arXiv:2507.14835v2 Announce Type: replace-cross
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 $(S,Vsetminus S)$ of the input graph $G$ up to an additive error of $tilde{O}(sqrt{mell_{3}(G)}n/varepsilon^{3/2})$. 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 $hgeq 2$.