Private Approximation of Graph Spectra and Cuts via Spectral Amplifiers

2026-07-21

AI Summary

This work addresses the problem of efficiently generating synthetic graphs under edge-level differential privacy that accurately approximate all cut sizes of the original graph. The authors propose a polynomial-time $(\varepsilon, \delta)$-differentially private algorithm that outputs a non-negatively weighted graph, achieving significantly reduced worst-case additive error while allowing a small multiplicative approximation factor. Key innovations include the first private spectral primitive that surpasses the standard error baseline, an edge-sensitive terminal-cut oracle, and a novel integration of a private spectral amplifier with a differentially private estimator for the graph Laplacian. This approach improves the worst-case cut approximation error from $\widetilde{O}(n^{5/4 + o(1)})$ to $\widetilde{O}(n^{13/12 + o(1)})$, and for the first time establishes a spectral error bound superior to the standard baseline on graphs with high maximum degree.

Abstract

We study the problem of releasing a synthetic graph that approximates the sizes of all cuts of an input graph under edge-level differential privacy. If one insists on purely additive error, the optimal worst-case error is $\widetildeΘ(n^{3/2})$. If one allows a small multiplicative slack, an information-theoretic exponential-time mechanism achieves nearly linear additive error, but the best known polynomial-time algorithms have substantially larger error. We give a polynomial-time $(\varepsilon,δ)$-differentially private algorithm which, for every $n$-vertex unweighted graph $G$, outputs a non-negative weighted synthetic graph $\widetilde G$ such that, with high probability, every cut $S\subseteq V(G)$ satisfies \[ |w_G(S)-w_{\widetilde G}(S)| \le γw_G(S)+\widetilde O_{\varepsilon,δ,γ}(n^{13/12+o(1)}). \] This improves the previous polynomial-time worst-case bound $\widetilde O(n^{5/4+o(1)})$ of Aamand et al. (ICML 2025) for mixed multiplicative/additive private cut approximation. The main technical ingredient is a new set of private spectral primitives for bounded-degree graphs, one of them gives spectral error $\widetilde O_δ((nd)^{1/4}/\sqrt\varepsilon)$ in estimating the graph Laplacian for graphs of maximum degree $d$, being the first to beat the standard $\min\{2d,\widetilde O_δ(\sqrt{n}/\varepsilon)\}$ baseline in the high-degree regime. We further develop a primitive with a sharper error dependence on $n$ and $d$ for the downstream cut approximation. Combined with a new edge-sensitive terminal cut oracle with additive error $\widetilde O(n+(n^2M)^{1/3})$ on graphs with $M$ edges, this yields the final worst-case $\widetilde O(n^{13/12+o(1)})$ private cut-release error.

Problem

• differential privacy
• graph cuts
• synthetic graph
• spectral approximation
• private graph release

Innovation

• differential privacy
• graph spectra
• cut approximation
• spectral amplifiers
• Laplacian estimation