Improved Robust Estimation for ErdH{o}s-R'enyi Graphs: The Sparse Regime and Optimal Breakdown Point

2025-03-05

AI Summary

This paper studies robust estimation of the edge density $d^circ$ in sparse Erdős–Rényi graphs $G(n, d^circ/n)$ under adversarial corruption of an $eta$-fraction of node adjacencies. We propose the first polynomial-time algorithm that breaks the longstanding $d^circ = o(log n)$ barrier for sparse regimes, achieving information-theoretically optimal error (up to a $log(1/eta)$ factor) and optimal breakdown point $eta = 1/2$. Our method leverages the Sum-of-Squares (SoS) hierarchy to construct constant-degree concentration certificates, enabling robust analysis of adjacency counts in sparse graphs. The estimator attains error $Oig(ig[sqrt{log n/n} + eta sqrt{log(1/eta)}ig]sqrt{d^circ} + eta log(1/eta)ig)$ for all $d^circ = Omega(1)$, significantly improving upon both existing efficient and computationally unbounded estimators.

Technology Category

• Reasoning under Uncertainty: Graphical Models
• Machine Learning: Graph-based Machine Learning
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We study the problem of robustly estimating the edge density of ErdH{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 $O([sqrt{log(n) / n} + etasqrt{log(1/eta)} ] cdot sqrt{d^circ} + eta log(1/eta))$. 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.

Problem

• Robust estimation of edge density in Erdős-Rényi graphs under adversarial edge modifications.
• Development of a polynomial-time algorithm with optimal breakdown point and error guarantees.
• Extension of sum-of-squares certificates to sparse regimes for improved algorithmic performance.

Innovation

• Polynomial-time algorithm for robust edge density estimation
• Sum-of-squares hierarchy for sparse graph analysis
• Optimal breakdown point achieved with minimal error