xiaowei ying, xintao wu dept. software and information systems univ. of n.c. – charlotte 2008 siam...
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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
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Randomizing Social Network: a Spectrum Preserving Approach, SDM08
2008 SIAM Conference on Data Mining, April 25th Atlanta, Georgia
Framework
• Background & Motivation
• Graph Spectrum & Structure
• Spectrum & Perturbation
• Spectrum Preserving
Randomization
• Privacy Protection
• Conclusion & Future Work2
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2008 SIAM Conference on Data Mining, April 25th Atlanta, Georgia
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".
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2008 SIAM Conference on Data Mining, April 25th Atlanta, Georgia
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].
4 Randomizing Social Network: a Spectrum Preserving Approach, SDM08
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2008 SIAM Conference on Data Mining, April 25th Atlanta, Georgia
Background & MotivationGraph Randomization/Perturbation:1.Random Add/Del edges (no. of edges
unchanged)
2.Random Switch edges (nodes’ degree unchanged)
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2008 SIAM Conference on Data Mining, April 25th Atlanta, Georgia
Background & MotivationGraph perturbation is resilient to subgraph attacks (refer to our paper for more details).
6 Randomizing Social Network: a Spectrum Preserving Approach, SDM08
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2008 SIAM Conference on Data Mining, April 25th Atlanta, Georgia
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.
7 Randomizing Social Network: a Spectrum Preserving Approach, SDM08
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Randomizing Social Network: a Spectrum Preserving Approach, SDM08
2008 SIAM Conference on Data Mining, April 25th Atlanta, Georgia
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
9 Randomizing Social Network: a Spectrum Preserving Approach, SDM08
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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:
10 Randomizing Social Network: a Spectrum Preserving Approach, SDM08
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2008 SIAM Conference on Data Mining, April 25th Atlanta, GeorgiaGraph Spectrum and StructureLaplacian Matrix:
Laplacian Spectrum
11 Randomizing Social Network: a Spectrum Preserving Approach, SDM08
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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.
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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.
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Randomizing Social Network: a Spectrum Preserving Approach, SDM08
2008 SIAM Conference on Data Mining, April 25th Atlanta, Georgia
Spectrum & Perturbation
•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, Georgia
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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2008 SIAM Conference on Data Mining, April 25th Atlanta, Georgia
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
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2008 SIAM Conference on Data Mining, April 25th Atlanta, Georgia
Spectrum & PerturbationGeneral bound on spectrum in
perturbation:Do the randomization for k times (refer to our
paper for more details)
17 Randomizing Social Network: a Spectrum Preserving Approach, SDM08
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Randomizing Social Network: a Spectrum Preserving Approach, SDM08
2008 SIAM Conference on Data Mining, April 25th Atlanta, Georgia
Spectrum Preserving Randomization
•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, 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?
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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)
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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)
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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
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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.
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Randomizing Social Network: a Spectrum Preserving Approach, SDM08
2008 SIAM Conference on Data Mining, April 25th Atlanta, Georgia
Privacy Protection
•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, Georgia
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
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2008 SIAM Conference on Data Mining, April 25th Atlanta, Georgia
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
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2008 SIAM Conference on Data Mining, April 25th Atlanta, Georgia
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
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Randomizing Social Network: a Spectrum Preserving Approach, SDM08
2008 SIAM Conference on Data Mining, April 25th Atlanta, Georgia
Conclusion & Future Work
•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, Georgia
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.
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2008 SIAM Conference on Data Mining, April 25th Atlanta, Georgia
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.
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Randomizing Social Network: a Spectrum Preserving Approach, SDM08
Thank You! Questions?
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2008 SIAM Conference on Data Mining, April 25th Atlanta, Georgia
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.
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2008 SIAM Conference on Data Mining, April 25th Atlanta, Georgia
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).
33 Randomizing Social Network: a Spectrum Preserving Approach, SDM08