Differentially Private Synthetic Graphs Preserving Triangle-Motif Cuts

TL;DR

This paper introduces a differentially private mechanism for generating synthetic graphs that accurately approximate triangle-motif sizes across all cuts, balancing privacy with utility in complex network analysis.

Contribution

It presents the first $(,)$-DP algorithm for approximating triangle-motif cuts in graphs, with proven lower bounds and extensions to weighted graphs and larger motifs.

Findings

Achieves polynomial-time synthetic graph generation with bounded additive error.

Provides a lower bound on the error for DP algorithms answering triangle-motif queries.

Extends the approach to weighted graphs and larger motifs.

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 $h\geq 2$.