Near-optimal node-private community estimation in polynomial-time

2026-07-10

AI Summary

This work addresses the problem of exact community recovery under node-level differential privacy in the stochastic block model. The authors propose a polynomial-time algorithm that achieves near-statistically optimal performance with high probability by constructing an explicit Lipschitz surrogate for the penalized likelihood function and designing an efficient accept-reject sampling scheme to implement the exponential mechanism. This approach is the first to attain exact recovery in polynomial time while matching the known lower bound on the privacy cost. Notably, it remains effective even when the number of communities \(K\) grows logarithmically with the network size \(n\) and the privacy parameter scales as \(\varepsilon = \Theta(\log n)\), achieving minimax-optimal exact recovery under these conditions.

Abstract

In this paper, we resolve an open question of Klopp & Zadik (2026) by providing a high-probability polynomial-time, node-private algorithm which nearly matches the performance of their exponential-time node-private algorithm for exact recovery in stochastic block models. Our result involves an explicitly constructed Lipschitz surrogate for the penalized likelihood function, as well as a carefully devised accept-reject algorithm that samples community labels from the corresponding exponential mechanism in polynomial-time. We rigorously analyze the privacy, runtime, and utility of our proposed algorithm, showing that even when the number of communities K grows logarithmically with the number of nodes n, we can achieve the minimax rates for exact recovery with the privacy parameter epsilon growing as log(n), thus matching known lower bounds on the cost of privacy for this setting.

Problem

• node-private
• community recovery
• stochastic block model
• polynomial-time
• exact recovery

Innovation

• node-private
• polynomial-time algorithm
• stochastic block model
• Lipschitz surrogate
• exponential mechanism