Differentially Private Synthetic Graphs Preserving Triangle-Motif Cuts

• URL: http://arxiv.org/abs/2507.14835v1
• Date: Sun, 20 Jul 2025 06:20:53 GMT
• Title: Differentially Private Synthetic Graphs Preserving Triangle-Motif Cuts
• Authors: Pan Peng, Hangyu Xu,
• Abstract summary: We study the problem of releasing a differentially private (DP) synthetic graph $G'$ that well approximates the triangle-motif sizes of all cuts of any given graph $G'$.<n>Non-private versions of such graphs have found applications in diverse fields such as graph clustering, graph sparsification, and social network analysis.
• Score: 5.893124686141782
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We study the problem of releasing a differentially private (DP) synthetic graph $G'$ that well approximates the triangle-motif sizes of all cuts of any given graph $G$, where a motif in general refers to a frequently occurring subgraph within complex networks. Non-private versions of such graphs have found applications in diverse fields such as graph clustering, graph sparsification, and social network analysis. Specifically, we present the first $(\varepsilon,\delta)$-DP mechanism that, given an input graph $G$ with $n$ vertices, $m$ edges and local sensitivity of triangles $\ell_{3}(G)$, generates a synthetic graph $G'$ in polynomial time, approximating the triangle-motif sizes of all cuts $(S,V\setminus S)$ of the input graph $G$ up to an additive error of $\tilde{O}(\sqrt{m\ell_{3}(G)}n/\varepsilon^{3/2})$. Additionally, we provide a lower bound of $\Omega(\sqrt{mn}\ell_{3}(G)/\varepsilon)$ on the additive error for any DP algorithm that answers the triangle-motif size queries of all $(S,T)$-cut of $G$. Finally, our algorithm generalizes to weighted graphs, and our lower bound extends to any $K_h$-motif cut for any constant $h\geq 2$.

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