Near-Optimal Differentially Private Graph Algorithms via the Multidimensional AboveThreshold Mechanism

2025-08-04

AI Summary

Differential privacy in graph algorithms suffers from error amplification—conventional Sparse Vector Technique (SVT) incurs linear error growth O(n) in the number of vertices due to per-vertex noise injection. Method: We propose the Multidimensional AboveThreshold (MAT) mechanism—the first extension of SVT to multidimensional vector queries—thereby avoiding per-vertex noise accumulation. MAT integrates local edge-level differential privacy, tree decomposition, and greedy iteration. Results: For k-core decomposition, MAT achieves an additive error of O(ε⁻¹ log n) in O(n) rounds, matching the asymptotically tight lower bound; it also yields a (2+η)-approximation algorithm with O(log²n) rounds. The framework unifies and improves private algorithms for multiple graph problems—including densest subgraph detection, low-outdegree ordering, and graph coloring—delivering substantial gains in both accuracy and efficiency.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Privacy-enhancing technologies
• Web Mining and Content Analysis: Content-based information diffusion

Abstract

Many differentially private and classical non-private graph algorithms rely crucially on determining whether some property of each vertex meets a threshold. For example, for the $k$-core decomposition problem, the classic peeling algorithm iteratively removes a vertex if its induced degree falls below a threshold. The sparse vector technique (SVT) is generally used to transform non-private threshold queries into private ones with only a small additive loss in accuracy. However, a naive application of SVT in the graph setting leads to an amplification of the error by a factor of $n$ due to composition, as SVT is applied to every vertex. In this paper, we resolve this problem by formulating a novel generalized sparse vector technique which we call the Multidimensional AboveThreshold (MAT) Mechanism which generalizes SVT (applied to vectors with one dimension) to vectors with multiple dimensions. As an application, we solve a number of important graph problems with better bounds than previous work. We apply our MAT mechanism to obtain a set of improved bounds for a variety of problems including $k$-core decomposition, densest subgraph, low out-degree ordering, and vertex coloring. We give a tight local edge DP algorithm for $k$-core decomposition with $O(ε^{-1}log n)$ additive error and no multiplicative error in $O(n)$ rounds. We also give a new $(2+η)$-factor multiplicative, $O(ε^{-1}log n)$ additive error algorithm in $O(log^2 n)$ rounds for any constant $η> 0$. Both of these results are asymptotically tight against our new lower bound of $Ω(log n)$ for any constant-factor approximation algorithm for $k$-core decomposition. Our new algorithms for $k$-core also directly lead to new algorithms for densest subgraph and low out-degree ordering. Our novel private defective coloring algorithms uses number of colors proportional to the arboricity of the graph.

Problem

• Improving differentially private graph algorithms via MAT mechanism
• Reducing error amplification in private graph threshold queries
• Achieving tight bounds for k-core decomposition and related problems

Innovation

• Multidimensional AboveThreshold Mechanism generalizes SVT
• MAT Mechanism reduces error amplification in graphs
• Novel private defective coloring uses arboricity-based colors