D of subsets, the matrix M will find them distinguishable.

Constructing the Matrix• Given t (set in the analysis), let M

be a 2t x n matrix defined randomly:– For each column j, choose t/d rows of

M at random and set these entries to 1.

– that is, we “inject” j into those t/d tests

Technique for Social Networks• Insert a small set of network

members• Form connections with random

network members• Test common-friends condition for the fictional members

Image from http://www.politicsforum.org/images/flame_warriors/flame_53.php

Exploiting Sparse Data Sets

0 500 1000 15000

50

100

150Facebook-Uniform

0 500 1000 15000

1000

2000

3000

4000Facebook-UNC

0 50 100 1500

100

200

300

400LiveJournal

0 50 1000

20

40

60Genomic

0 5 10 15 200

500

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1500

2000Power Grid

0 500 1000 15000

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200Netflix

Histogram of differences from R:

Table of sizes, lengths, and differences from R:

Number of Tests Needed in Theory

1st column: To clone entire database with high probability2nd column: To clone sparsest 50% of database with high probability3rd column: To clone entire database with probability 1

Different Choices for “d” • Tradeoff:

– The smaller the “d”, the faster we can recover sparse vectors– With very small “d”, it can take a long time to recover the

vectors that are not so sparse.

• But most vectors are sparse so we generally want a pretty small “d”

1000 2000 3000 4000 50000

0.5

1

1.5

2x 10

4

Tests

L1 E

rror

d̂ = 998

d̂ = 165

d̂ = 98

d̂ = 60 Attack on a Netflix user who has rated 98 movies.

With smaller “d”, the rate of convergence is faster.

0 500 10000

1

2

3

4x 10

4 Facebook-UNC

d̂

Tes

ts

meanmedian

20 40 60 80800

1000

1200

1400

1600

1800Genomic

d̂

Tes

ts

mean0 5 10 15 20

0

100

200

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400Power Grid

d̂

Tes

ts

meanmedian

0 100 200 3000

2000

4000

6000

8000

10000LiveJournal

d̂T

ests

meanmedian

200 400 600 8000

0.5

1

1.5

2x 10

4 Netflix

d̂

Tes

tsmeanmedian

Different choices for “d” • Here we vary “d” on the x-axis and we plot the mean and

median number of tests required across the vectors in the database.

0 200 400 6000

1

2

3

4

5x 10

4

Distance from R

Tes

ts

Facebook-UNC

d̂=25

d̂=62

d̂=84

d̂=1000

0 50 1000

2000

4000

6000

8000

10000

Distance from R

Tes

ts

Genomic

d̂=22

0 5 10 15 200

1000

2000

3000

Distance from R

Tes

ts

Power Grid

d̂ = 1.5

d̂ = 2

d̂ = 5

d̂ = 19

0 100 200 3000

1

2

3

4x 10

4

Distance from R

Te

sts

LiveJournal

d̂=6

d̂=17

d̂=100

d̂=320

0 500 10000

1

2

3

4

5x 10

4

Distance from R

Tes

ts

Netflix

d̂=165

Distance from R • More tests are needed for vectors which are further from the

reference R (but note most vectors are close to R).

• We also see the tradeoff between various “d”

Thresholding Behavior• There are critical values of our

estimated value for d:

Conclusion and Future Work• We have presented a

way to turn privacy leaks into floods, with a number of applications:– Social networks– DNA databases– Ratings vectors

• Future work: extend our approach to non-binary vectors (e.g., friends and foes)