Edge Differential Privacy for Algebraic Connectivity of Graphs

TL;DR

This paper introduces a method to privately release a graph's algebraic connectivity using edge differential privacy, balancing privacy with accuracy for multi-agent system analysis.

Contribution

It proposes a novel approach to protect sensitive graph topology information while accurately estimating algebraic connectivity under differential privacy constraints.

Findings

Bounded Laplace noise improves privacy-accuracy trade-off.

Private algebraic connectivity accurately estimates convergence rates.

Method effectively conceals sensitive graph connections.

Abstract

Graphs are the dominant formalism for modeling multi-agent systems. The algebraic connectivity of a graph is particularly important because it provides the convergence rates of consensus algorithms that underlie many multi-agent control and optimization techniques. However, sharing the value of algebraic connectivity can inadvertently reveal sensitive information about the topology of a graph, such as connections in social networks. Therefore, in this work we present a method to release a graph's algebraic connectivity under a graph-theoretic form of differential privacy, called edge differential privacy. Edge differential privacy obfuscates differences among graphs' edge sets and thus conceals the absence or presence of sensitive connections therein. We provide privacy with bounded Laplace noise, which improves accuracy relative to conventional unbounded noise. The private algebraic connectivity values are analytically shown to provide accurate estimates of consensus convergence rates, as well as accurate bounds on the diameter of a graph and the mean distance between its nodes. Simulation results confirm the utility of private algebraic connectivity in these contexts.