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.

翻译：本文提出了一种在局部差分隐私框架下对退化有界输入图进行环计数的算法。已有大量研究聚焦于在该隐私定义下进行三角形计数，并证明这些算法的期望\(\ell_2\)误差为\(\Omega(n^{1.5})\)，其中\(n\)为图中节点数。当以长度为四的环（\(C_4\)）数量作为参数时，现有最优三角形计数算法的误差为\(O(n^{1.5} + \sqrt{C_4}) = O(n^2)\)。本文提出一种算法，其期望\(\ell_2\)误差为\(O(\delta^{1.5} n^{0.5} + \delta^{0.5} d_{\max}^{0.5} n^{0.5})\)，其中\(\delta\)为图的退化度，\(d_{\max}\)为最大节点度。对于实际社交网络中常见的退化有界图（\(\delta \in \Theta(1)\)），本算法可实现期望\(\ell_2\)误差\(O(d_{\max}^{0.5} n^{0.5}) = O(n)\)。算法的核心思想是在对全部节点度数进行近似排序的预处理步骤后，实现对三角形的精确计数。该方法可扩展至近似计算长度为\(k\)的环数量，并保持类似的\(\ell_2\)误差，即对于退化有界图，误差为$O(\delta^{(k-2)/2} d_{\max}^{0.5} n^{(k-2)/2} + \delta^{k/2} n^{(k-2)/2})$或$O(d_{\max}^{0.5} n^{(k-2)/2}) = O(n^{(k-1)/2})$。