← 研踪 PaperTrailarXiv:2611.05038 · v1
The Power of Interaction in Locally Edge-Private Subgraph Counting
首次提交 2026-11-03 23:10
版本更新 2026-11-03 23:10
雷达观测 2026-11-04 23:39
摘要
We study (non-induced) subgraph counting under interactive local edge differential privacy (LEDP). Given a fixed pattern graph $H$ with $k$ vertices, $\ell\ge1$ edges, and $q$ connected components, our goal is to approximately count its copies in an arbitrary $n$-vertex undirected graph while preserving edge privacy. We give two complementary accuracy guarantees. First, for every $H$, we give Linear Noise Cancellation (LNC), a two-round $\varepsilon$-LEDP algorithm with additive error and root mean square error both $O(n^{k-3/2}/\varepsilon^\ell)$ on every $n$-vertex graph. For fixed sufficiently small $\varepsilon$, this improves the optimal noninteractive additive error for triangles by a factor of $\sqrt n$ and gives the same improvement in worst-case root mean square error for all cliques on at least three vertices. To our knowledge, this is the first separation between interactive and noninteractive LEDP subgraph counting, showing that a single additional round can improve worst-case accuracy polynomially. Second, we determine how accuracy improves when the graph has maximum degree at most a public bound $d$. Our Predict and Correct (PC) framework gives an $(\ell+2)$-round $\varepsilon$-LEDP algorithm with additive error $O(n^{q-1/2}d^{k-q-1}\log^{3\ell-2}n/\varepsilon^\ell)$. For connected patterns, its polynomial factor is $\sqrt n\,d^{k-2}$, giving sublinear error on constant-degree graphs at fixed privacy. We prove matching lower bounds in $n$ and $d$, up to logarithmic factors at fixed privacy, even under arbitrary interaction and approximate privacy with sufficiently small $δ$. At $d=Θ(n)$, the lower bound also establishes the optimality of LNC's dependence on $n$.
作者备注：64 pages
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v1 · 2026-11-03 23:10 · 最新观测版本
The Power of Interaction in Locally Edge-Private Subgraph Counting
We study (non-induced) subgraph counting under interactive local edge differential privacy (LEDP). Given a fixed pattern graph $H$ with $k$ vertices, $\ell\ge1$ edges, and $q$ connected components, our goal is to approximately count its copies in an arbitrary $n$-vertex undirected graph while preserving edge privacy. We give two complementary accuracy guarantees. First, for every $H$, we give Linear Noise Cancellation (LNC), a two-round $\varepsilon$-LEDP algorithm with additive error and root mean square error both $O(n^{k-3/2}/\varepsilon^\ell)$ on every $n$-vertex graph. For fixed sufficiently small $\varepsilon$, this improves the optimal noninteractive additive error for triangles by a factor of $\sqrt n$ and gives the same improvement in worst-case root mean square error for all cliques on at least three vertices. To our knowledge, this is the first separation between interactive and noninteractive LEDP subgraph counting, showing that a single additional round can improve worst-case accuracy polynomially. Second, we determine how accuracy improves when the graph has maximum degree at most a public bound $d$. Our Predict and Correct (PC) framework gives an $(\ell+2)$-round $\varepsilon$-LEDP algorithm with additive error $O(n^{q-1/2}d^{k-q-1}\log^{3\ell-2}n/\varepsilon^\ell)$. For connected patterns, its polynomial factor is $\sqrt n\,d^{k-2}$, giving sublinear error on constant-degree graphs at fixed privacy. We prove matching lower bounds in $n$ and $d$, up to logarithmic factors at fixed privacy, even under arbitrary interaction and approximate privacy with sufficiently small $δ$. At $d=Θ(n)$, the lower bound also establishes the optimality of LNC's dependence on $n$.
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