Document Open Access Logo

Cycle Counting Under Local Differential Privacy for Degeneracy-Bounded Graphs

Authors Quentin Hillebrand , Vorapong Suppakitpaisarn , Tetsuo Shibuya

PDF

File

Thumbnail PDF
PDF
LIPIcs.STACS.2025.49.pdf
  • Filesize: 0.95 MB
  • 22 pages

Document Identifiers

Subject Classification

ACM Subject Classification
  • Security and privacy → Privacy-preserving protocols
  • Theory of computation → Graph algorithms analysis
Keywords
  • Differential privacy
  • triangle counting
  • degeneracy
  • arboricity
  • graph theory
  • parameterized accuracy

Metrics

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

Abstract

We propose an algorithm for counting the number of cycles under local differential privacy for degeneracy-bounded input graphs. Numerous studies have focused on counting the number of triangles under the privacy notion, demonstrating that the expected 𝓁₂-error of these algorithms is Ω(n^{1.5}), where n is the number of nodes in the graph. When parameterized by the number of cycles of length four (C₄), the best existing triangle counting algorithm has an error of O(n^{1.5} + √C₄) = O(n²). In this paper, we introduce an algorithm with an expected 𝓁₂-error of O(δ^1.5 n^0.5 + δ^0.5 d_max^0.5 n^0.5), where δ is the degeneracy and d_{max} is the maximum degree of the graph. For degeneracy-bounded graphs (δ ∈ Θ(1)) commonly found in practical social networks, our algorithm achieves an expected 𝓁₂-error of O(d_{max}^{0.5} n^{0.5}) = O(n). Our algorithm’s core idea is a precise count of triangles following a preprocessing step that approximately sorts the degree of all nodes. This approach can be extended to approximate the number of cycles of length k, maintaining a similar 𝓁₂-error, namely O(δ^{(k-2)/2} d_max^0.5 n^{(k-2)/2} + δ^{k/2} n^{(k-2)/2}) or O(d_max^0.5 n^{(k-2)/2}) = O(n^{(k-1)/2}) for degeneracy-bounded graphs.

Cite As Get BibTex

Quentin Hillebrand, Vorapong Suppakitpaisarn, and Tetsuo Shibuya. Cycle Counting Under Local Differential Privacy for Degeneracy-Bounded Graphs. In 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 327, pp. 49:1-49:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.STACS.2025.49

Author Details

Quentin Hillebrand
  • The University of Tokyo, Japan
Vorapong Suppakitpaisarn
  • The University of Tokyo, Japan
Tetsuo Shibuya
  • The University of Tokyo, Japan

Funding

  • Hillebrand, Quentin: partially supported by KAKENHI Grant 20H05965, and by JST SPRING Grant Number JPMJSP2108.
  • Suppakitpaisarn, Vorapong: partially supported by KAKENHI Grant 21H05845 and 23H04377.
  • Shibuya, Tetsuo: partially supported by KAKENHI Grant 20H05967, 21H05052, and 23H03345.

Acknowledgements

The authors wish to express their thanks to the anonymous reviewers whose valuable feedback greatly enhanced the quality of this paper.

References

  1. Albert-László Barabási and Réka Albert. Emergence of scaling in random networks. science, 286(5439):509-512, 1999. Google Scholar
  2. Suman K. Bera, Lior Gishboliner, Yevgeny Levanzov, C. Seshadhri, and Asaf Shapira. Counting subgraphs in degenerate graphs. ACM Journal of the ACM (JACM), 69(3):23:1-23:21, 2022. URL: https://doi.org/10.1145/3520240.
  3. Louis Betzer, Vorapong Suppakitpaisarn, and Quentin Hillebrand. Publishing number of walks and katz centrality under local differential privacy. In The 40th Conference on Uncertainty in Artificial Intelligence, 2024. URL: https://openreview.net/forum?id=76UkTmdmkB.
  4. Jeremiah Blocki, Avrim Blum, Anupam Datta, and Or Sheffet. Differentially private data analysis of social networks via restricted sensitivity. In 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. Marco Bressan. Faster algorithms for counting subgraphs in sparse graphs. Algorithmica, 83(8):2578-2605, 2021. URL: https://doi.org/10.1007/s00453-021-00811-0.
  6. Norishige Chiba and Takao Nishizeki. Arboricity and subgraph listing algorithms. SIAM J. Comput., 14(1):210-223, 1985. URL: https://doi.org/10.1137/0214017.
  7. Graham Cormode, Somesh Jha, Tejas Kulkarni, Ninghui Li, Divesh Srivastava, and Tianhao Wang. Privacy at scale: Local differential privacy in practice. In Proceedings of the 2018 International Conference on Management of Data, SIGMOD Conference 2018, Houston, TX, USA, June 10-15, 2018, pages 1655-1658. ACM, 2018. URL: https://doi.org/10.1145/3183713.3197390.
  8. Damien Desfontaines and Balázs Pejó. Sok: Differential privacies. Proc. Priv. Enhancing Technol., 2020(2):288-313, 2020. URL: https://doi.org/10.2478/popets-2020-0028.
  9. Laxman Dhulipala, George Z. Li, and Quanquan C. Liu. Near-optimal differentially private k-core decomposition. CoRR, abs/2312.07706, 2023. URL: https://doi.org/10.48550/arXiv.2312.07706.
  10. Laxman Dhulipala, Quanquan C. Liu, Sofya Raskhodnikova, Jessica Shi, Julian Shun, and Shangdi Yu. Differential privacy from locally adjustable graph algorithms: k-core decomposition, low out-degree ordering, and densest subgraphs. In 63rd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2022, Denver, CO, USA, October 31 - November 3, 2022, pages 754-765. IEEE, 2022. URL: https://doi.org/10.1109/FOCS54457.2022.00077.
  11. Michael Dinitz, Satyen Kale, Silvio Lattanzi, and Sergei Vassilvitskii. Improved differentially private densest subgraph: Local and purely additive. CoRR, abs/2308.10316, 2023. URL: https://doi.org/10.48550/arXiv.2308.10316.
  12. Pål Grønås Drange, Patrick Greaves, Irene Muzi, and Felix Reidl. Computing complexity measures of degenerate graphs. In 18th International Symposium on Parameterized and Exact Computation, IPEC 2023, September 6-8, 2023, Amsterdam, The Netherlands, volume 285 of LIPIcs, pages 14:1-14:21. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.IPEC.2023.14.
  13. Cynthia Dwork. Differential privacy. In Automata, Languages and Programming, 33rd International Colloquium, ICALP 2006, Venice, Italy, July 10-14, 2006, Proceedings, Part II, volume 4052 of Lecture Notes in Computer Science, pages 1-12. Springer, 2006. URL: https://doi.org/10.1007/11787006_1.
  14. Cynthia Dwork, Frank McSherry, Kobbi Nissim, and Adam D. Smith. Calibrating noise to sensitivity in private data analysis. In Theory of Cryptography, Third Theory of Cryptography Conference, TCC 2006, New York, NY, USA, March 4-7, 2006, Proceedings, volume 3876 of Lecture Notes in Computer Science, pages 265-284. Springer, 2006. URL: https://doi.org/10.1007/11681878_14.
  15. 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.
  16. Talya Eden, Quanquan C. Liu, Sofya Raskhodnikova, and Adam D. Smith. Triangle counting with local edge differential privacy. In 50th International Colloquium on Automata, Languages, and Programming, ICALP 2023, July 10-14, 2023, Paderborn, Germany, volume 261 of LIPIcs, pages 52:1-52:21. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.ICALP.2023.52.
  17. Alexandre V. Evfimievski, Johannes Gehrke, and Ramakrishnan Srikant. Limiting privacy breaches in privacy preserving data mining. In Proceedings of the Twenty-Second ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems, June 9-12, 2003, San Diego, CA, USA, pages 211-222. ACM, 2003. URL: https://doi.org/10.1145/773153.773174.
  18. Anupam Gupta, Katrina Ligett, Frank McSherry, Aaron Roth, and Kunal Talwar. Differentially private combinatorial optimization. In 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 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. Quentin Hillebrand, Vorapong Suppakitpaisarn, and Tetsuo Shibuya. Communication cost reduction for subgraph counting under local differential privacy via hash functions. CoRR, abs/2312.07055, 2023. URL: https://doi.org/10.48550/arXiv.2312.07055.
  21. Quentin Hillebrand, Vorapong Suppakitpaisarn, and Tetsuo Shibuya. Unbiased locally private estimator for polynomials of laplacian variables. In Proceedings of the 29th ACM SIGKDD Conference on Knowledge Discovery and Data Mining, KDD 2023, Long Beach, CA, USA, August 6-10, 2023, pages 741-751. ACM, 2023. URL: https://doi.org/10.1145/3580305.3599537.
  22. Jacob Imola, Takao Murakami, and Kamalika Chaudhuri. Locally differentially private analysis of graph statistics. In 30th USENIX Security Symposium, USENIX Security 2021, August 11-13, 2021, pages 983-1000. USENIX Association, 2021. URL: https://www.usenix.org/conference/usenixsecurity21/presentation/imola.
  23. Jacob Imola, Takao Murakami, and Kamalika Chaudhuri. Communication-efficient triangle counting under local differential privacy. In 31st USENIX Security Symposium, USENIX Security 2022, Boston, MA, USA, August 10-12, 2022, pages 537-554. USENIX Association, 2022. URL: https://www.usenix.org/conference/usenixsecurity22/presentation/imola.
  24. Jacob Imola, Takao Murakami, and Kamalika Chaudhuri. Differentially private triangle and 4-cycle counting in the shuffle model. In Proceedings of the 2022 ACM SIGSAC Conference on Computer and Communications Security, CCS 2022, Los Angeles, CA, USA, November 7-11, 2022, pages 1505-1519. ACM, 2022. URL: https://doi.org/10.1145/3548606.3560659.
  25. George Manoussakis. Listing all fixed-length simple cycles in sparse graphs in optimal time. In Fundamentals of Computation Theory - 21st International Symposium, FCT 2017, Bordeaux, France, September 11-13, 2017, Proceedings, volume 10472 of Lecture Notes in Computer Science, pages 355-366. Springer, Springer, 2017. URL: https://doi.org/10.1007/978-3-662-55751-8_28.
  26. Frank McSherry and Kunal Talwar. Mechanism design via differential privacy. In 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2007), October 20-23, 2007, Providence, RI, USA, Proceedings, pages 94-103. IEEE Computer Society, 2007. URL: https://doi.org/10.1109/FOCS.2007.41.
  27. Jaroslav Nesetril and Patrice Ossona de Mendez. Sparsity - Graphs, Structures, and Algorithms, volume 28 of Algorithms and combinatorics. Springer, 2012. URL: https://doi.org/10.1007/978-3-642-27875-4.
  28. Iyiola E. Olatunji, Thorben Funke, and Megha Khosla. Releasing graph neural networks with differential privacy guarantees. Trans. Mach. Learn. Res., 2023, 2023. URL: https://openreview.net/forum?id=wk8oXR0kFA.
  29. Zhan Qin, Ting Yu, Yin Yang, Issa Khalil, Xiaokui Xiao, and Kui Ren. Generating synthetic decentralized social graphs with local differential privacy. In Proceedings of the 2017 ACM SIGSAC Conference on Computer and Communications Security, CCS 2017, Dallas, TX, USA, October 30 - November 03, 2017, pages 425-438. ACM, 2017. URL: https://doi.org/10.1145/3133956.3134086.
  30. Sofya Raskhodnikova and Adam D. Smith. Differentially private analysis of graphs. In Encyclopedia of Algorithms, pages 543-547. Springer, 2016. URL: https://doi.org/10.1007/978-1-4939-2864-4_549.
  31. Sina Sajadmanesh and Daniel Gatica-Perez. Locally private graph neural networks. In CCS '21: 2021 ACM SIGSAC Conference on Computer and Communications Security, Virtual Event, Republic of Korea, November 15 - 19, 2021, pages 2130-2145. ACM, 2021. URL: https://doi.org/10.1145/3460120.3484565.
  32. Yue Wang, Xintao Wu, and Donghui Hu. Using randomized response for differential privacy preserving data collection. In Proceedings of the Workshops of the EDBT/ICDT 2016 Joint Conference, EDBT/ICDT Workshops 2016, Bordeaux, France, March 15, 2016, volume 1558 of CEUR Workshop Proceedings. CEUR-WS.org, 2016. URL: https://ceur-ws.org/Vol-1558/paper35.pdf.
  33. Stanley L. Warner. Randomized response: A survey technique for eliminating evasive answer bias. Journal of the American Statistical Association, 60(309):63-69, 1965. Google Scholar
  34. Qingqing Ye, Haibo Hu, Man Ho Au, Xiaofeng Meng, and Xiaokui Xiao. Towards locally differentially private generic graph metric estimation. In 36th IEEE International Conference on Data Engineering, ICDE 2020, Dallas, TX, USA, April 20-24, 2020, pages 1922-1925. IEEE, 2020. URL: https://doi.org/10.1109/ICDE48307.2020.00204.
  35. Qingqing Ye, Haibo Hu, Man Ho Au, Xiaofeng Meng, and Xiaokui Xiao. LF-GDPR: A framework for estimating graph metrics with local differential privacy. IEEE Trans. Knowl. Data Eng., 34(10):4905-4920, 2022. URL: https://doi.org/10.1109/TKDE.2020.3047124.
  36. Hailong Zhang, Sufian Latif, Raef Bassily, and Atanas Rountev. Differentially-private control-flow node coverage for software usage analysis. In 29th USENIX Security Symposium, USENIX Security 2020, August 12-14, 2020, pages 1021-1038. USENIX Association, 2020. URL: https://www.usenix.org/conference/usenixsecurity20/presentation/zhang-hailong.
  37. Xiao Zhou and Takao Nishizeki. Graph coloring algorithms. IEICE Transactions on Information and Systems, 83(3):407-417, 2000. Google Scholar
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
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact