Seeded graph matching: Efficient algorithms and theoretical guarantees

In this paper, a new information theoretic framework for graph matching is introduced. Using this framework, the graph isomorphism and seeded graph matching problems are studied. The maximum degree algorithm for graph isomorphism is analyzed and sufficient conditions for successful matching are rederived using type analysis. Furthermore, a new seeded matching algorithm with polynomial time complexity is introduced. The algorithm uses ‘typicality matching’ and techniques from point-to-point communications for reliable matching. Assuming an Erdös-Renyi model on the correlated graph pair, it is shown that successful matching is guaranteed when the number of seeds grows logarithmically with the number of vertices in the graphs. The logarithmic coefficient is shown to be inversely proportional to the mutual information between the edge variables in the two graphs.

[1]  Michele Garetto,et al.  Social Network De-Anonymization Under Scale-Free User Relations , 2016, IEEE/ACM Transactions on Networking.

[2]  Xinbing Wang,et al.  De-anonymization of Social Networks with Communities: When Quantifications Meet Algorithms , 2017, ArXiv.

[3]  Paul Erdös,et al.  Random Graph Isomorphism , 1980, SIAM J. Comput..

[4]  David G. Kirkpatrick,et al.  A Theoretical Analysis of Various Heuristics for the Graph Isomorphism Problem , 1980, SIAM J. Comput..

[5]  Matthias Grossglauser,et al.  A Bayesian method for matching two similar graphs without seeds , 2013, 2013 51st Annual Allerton Conference on Communication, Control, and Computing (Allerton).

[6]  Daniel Cullina,et al.  Improved Achievability and Converse Bounds for Erdos-Renyi Graph Matching , 2016, SIGMETRICS.

[7]  Matthias Grossglauser,et al.  When can two unlabeled networks be aligned under partial overlap? , 2015, 2015 53rd Annual Allerton Conference on Communication, Control, and Computing (Allerton).

[8]  Elza Erkip,et al.  An information theoretic framework for active de-anonymization in social networks based on group memberships , 2017, 2017 55th Annual Allerton Conference on Communication, Control, and Computing (Allerton).

[9]  Guillermo Sapiro,et al.  Robust Multimodal Graph Matching: Sparse Coding Meets Graph Matching , 2013, NIPS.

[10]  Gopal Pandurangan,et al.  Improved Random Graph Isomorphism Tomek Czajka , 2006 .

[11]  Onaran Efe,et al.  Optimal de-anonymization in random graphs with community structure , 2016 .

[12]  Ehsan Kazemi,et al.  Network Alignment: Theory, Algorithms, and Applications , 2016 .

[13]  E. Wright.  Graphs on unlabelled nodes with a given number of edges , 1971 .

[14]  Carey E. Priebe,et al.  Seeded graph matching for correlated Erdös-Rényi graphs , 2014, J. Mach. Learn. Res..

[15]  B. Bollobás.  The evolution of random graphs , 1984 .

[16]  P. Erdos,et al.  On the evolution of random graphs , 1984 .

[17]  Prateek Mittal,et al.  Seed-Based De-Anonymizability Quantification of Social Networks , 2016, IEEE Transactions on Information Forensics and Security.

[18]  Matthias Grossglauser,et al.  Growing a Graph Matching from a Handful of Seeds , 2015, Proc. VLDB Endow..

[19]  Shouling Ji,et al.  Structural Data De-anonymization: Quantification, Practice, and Implications , 2014, CCS.

[20]  Matthias Grossglauser,et al.  On the performance of percolation graph matching , 2013, COSN '13.