  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.
