Optimal Bounds on Private Graph Approximation

We propose an efficient ε-differentially private algorithm, that given a simple weighted nvertex, m-edge graph G with a maximum unweighted degree ∆(G) ≤ n -1, outputs a synthetic graph which approximates the spectrum with O(min∆(G), √ n) bound on the purely additive error. To the best of our knowledge, this is the first ε-differentially private algorithm with a non-trivial additive error for approximating the spectrum of the graph. One of the subroutines of our algorithm also precisely simulates the exponential mechanism over a non-convex set, which could be of independent interest given the recent interest in sampling from a log-concave distribution defined over a convex set. As a direct application of our result, we give the first nontrivial bound on approximating all-pairs effective resistances by a synthetic graph, which also implies approximating hitting/commute time and cover time of random walks on the graph. Given the significance of effective resistance in understanding the statistical properties of a graph, we believe our result would have further implications.

Spectral approximation also allows us to approximate all possible (S, T)-cuts, but it incurs an error that depends on the maximum degree, ∆(G). We further show that using our sampler, we can also output a synthetic graph that approximates the sizes of all (S, T)-cuts on n vertices weighted graph G with m edges while preserving (ε, δ)-differential privacy and an additive error of O( √ mn/ε). We also give a matching lower bound (with respect to all the parameters) on the private cut approximation for weighted graphs. This removes the gap of W avg in the upper and lower bound in Eliáš, Kapralov, Kulkarni, and Lee (SODA 2020), where W avg is the average edge weight.