Improved Robust Estimation for Erdős-Rényi Graphs: The Sparse Regime and Optimal Breakdown Point

• URL: http://arxiv.org/abs/2503.03923v1
• Date: Wed, 05 Mar 2025 21:45:17 GMT
• Title: Improved Robust Estimation for Erdős-Rényi Graphs: The Sparse Regime and Optimal Breakdown Point
• Authors: Hongjie Chen, Jingqiu Ding, Yiding Hua, Stefan Tiegel,
• Abstract summary: We study the problem of robustly estimating the edge density of ErdHos-R'enyi random graphs $G(n, dcirc/n)$.<n>Our algorithm is based on the sum-of-squares hierarchy.
• Score: 3.793609515750114
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We study the problem of robustly estimating the edge density of Erd\H{o}s-R\'enyi random graphs $G(n, d^\circ/n)$ when an adversary can arbitrarily add or remove edges incident to an $\eta$-fraction of the nodes. We develop the first polynomial-time algorithm for this problem that estimates $d^\circ$ up to an additive error $O([\sqrt{\log(n) / n} + \eta\sqrt{\log(1/\eta)} ] \cdot \sqrt{d^\circ} + \eta \log(1/\eta))$. Our error guarantee matches information-theoretic lower bounds up to factors of $\log(1/\eta)$. Moreover, our estimator works for all $d^\circ \geq \Omega(1)$ and achieves optimal breakdown point $\eta = 1/2$. Previous algorithms [AJK+22, CDHS24], including inefficient ones, incur significantly suboptimal errors. Furthermore, even admitting suboptimal error guarantees, only inefficient algorithms achieve optimal breakdown point. Our algorithm is based on the sum-of-squares (SoS) hierarchy. A key ingredient is to construct constant-degree SoS certificates for concentration of the number of edges incident to small sets in $G(n, d^\circ/n)$. Crucially, we show that these certificates also exist in the sparse regime, when $d^\circ = o(\log n)$, a regime in which the performance of previous algorithms was significantly suboptimal.

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