Node and Edge Differential Privacy for Graph Laplacian Spectra: Mechanisms and Scaling Laws

TL;DR

This paper introduces differential privacy mechanisms for graph Laplacian spectra, comparing edge and node privacy formulations, and provides bounds and empirical validation for private spectral accuracy.

Contribution

It develops a framework for privatizing graph Laplacian spectra using differential privacy, with a focus on edge privacy and algebraic connectivity, including analytical bounds and empirical results.

Findings

Edge privacy is more suitable for engineering applications.

Analytical bounds on spectral accuracy are established.

Numerical examples confirm practical effectiveness.

Abstract

This paper develops a framework for privatizing the spectrum of the graph Laplacian of an undirected graph using differential privacy. We consider two privacy formulations. The first obfuscates the presence of edges in the graph and the second obfuscates the presence of nodes. We compare these two privacy formulations and show that the privacy formulation that considers edges is better suited to most engineering applications. We use the bounded Laplace mechanism to provide differential privacy to the eigenvalues of a graph Laplacian, and we pay special attention to the algebraic connectivity, which is the Laplacian's second smallest eigenvalue. Analytical bounds are presented on the the accuracy of the mechanisms and on certain graph properties computed with private spectra. A suite of numerical examples confirms the accuracy of private spectra in practice.