Document Open Access Logo

Privately Estimating Graph Parameters in Sublinear Time

Authors Jeremiah Blocki, Elena Grigorescu, Tamalika Mukherjee

PDF
Thumbnail PDF

File

LIPIcs.ICALP.2022.26.pdf
  • Filesize: 0.84 MB
  • 19 pages

Document Identifiers

Author Details

Jeremiah Blocki
  • Department of Computer Science, Purdue University, West Lafayette, IN, USA
Elena Grigorescu
  • Department of Computer Science, Purdue University, West Lafayette, IN, USA
Tamalika Mukherjee
  • Department of Computer Science, Purdue University, West Lafayette, IN, USA

Funding

  • Blocki, Jeremiah: supported in part by NSF CNS-1931443 and NSF CCF-1910659.
  • Grigorescu, Elena: supported in part by NSF CCF-1910659 and NSF CCF-1910411.
  • Mukherjee, Tamalika: supported in part by NSF CCF-1910659 and NSF CCF-1910411.

Acknowledgements

We thank several anonymous reviewers for their valuable feedback on a preliminary version of this work. We thank Soheil Behnezhad for bringing to our attention the subtlety in the analysis of [Krzysztof Onak et al., 2012], first discovered by [Yu Chen et al., 2020], and finally resolved in his recent work [Soheil Behnezhad, 2021]. We also thank Jakub Tetek for bringing up several points for clarification, which we have incorporated in the current version.

Cite As Get BibTex

Jeremiah Blocki, Elena Grigorescu, and Tamalika Mukherjee. Privately Estimating Graph Parameters in Sublinear Time. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 26:1-26:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.ICALP.2022.26

Abstract

We initiate a systematic study of algorithms that are both differentially-private and run in sublinear time for several problems in which the goal is to estimate natural graph parameters. Our main result is a differentially-private (1+ρ)-approximation algorithm for the problem of computing the average degree of a graph, for every ρ > 0. The running time of the algorithm is roughly the same (for sparse graphs) as its non-private version proposed by Goldreich and Ron (Sublinear Algorithms, 2005). We also obtain the first differentially-private sublinear-time approximation algorithms for the maximum matching size and the minimum vertex cover size of a graph.
An overarching technique we employ is the notion of coupled global sensitivity of randomized algorithms. Related variants of this notion of sensitivity have been used in the literature in ad-hoc ways. Here we formalize the notion and develop it as a unifying framework for privacy analysis of randomized approximation algorithms.

Subject Classification

ACM Subject Classification
  • Security and privacy → Privacy-preserving protocols
  • Theory of computation → Streaming, sublinear and near linear time algorithms
Keywords
  • differential privacy
  • sublinear time
  • graph algorithms

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Related Versions

References

  1. Daniel Alabi, Audra McMillan, Jayshree Sarathy, Adam D. Smith, and Salil P. Vadhan. Differentially private simple linear regression. CoRR, abs/2007.05157, 2020. URL: http://arxiv.org/abs/2007.05157.
  2. Rémi Bardenet and Odalric-Ambrym Maillard. Concentration inequalities for sampling without replacement. Bernoulli, 21(3):1361-1385, 2015. Google Scholar
  3. Soheil Behnezhad. Time-optimal sublinear algorithms for matching and vertex cover. CoRR, abs/2106.02942, 2021. URL: http://arxiv.org/abs/2106.02942.
  4. Jeremiah Blocki, Avrim Blum, Anupam Datta, and Or Sheffet. Differentially private data analysis of social networks via restricted sensitivity. In Robert D. Kleinberg, editor, Innovations in Theoretical Computer Science, ITCS '13, Berkeley, CA, USA, January 9-12, 2013, pages 87-96. ACM, 2013. URL: https://doi.org/10.1145/2422436.2422449.
  5. Jeremiah Blocki, Elena Grigorescu, and Tamalika Mukherjee. Privately estimating graph parameters in sublinear time. CoRR, abs/2202.05776, 2022. URL: http://arxiv.org/abs/2202.05776.
  6. Christian Borgs, Jennifer T. Chayes, and Adam D. Smith. Private graphon estimation for sparse graphs. In Corinna Cortes, Neil D. Lawrence, Daniel D. Lee, Masashi Sugiyama, and Roman Garnett, editors, Advances in Neural Information Processing Systems 28: Annual Conference on Neural Information Processing Systems 2015, December 7-12, 2015, Montreal, Quebec, Canada, pages 1369-1377, 2015. URL: https://proceedings.neurips.cc/paper/2015/hash/7250eb93b3c18cc9daa29cf58af7a004-Abstract.html.
  7. Christian Borgs, Jennifer T. Chayes, Adam D. Smith, and Ilias Zadik. Revealing network structure, confidentially: Improved rates for node-private graphon estimation. In Mikkel Thorup, editor, 59th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2018, Paris, France, October 7-9, 2018, pages 533-543. IEEE Computer Society, 2018. URL: https://doi.org/10.1109/FOCS.2018.00057.
  8. Kamalika Chaudhuri and Staal A. Vinterbo. A stability-based validation procedure for differentially private machine learning. In Christopher J. C. Burges, Léon Bottou, Zoubin Ghahramani, and Kilian Q. Weinberger, editors, Advances in Neural Information Processing Systems 26: 27th Annual Conference on Neural Information Processing Systems 2013. Proceedings of a meeting held December 5-8, 2013, Lake Tahoe, Nevada, United States, pages 2652-2660, 2013. URL: https://proceedings.neurips.cc/paper/2013/hash/e6d8545daa42d5ced125a4bf747b3688-Abstract.html.
  9. Shixi Chen and Shuigeng Zhou. Recursive mechanism: towards node differential privacy and unrestricted joins. In Kenneth A. Ross, Divesh Srivastava, and Dimitris Papadias, editors, Proceedings of the ACM SIGMOD International Conference on Management of Data, SIGMOD 2013, New York, NY, USA, June 22-27, 2013, pages 653-664. ACM, 2013. URL: https://doi.org/10.1145/2463676.2465304.
  10. Yu Chen, Sampath Kannan, and Sanjeev Khanna. Sublinear algorithms and lower bounds for metric TSP cost estimation. In Artur Czumaj, Anuj Dawar, and Emanuela Merelli, editors, 47th International Colloquium on Automata, Languages, and Programming, ICALP 2020, July 8-11, 2020, Saarbrücken, Germany (Virtual Conference), volume 168 of LIPIcs, pages 30:1-30:19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.ICALP.2020.30.
  11. Anirban Dasgupta, Ravi Kumar, and Tamás Sarlós. On estimating the average degree. In Chin-Wan Chung, Andrei Z. Broder, Kyuseok Shim, and Torsten Suel, editors, 23rd International World Wide Web Conference, WWW '14, Seoul, Republic of Korea, April 7-11, 2014, pages 795-806. ACM, 2014. URL: https://doi.org/10.1145/2566486.2568019.
  12. Cynthia Dwork, Frank McSherry, Kobbi Nissim, and Adam Smith. Calibrating noise to sensitivity in private data analysis. Journal of Privacy and Confidentiality, 7(3):17-51, May 2017. Google Scholar
  13. Cynthia Dwork and Aaron Roth. The algorithmic foundations of differential privacy. Found. Trends Theor. Comput. Sci., 9(3-4):211-407, 2014. URL: https://doi.org/10.1561/0400000042.
  14. Uriel Feige. On sums of independent random variables with unbounded variance and estimating the average degree in a graph. SIAM J. Comput., 35(4):964-984, 2006. URL: https://doi.org/10.1137/S0097539704447304.
  15. Hendrik Fichtenberger, Monika Henzinger, and Wolfgang Ost. Differentially private algorithms for graphs under continual observation. In Petra Mutzel, Rasmus Pagh, and Grzegorz Herman, editors, 29th Annual European Symposium on Algorithms, ESA 2021, September 6-8, 2021, Lisbon, Portugal (Virtual Conference), volume 204 of LIPIcs, pages 42:1-42:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.ESA.2021.42.
  16. Johannes Gehrke, Edward Lui, and Rafael Pass. Towards privacy for social networks: A zero-knowledge based definition of privacy. In Yuval Ishai, editor, Theory of Cryptography - 8th Theory of Cryptography Conference, TCC 2011, Providence, RI, USA, March 28-30, 2011. Proceedings, volume 6597 of Lecture Notes in Computer Science, pages 432-449. Springer, 2011. URL: https://doi.org/10.1007/978-3-642-19571-6_26.
  17. Oded Goldreich and Dana Ron. Approximating average parameters of graphs. Random Struct. Algorithms, 32(4):473-493, 2008. URL: https://doi.org/10.1002/rsa.20203.
  18. Anupam Gupta, Katrina Ligett, Frank McSherry, Aaron Roth, and Kunal Talwar. Differentially private combinatorial optimization. In Moses Charikar, editor, Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2010, Austin, Texas, USA, January 17-19, 2010, pages 1106-1125. SIAM, 2010. URL: https://doi.org/10.1137/1.9781611973075.90.
  19. Michael Hay, Chao Li, Gerome Miklau, and David D. Jensen. Accurate estimation of the degree distribution of private networks. In Wei Wang, Hillol Kargupta, Sanjay Ranka, Philip S. Yu, and Xindong Wu, editors, ICDM 2009, The Ninth IEEE International Conference on Data Mining, Miami, Florida, USA, 6-9 December 2009, pages 169-178. IEEE Computer Society, 2009. URL: https://doi.org/10.1109/ICDM.2009.11.
  20. Wassily Hoeffding. Probability inequalities for sums of bounded random variables. In The collected works of Wassily Hoeffding, pages 409-426. Springer, 1994. Google Scholar
  21. Vishesh Karwa, Sofya Raskhodnikova, Adam D. Smith, and Grigory Yaroslavtsev. Private analysis of graph structure. ACM Trans. Database Syst., 39(3):22:1-22:33, 2014. URL: https://doi.org/10.1145/2611523.
  22. Shiva Prasad Kasiviswanathan, Kobbi Nissim, Sofya Raskhodnikova, and Adam D. Smith. Analyzing graphs with node differential privacy. In Amit Sahai, editor, Theory of Cryptography - 10th Theory of Cryptography Conference, TCC 2013, Tokyo, Japan, March 3-6, 2013. Proceedings, volume 7785 of Lecture Notes in Computer Science, pages 457-476. Springer, 2013. URL: https://doi.org/10.1007/978-3-642-36594-2_26.
  23. Wentian Lu and Gerome Miklau. Exponential random graph estimation under differential privacy. In Sofus A. Macskassy, Claudia Perlich, Jure Leskovec, Wei Wang, and Rayid Ghani, editors, The 20th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD '14, New York, NY, USA - August 24 - 27, 2014, pages 921-930. ACM, 2014. URL: https://doi.org/10.1145/2623330.2623683.
  24. Huy N. Nguyen and Krzysztof Onak. Constant-time approximation algorithms via local improvements. In 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008, October 25-28, 2008, Philadelphia, PA, USA, pages 327-336. IEEE Computer Society, 2008. URL: https://doi.org/10.1109/FOCS.2008.81.
  25. Kobbi Nissim, Sofya Raskhodnikova, and Adam D. Smith. Smooth sensitivity and sampling in private data analysis. In David S. Johnson and Uriel Feige, editors, Proceedings of the 39th Annual ACM Symposium on Theory of Computing, San Diego, California, USA, June 11-13, 2007, pages 75-84. ACM, 2007. URL: https://doi.org/10.1145/1250790.1250803.
  26. Krzysztof Onak, Dana Ron, Michal Rosen, and Ronitt Rubinfeld. A near-optimal sublinear-time algorithm for approximating the minimum vertex cover size. In Yuval Rabani, editor, Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012, Kyoto, Japan, January 17-19, 2012, pages 1123-1131. SIAM, 2012. URL: https://doi.org/10.1137/1.9781611973099.88.
  27. Michal Parnas and Dana Ron. Approximating the minimum vertex cover in sublinear time and a connection to distributed algorithms. Theor. Comput. Sci., 381(1-3):183-196, 2007. URL: https://doi.org/10.1016/j.tcs.2007.04.040.
  28. Sofya Raskhodnikova and Adam D. Smith. Efficient lipschitz extensions for high-dimensional graph statistics and node private degree distributions. CoRR, abs/1504.07912, 2015. URL: http://arxiv.org/abs/1504.07912.
  29. Dana Ron. Sublinear-time algorithms for approximating graph parameters. In Computing and Software Science, volume 10000 of Lecture Notes in Computer Science, pages 105-122. Springer, 2019. Google Scholar
  30. Adam Sealfon and Jonathan R. Ullman. Efficiently estimating erdos-renyi graphs with node differential privacy. J. Priv. Confidentiality, 11(1), 2021. URL: https://doi.org/10.29012/jpc.745.
  31. C. Seshadhri. A simpler sublinear algorithm for approximating the triangle count. CoRR, abs/1505.01927, 2015. URL: http://arxiv.org/abs/1505.01927.
  32. Harry Sivasubramaniam, Haonan Li, and Xi He. Differentially private sublinear average degree approximation. Google Scholar
  33. Shuang Song, Susan Little, Sanjay Mehta, Staal A. Vinterbo, and Kamalika Chaudhuri. Differentially private continual release of graph statistics. CoRR, abs/1809.02575, 2018. URL: http://arxiv.org/abs/1809.02575.
  34. Douglas Brent West et al. Introduction to graph theory, volume 2. Prentice hall Upper Saddle River, 2001. Google Scholar
  35. Yuichi Yoshida, Masaki Yamamoto, and Hiro Ito. Improved constant-time approximation algorithms for maximum matchings and other optimization problems. SIAM J. Comput., 41(4):1074-1093, 2012. URL: https://doi.org/10.1137/110828691.
  36. Jun Zhang, Graham Cormode, Cecilia M. Procopiuc, Divesh Srivastava, and Xiaokui Xiao. Private release of graph statistics using ladder functions. In Timos K. Sellis, Susan B. Davidson, and Zachary G. Ives, editors, Proceedings of the 2015 ACM SIGMOD International Conference on Management of Data, Melbourne, Victoria, Australia, May 31 - June 4, 2015, pages 731-745. ACM, 2015. URL: https://doi.org/10.1145/2723372.2737785.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact