Title: Improved Robust Estimation for Erdős-Rényi Graphs: The Sparse Regime and Optimal Breakdown Point
Venue: NeurIPS 2025
Accepted: yes
Keywords: Robust statistics

random graph estimation

sum-of-squares
Abstract: We study the problem of robustly estimating the edge density of Erdos Renyi random graphs $\mathbb{G}(n, d^\circ/n)$ 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 $O\left({[\sqrt{\log(n) / n}  + \eta\sqrt{\log(1/\eta)} ] \cdot \sqrt{d^\circ} + \eta \log(1/\eta)}\right)$.
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 [Acharya et al 2022, Chen et al 2024], 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 $\mathbb{G}(n, d^\circ/n)$.
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.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper addresses the problem of robustly estimating the edge density parameter $d^\\circ$ of Erdős-Rényi random graphs $G(n, d^\\circ/n)$ under adversarial node corruptions. ​ The authors propose the first polynomial-time algorithm that achieves near-optimal error rates and reaches the optimal breakdown point of $ \\eta = 1/2 $. ​ The algorithm leverages the sum-of-squares (SoS) hierarchy to construct certificates for concentration properties of edges in sparse graphs, overcoming limitations of previous methods in the sparse regime ($d^\\circ = o(\\log n)$). ​ The paper provides formal proofs, theoretical guarantees, and a clean analysis of the algorithm's performance.\n\nSTRENGTHS AND WEAKNESSES:\nThe paper is well-written and well-motivated. However, the lack of any sort of experiments (even synthetic) is glaring\n\nQUESTIONS:\nBelow are my detailed comments/questions:\n\n1. For the sake of completeness, it will be very useful for the authors to explain why robust mean estimation methods will not work in this setting. The paper has referred to dependencies of the degrees of the uncorrupted graph. While that is true, I cannot understand the limitation of the following simple trick: If I view the set of edges as a (n choose 2) binary vector, then each entry of this vector is iid generated with prob. $d^\\circ/n$. Therefore, estimating the mean is a simple 1D mean estimation problem - if some elements are corrupted, then why will median of means not work? median of means is optimal for robust mean estimation in 1D. \n\n2. While the results are strong, the lack of experiments is a glaring problem. Given that the setting is simple, the lack of experiments (even synthetic) hurts the value of the paper. Especially so since the authors have highlighted the efficiency and statistical guarantees of existing work as a limitation.\n\nLIMITATIONS:\nSimple Experiments even on synthetic datasets", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 4, "ORIGINALITY": 4, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies a robust statistics problem on Erdos-Renyi random graphs.  Suppose that we draw an ER graph with expected degree $d$, but then we allow an adversary to choose an arbitrary subset of $\\eta n$ nodes and completely change the edges incident on them (not just between them).  Can we still estimate the original parameter $d$?  This paper gives state of the art results, showing a polytime algorithm that essentially matches the information-theoretic lower bound (up to a log factor) for all possible regimes.  This is in contrast to previous work, which either gave optimal bounds for only some parameter regimes or gave suboptimal bounds.  Their algorithm is based on a sum-of-squares proof, where they first given an inefficient algorithm, but then identify from this algorithm a certificate that can be expressed as sum-of-squares.  Since there is a well-known duality between SoS proofs and certain semidefinite programs, this means that there is an SDP-based polytime algorithm for the problem.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n- This paper seems to basically close this problem (up to the log factor), as they get almost optimal bounds in all regimes where bounds are possible.\n- The paper is well-written and understandable.\n\nWeaknesses:\n- While I understand that this is a theory paper, for NeurIPS I would expect there to be either experiments or some type of argument about practicality.  But my understanding is that SDP solvers are still quite slow, so the algorithm in this paper might not actually be practical.\n- There has been previous work on this problem, so it seems to be of interest to the community, but I wish the authors had bothered to motivate the problem a little bit.  Why this corruption model?  Why study ER graphs?  Naivel, I would have expected something like \"real-world graphs are often preferential-attachment or something similar, and so we focus on trying to statistically recover the parameters of those models in the presence of adversarial noise.  The exact model of adversarial noise depends on the application: one could imagine edges being corrupted, nodes being corrupted, or some combination of the two.  Here are motivating examples of why we might want to consider each of these types of corruption in each of these random graph models: ...\".\n\nQUESTIONS:\n- Is this algorithm practical?\n- What's the motivation for this precise problem?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}

comments: SUMMARY:
The paper studies the problem of estimating the edge density of ER random graphs $G(n, d/n)$ under node corruptions. The paper proves that their method can estimate d within an additive error of $O((\sqrt{\log{(n)}/n}+\eta \sqrt{\log{(1/\eta)}})\sqrt{d}+\eta \log{(1/\eta)})$. This upper bound differs from the lower bound for this problem by a factor of $\log{(1/\eta)}$. The paper also emphasizes that the proposed algorithm works for all $d \ge \Omega(1)$ and can achieve optimal breakdown point $\eta =1/2$.

STRENGTHS AND WEAKNESSES:
Strengths

S1. The paper studies the problem of estimating the edge density of Erdős–Rényi (ER) random graphs — a fundamental problem with both theoretical and practical significance. It improves the approximation error bound for this task. The prior bound in [AJK+] is suboptimal and does not recover the non-robust case when there are no corruptions. I think the paper is strong and is worthy of acceptance.

S2. The paper is well written. Its structure is clear and easy to follow. Prior work is thoroughly reviewed, and the high-level intuition behind the proposed method is well explained.

S3. As mentioned in the paper, the proof is relatively simple, which actually enhances the contribution of the paper


Weaknesses

W1. The additive error bound presented in the paper is not very clean and still leaves a gap compared to the optimal result. There remains room for further exploration in this direction.

W2. The paper does not include any empirical study. However, since this is a theoretical paper, this is acceptable in my view.

QUESTIONS:
I think this paper is a clear accept and have no specific questions.

LIMITATIONS:
There is not much concern about the potential negative societal impact of this paper.
TITLE: Official Review
CLARITY: 3
IMPACT: 4
ORIGINALITY: 4
DATE: 2025-07-03

comments: SUMMARY:
Consider a graph G generated from the classic Erdos-Renyi distribution G(n,p), where we think of p = d/n. So the average degree will be d. An adversary corrupts an $\eta$-fraction of the nodes, by arbitrarily changing edges incident to these "corrupted nodes". The aim is to estimate d. Previous work showed that many standard estimators fail. There has been recent work getting estimators for this regime.

The main problem is that when d = Theta(1), no previous algorithm gets the optimal bound. The first result [AJK+22] has an additive O(log n) error. The next result [CDHS24] has an additive error of d/10, regardless of d. Also, these algorithms fail when $\eta$ is larger than some small constant.

This paper gives an algorithm that avoids all of these issues, and gets (nearly) optimal error. The bounds are technical, but the main point is that it allows for $\eta < 1/2$, d to be a constant, and the errors goes to zero as d becomes larger.

The main idea is to use tools for the Sum-of-Squares (SoS) machinery that builds a semi-definite program (SDP) to solve this problem.

STRENGTHS AND WEAKNESSES:
The primary strength is getting an estimation result for the constant d case, with optimal threshold $\eta < 1/2$, and a multiplicative error that goes to zero as d increases. 

The primary weakness is that I doubt that this is a realistic algorithm. The polynomial running time is probably quite prohibitive. So this result is of somewhat narrow interest to the people who work on this problem.

QUESTIONS:
What is the running time of the algorithm?

LIMITATIONS:
Yes
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 4
DATE: 2025-07-03