Differentially Private Synthetic Graphs Preserving Triangle-Motif Cuts

Abstract

We study the problem of releasing a differentially private (DP) synthetic graph $G^{\prime }$ 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 $(\epsilon ,\delta )$-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 $G^{\prime }$ 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 $\tilde{O}(\sqrt{m{\ell }_3(G)}n/{\epsilon }^{3/2})$. Additionally, we provide a lower bound of $\Omega (\sqrt{mn}{\ell }_3(G)/\epsilon )$ 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 $h\ge 2$.