Cycle Counting under Local Differential Privacy for Degeneracy-bounded Graphs

• URL: http://arxiv.org/abs/2409.16688v2
• Date: Fri, 27 Sep 2024 03:48:50 GMT
• Title: Cycle Counting under Local Differential Privacy for Degeneracy-bounded Graphs
• Authors: Quentin Hillebrand, Vorapong Suppakitpaisarn, Tetsuo Shibuya,
• Abstract summary: We propose an algorithm for counting the number of cycles under local differential privacy for degeneracy-bounded input graphs. Our algorithm achieves an expected $ell$-error of $O(d_max0.5 n0.5) = O(n)$ for degeneracy-bounded graphs.
• Score: 2.9123921488295768
• License: http://creativecommons.org/licenses/by/4.0/
• 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 $\ell_2$-error of these algorithms is $\Omega(n^{1.5})$, where $n$ is the number of nodes in the graph. When parameterized by the number of cycles of length four ($C_4$), the best existing triangle counting algorithm has an error of $O(n^{1.5} + \sqrt{C_4}) = O(n^2)$. In this paper, we introduce an algorithm with an expected $\ell_2$-error of $O(\delta^{1.5} n^{0.5} + \delta^{0.5} d_{\max}^{0.5} n^{0.5})$, where $\delta$ is the degeneracy and $d_{\max}$ is the maximum degree of the graph. For degeneracy-bounded graphs ($\delta \in \Theta(1)$) commonly found in practical social networks, our algorithm achieves an expected $\ell_2$-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 $\ell_2$-error, namely $O(\delta^{(k-2)/2} d_{\max}^{0.5} n^{(k-2)/2} + \delta^{k/2} n^{(k-2)/2})$ or $O(d_{\max}^{0.5} n^{(k-2)/2}) = O(n^{(k-1)/2})$ for degeneracy-bounded graphs.

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