Differentially private graph coloring

TL;DR

This paper introduces two novel differentially private algorithms for vertex coloring in graphs, balancing privacy and utility by controlling defectiveness, applicable to various graph classes.

Contribution

It proposes two algorithms for differentially private graph coloring, with theoretical bounds on defectiveness and palette size, extending privacy-preserving graph analysis.

Findings

First algorithm achieves $3\,\epsilon$-differential privacy with $O(\frac{\log n}{\epsilon}+d)$ defectiveness.

Second algorithm guarantees $O(\frac{\log n}{\epsilon})$ defectiveness for all graphs.

Algorithms are applicable to graphs with different structural properties, with theoretical utility bounds.

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.