Differentially Private Synthetic Graphs Preserving Triangle-Motif Cuts

2025-07-20

AI Summary

This work addresses the problem of efficiently publishing a synthetic graph (G') that accurately approximates the number of triangle motifs across all cuts in an original graph (G), under ((varepsilon,delta))-differential privacy. The proposed method introduces the first polynomial-time algorithm: it leverages local triangle sensitivity analysis to design a privacy-preserving mechanism and constructs a synthetic graph satisfying the privacy constraint. Theoretically, the paper establishes the first lower bound (Omegaig(sqrt{mn},ell_3(G)/varepsilonig)) on the error for any differentially private algorithm on this task, and proves that the proposed algorithm achieves an upper bound of (widetilde{O}ig(sqrt{m,ell_3(G)},n/varepsilon^{3/2}ig)). The approach naturally extends to weighted graphs and more general (K_h)-motif cut queries. These results enable differentially private graph clustering, sparsification, and social network analysis.

Technology Category

• Machine Learning: Privacy
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Data transparency and provenance
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

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 $(varepsilon,δ)$-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 $ ilde{O}(sqrt{mell_{3}(G)}n/varepsilon^{3/2})$. Additionally, we provide a lower bound of $Ω(sqrt{mn}ell_{3}(G)/varepsilon)$ 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 $hgeq 2$.

Problem

• Release DP synthetic graph approximating triangle-motif cuts
• Provide polynomial-time mechanism for triangle-motif size queries
• Generalize results to weighted graphs and other motif cuts

Innovation

• DP mechanism for synthetic graph generation
• Polynomial-time triangle-motif size approximation
• Generalization to weighted graphs and motifs