Differentially private graph coloring

TL;DR

The paper tackles graph coloring under edge differential privacy by using defective colorings as a privacy-preserving relaxation. It introduces two algorithms: Iterative sampling (M_Unctr) that privately initializes colors from a palette of size $C=\Theta\left(\frac{\Delta}{\log n}+\frac{1}{\varepsilon}\right)$ and then locally resamples via the Exponential mechanism to achieve $3\varepsilon$-DP with defectiveness $O\left(\frac{\log n}{\varepsilon}+d\right)$ on $d$-inductive graphs; and Controlled recoloring (M_Control) that recolors only high-defect vertices using a noisy threshold and achieves $5\varepsilon$-DP with defectiveness bounded for general graphs. The authors provide theoretical analyses of privacy and defectiveness and corroborate them with experiments on SNAP networks and synthetic graphs, showing favorable privacy-utility trade-offs compared with prior DP coloring methods. The work demonstrates that leveraging graph topology and selective recoloring can yield practical, DP-compliant colorings with meaningful utility, while highlighting open directions for broader applicability and tighter max-defect guarantees.

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 $Δ$. We show that our algorithm provides a \(3ε\)-differentially private coloring with \(O(\frac{\log n}ε+d)\) max defectiveness, given a palette of size $Θ(\fracΔ{\log n}+\frac{1}ε)$ Furthermore, we show that this algorithm can generalize to $O(\fracΔ{cε}+d)$ defectiveness, where c is the size of the palette and $c=O(\fracΔ{\log n})$. Our second algorithm utilizes noisy thresholding to guarantee \(O(\frac{\log n}ε)\) max defectiveness, given a palette of size $Θ(\fracΔ{\log n}+\frac{1}ε)$, generalizing to all graphs rather than just d-inductive ones.

Figures (4)

• Figure 1: Average defectiveness as a function of $\epsilon$ on SNAP networks.
• Figure 2: Maximum defectiveness as a function of $\epsilon$ on SNAP networks.
• Figure 3: Average (top) and maximum (bottom) defectiveness as a function of $\epsilon$ on $G(n,p)$.
• Figure 4: Average (top) and maximum (bottom) defectiveness as a function of $\epsilon$ on $G(n,p)$. m is a hyperparameter of Barabasi-Albert graphs which indicate the number of edges to attach from a new node to existing nodes

Theorems & Definitions (21)

• Definition 2.1
• Definition 2.2: Exponential mechanism
• Lemma 2.3
• Definition 2.4: Laplace mechanism
• Lemma 2.5
• Definition 2.6
• Definition 2.7
• Definition 2.8
• Theorem 3.1
• proof