  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.
