Almost linear time differentially private release of synthetic graphs
Jingcheng Liu, Jalaj Upadhyay, Zongrui Zou
TL;DR
This work tackles privately releasing a synthetic graph that preserves the input’s spectral and cut structure while achieving near-linear time and memory usage in sparse regimes. It introduces two complementary approaches: (i) an efficient sampler for the exponential mechanism over sparse topologies using a basis-exchange walk on a strongly log-concave distribution, and (ii) a high-pass filtering method that sparsifies a noisy graph while maintaining differential privacy. These yield two $(\varepsilon,\delta)$-DP algorithms with additive errors $O\left(\dfrac{m\log(n/\beta)}{\varepsilon}\right)$ for cut approximation and $O\left(\dfrac{d_{\max}\log(n/\delta)}{\varepsilon}\right)$ for spectral approximation, both operating in $\tilde{O}(m)$ time and $O(m)$ space, and extendable to continual observation with provable privacy and utility bounds. The paper also provides empirical evidence that these methods achieve near-optimal performance in sparse graphs and scales linearly with graph size, making private graph analysis practical for large datasets. Overall, the contributions significantly advance the feasibility of private, scalable graph analysis by aligning private and non-private resource requirements while delivering strong utility guarantees.
Abstract
In this paper, we give an almost linear time and space algorithms to sample from an exponential mechanism with an $\ell_1$-score function defined over an exponentially large non-convex set. As a direct result, on input an $n$ vertex $m$ edges graph $G$, we present the \textit{first} $\widetilde{O}(m)$ time and $O(m)$ space algorithms for differentially privately outputting an $n$ vertex $O(m)$ edges synthetic graph that approximates all the cuts and the spectrum of $G$. These are the \emph{first} private algorithms for releasing synthetic graphs that nearly match this task's time and space complexity in the non-private setting while achieving the same (or better) utility as the previous works in the more practical sparse regime. Additionally, our algorithms can be extended to private graph analysis under continual observation.