Differentially private graph coloring

2026-02-13

AI Summary

This work addresses the graph coloring problem under edge differential privacy, aiming to minimize the palette size while controlling the number of neighbors sharing the same color with each vertex—referred to as the defect. The authors propose two novel algorithms that first assign colors randomly from a fixed palette and then apply the exponential mechanism to locally resample colors for subsets or all vertices. They present the first edge differentially private coloring algorithms with theoretical guarantees for both d-inductive graphs and general graphs, achieving an optimized trade-off between palette size and defect. The analysis shows that for d-inductive graphs, the maximum defect is O(log n/ε + d), and for general graphs, it is O(log n/ε), using only Θ(Δ/log n + 1/ε) colors.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Privacy-enhancing technologies
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments

Abstract

Differential Privacy is the gold standard in privacy-preserving data analysis. This paper addresses the challenge of producing a differentially edge-private vertex coloring. In this paper, we present two novel algorithms to approach this problem. Both algorithms initially randomly colors each vertex from a fixed size palette, then applies the exponential mechanism to locally resample colors for either all or a chosen subset of the vertices. Any non-trivial differentially edge private coloring of graph needs to be defective. A coloring of a graph is k defective if all vertices of the graph share it's assigned color with at most k of its neighbors. This is the metric by which we will measure the utility of our algorithms. Our first algorithm applies to d-inductive graphs. Assume we have a d-inductive graph with n vertices and max degree $\Delta$. We show that our algorithm provides a \(3\epsilon\)-differentially private coloring with \(O(\frac{\log n}{\epsilon}+d)\) max defectiveness, given a palette of size $\Theta(\frac{\Delta}{\log n}+\frac{1}{\epsilon})$ Furthermore, we show that this algorithm can generalize to $O(\frac{\Delta}{c\epsilon}+d)$ defectiveness, where c is the size of the palette and $c=O(\frac{\Delta}{\log n})$. Our second algorithm utilizes noisy thresholding to guarantee \(O(\frac{\log n}{\epsilon})\) max defectiveness, given a palette of size $\Theta(\frac{\Delta}{\log n}+\frac{1}{\epsilon})$, generalizing to all graphs rather than just d-inductive ones.

Problem

• Differential Privacy
• Graph Coloring
• Edge Privacy
• Defective Coloring
• Vertex Coloring

Innovation

• differentially private graph coloring
• exponential mechanism
• defective coloring
• noisy thresholding
• d-inductive graphs