xiaowei ying, xintao wu dept. software and information systems univ. of n.c. – charlotte 2008 siam...

Xiaowei Ying, Xintao WuDept. Software and Information Systems

Univ. of N.C. – Charlotte

2008 SIAM Conference on Data Mining, April 25th Atlanta, Georgia

Randomizing Social Network:A Spectrum Preserving

Approach

Randomizing Social Network: a Spectrum Preserving Approach, SDM08


Framework

• Background & Motivation

• Graph Spectrum & Structure

• Spectrum & Perturbation

• Spectrum Preserving

Randomization

• Privacy Protection

• Conclusion & Future Work2


Background & MotivationSocial Network

3 Randomizing Social Network: a Spectrum Preserving Approach, SDM08

• Friendship in Karate club [Zachary, 77]• Biological association network of dolphins [Lusseau et al., 03]

• Collaboration network of scientists [Newman, 06]

Network of US political books

(105 nodes, 441 edges)Books about US politics sold by Amazon.com. Edges represent frequent co-purchasing of books by the same buyers. Nodes have been given colors of blue, white, or red to indicate whether they are "liberal", "neutral", or "conservative".


Background & MotivationPrivacy Issues in Social Network:

Social network contains much private relation information;

Anonymization is not enough for protecting the privacy. Subgraph attacks [Backstrom et al., WWW07, Hay et al., 07].



Background & MotivationGraph Randomization/Perturbation:1.Random Add/Del edges (no. of edges

unchanged)

2.Random Switch edges (nodes’ degree unchanged)

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2 3

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5

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2 3

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Background & MotivationGraph perturbation is resilient to subgraph attacks (refer to our paper for more details).



MotivationGraph Randomization/Perturbation: Data utility:

How will the graph structure change due to perturbation?

How to preserve graph structural features better?

Data privacy:Protection on the link privacy.




Graph Spectrum & Structure

•Background & Motivation•Graph Spectrum & Structure•Spectrum & Perturbation•Spectrum Preserving Randomization•Privacy Protection•Conclusion & Future Work

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2008 SIAM Conference on Data Mining, April 25th Atlanta, GeorgiaGraph Spectrum and StructureNumerous properties and measures of

networks (Graph G contains n nodes and m edges):Harmonic mean of shortest distance;

Transitivity(cluster coefficient)

Subgraph centrality;

Modularity (community structure);And many others


2008 SIAM Conference on Data Mining, April 25th Atlanta, GeorgiaGraph Spectrum and StructureAdjacency Matrix (Graph G contains n

nodes and m edges):

Adjacency SpectrumA is symmetric, it has n real eigenvalues:


2008 SIAM Conference on Data Mining, April 25th Atlanta, GeorgiaGraph Spectrum and StructureLaplacian Matrix:

Laplacian Spectrum


2008 SIAM Conference on Data Mining, April 25th Atlanta, GeorgiaGraph Spectrum and StructureMany real graph structural features are related to adjacency/Laplacian spectrum, e.g.:

No. of triangles:

Subgraph centrality:

Graph diameter:

k disconnected parts in the graph ⇔ k 0’s in the Laplacian spectrum.


2008 SIAM Conference on Data Mining, April 25th Atlanta, GeorgiaGraph Spectrum and StructureTwo important eigenvalues: and1.The maximum degree, chromatic number, clique number etc. are related to ;2.Epidemic threshold for virus propagates in the network is related to [Wang et al., KDD03];3. indicates the community structure of the graph:

clear community structure ⇔ ≈ 0.




Spectrum & Perturbation


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Spectrum & PerturbationGraph Perturbation:1.Random Add/Del edges (no. of edges doesn’t

change)

2.Random Switch edges (nodes’ degree doesn’t change)

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2 3

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2 3

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2 3

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Spectrum & PerturbationBoth topological and spectral graph features change along the perturbation, and they shows similar trends.

(Networks of US political books, 105 nodes and 441 edges)16 Randomizing Social Network: a Spectrum Preserving Approach,

SDM08


Spectrum & PerturbationGeneral bound on spectrum in

perturbation:Do the randomization for k times (refer to our

paper for more details)




Spectrum Preserving Randomization


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2008 SIAM Conference on Data Mining, April 25th Atlanta, GeorgiaSpectrum Preserving Randomization

Intuition: since spectrum is related to many graph topological features, can we preserve more structural features by controlling the movement of eigenvalues?


2008 SIAM Conference on Data Mining, April 25th Atlanta, GeorgiaSpectrum Preserving RandomizationSpectral Switch (apply to adjacency matrix):

To increase the eigenvalue:

To decrease the eigenvalue:

t (xt)

v (xv)

u (xu)

w (xw)

t (xt) u (xu)

w (xw) v (xv)


2008 SIAM Conference on Data Mining, April 25th Atlanta, GeorgiaSpectrum Preserving RandomizationSpectral Switch (apply to Laplacian matrix):

To decrease the eigenvalue:

To increase the eigenvalue:

t (yt)

v (yv)

u (yu)

w (yw)

t (yt) u (yu)

w (yw) v (yv)


2008 SIAM Conference on Data Mining, April 25th Atlanta, GeorgiaSpectrum Preserving RandomizationEvaluation:

(Networks of US political books, 105 nodes and 441 edges)22 Randomizing Social Network: a Spectrum Preserving Approach,

SDM08

2008 SIAM Conference on Data Mining, April 25th Atlanta, GeorgiaSpectrum Preserving Randomization

Similarly, we also develop Spectral Add/Del strategy(Refer to our paper for more details)

In summary, by controlling the movement of the eigenvalues, spectrum can preserving randomization strategies better preserve the graph structure.




Privacy Protection


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Privacy ProtectionPrivacy protection measure:A-prior probability (without the released

data):

Posterior probability (with released the data & perturbation parameters):

The absolute measure

The relative measure



Privacy ProtectionHow many times shall we do add/del or

switches?Objective: the minimum level of protection should be above some threshold:

For random add/del and random switch



Privacy ProtectionSpectral strategy and random strategy do not differ much in protecting the privacy:

In the graph, there exits both up-edge pairs and down-edge pairs.

Their proportions affect the privacy protection of spectral strategy

Further study in future work




Conclusion & Future Work


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Conclusion

1. Graph structure and spectrum are closely related, and perturbation can significantly change both.

2. Spectrum preserving randomization strategies can better preserve the graph structure;

3. Privacy protection issues for random perturbation.



Future Work

1. Further study on privacy issues of spectral strategy;

2. A more flexible algorithm for other eigenvalues;

3. Algorithms controlling the magnitude of eigenvalues’ change.



Thank You! Questions?

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Reference

1.L. Backstrom et al., Wherefore art thou r3579x?: anonymized social networks, hidden patterns, and structural steganography, 2007.

2.M. Hay et al., Anonymizing social networks, 2007.

3.Y. Wang et al., Epidemic spreading in real networks: An eigenvalue viewpoint, 2003.



Privacy ProtectionPrivacy protection measure:

We can proof, for a given (i, j)

Therefore, to calculation the measure is based on calculating the number of false edges (refer to our paper for more details).


xiaowei ying, xintao wu dept. software and information systems univ. of n.c. – charlotte 2008 siam...

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