We study robust community detection in the context of node-corrupted stochastic block model, where an adversary can arbitrarily modify all the edges incident to a fraction of the $n$ vertices. We present the first polynomial-time algorithm that achieves weak recovery at the Kesten-Stigum threshold even in the presence of a small constant fraction of corrupted nodes. Prior to this work, even state-of-the-art robust algorithms were known to break under such node corruption adversaries, when close to the Kesten-Stigum threshold. We further extend our techniques to the $Z_2$ synchronization problem, where our algorithm reaches the optimal recovery threshold in the presence of similar strong adversarial perturbations. The key ingredient of our algorithm is a novel identifiability proof that leverages the push-out effect of the Grothendieck norm of principal submatrices.

翻译：我们研究节点损坏随机块模型下的鲁棒社区检测问题，其中对手可以任意修改与$n$个顶点中一部分相连的所有边。我们提出首个多项式时间算法，该算法即使在存在小常数比例损坏节点的情况下，也能在Kesten-Stigum阈值处实现弱恢复。在此工作之前，即使是最先进的鲁棒算法，在接近Kesten-Stigum阈值时，也会因这种节点损坏对手而失效。我们进一步将技术扩展到$Z_2$同步问题，在该问题中，我们的算法在存在类似强对抗扰动的情况下达到了最优恢复阈值。算法的关键要素是一个新颖的可识别性证明，该证明利用了主子矩阵的Grothendieck范数的推出效应。