Rate-optimal community detection near the KS threshold via node-robust algorithms

Jingqiu Ding, Yiding Hua, Kasper Lindberg, David Steurer, Aleksandr Storozhenko
Proceedings of Thirty Ninth Conference on Learning Theory, PMLR 336:1965-2037, 2026.

Abstract

We study community detection in the \emph{symmetric $k$-stochastic block model}, where $n$ nodes are evenly partitioned into $k$ clusters with intra- and inter-cluster connection probabilities $p$ and $q$, respectively. Our main result is a polynomial-time algorithm that achieves the optimal misclassification rate $\exp(-(1 \pm o(1)) C/k)$, where $C = (\sqrt{pn} - \sqrt{qn})^2$, whenever $C \geq K k^2 \log k$ for some universal constant $K$, matching the Kesten–Stigum ({KS}) threshold up to a $\log k$ factor. Notably, this rate holds even when an adversary corrupts an $\eta \leq \exp(-(1 \pm o(1)) C/k)$ fraction of the nodes. To the best of our knowledge, this optimal error rate was previously only attainable either via computationally inefficient procedures (Zhang and Zhou, 2015) or via polynomial-time algorithms that require strictly stronger assumptions such as $C \geq K k^3$ (Gao et al., 2017). In the node-robust setting, the best known algorithm requires the substantially stronger condition $C \geq K k^{102}$ (Liu and Moitra, 2022). Our results close this gap by providing the first polynomial-time algorithm that achieves the optimal error rate near the {KS} threshold in both settings. Our work has two key technical contributions: (1) we robustify majority voting via the Sum-of-Squares framework, (2) we develop a novel graph bisectioning algorithm via robust majority voting, which allows us to significantly improve the misclassification rate to $1/\mathrm{poly}(k)$ for the initial estimation near the {KS} threshold.

Cite this Paper

BibTeX
@InProceedings{pmlr-v336-ding26b, title = {Rate-optimal community detection near the {KS} threshold via node-robust algorithms}, author = {Ding, Jingqiu and Hua, Yiding and Lindberg, Kasper and Steurer, David and Storozhenko, Aleksandr}, booktitle = {Proceedings of Thirty Ninth Conference on Learning Theory}, pages = {1965--2037}, year = {2026}, editor = {Hanneke, Steve and Lattimore, Tor}, volume = {336}, series = {Proceedings of Machine Learning Research}, month = {29 Jun--03 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v336/main/assets/ding26b/ding26b.pdf}, url = {https://proceedings.mlr.press/v336/ding26b.html}, abstract = {We study community detection in the \emph{symmetric $k$-stochastic block model}, where $n$ nodes are evenly partitioned into $k$ clusters with intra- and inter-cluster connection probabilities $p$ and $q$, respectively. Our main result is a polynomial-time algorithm that achieves the optimal misclassification rate $\exp(-(1 \pm o(1)) C/k)$, where $C = (\sqrt{pn} - \sqrt{qn})^2$, whenever $C \geq K k^2 \log k$ for some universal constant $K$, matching the Kesten–Stigum ({KS}) threshold up to a $\log k$ factor. Notably, this rate holds even when an adversary corrupts an $\eta \leq \exp(-(1 \pm o(1)) C/k)$ fraction of the nodes. To the best of our knowledge, this optimal error rate was previously only attainable either via computationally inefficient procedures (Zhang and Zhou, 2015) or via polynomial-time algorithms that require strictly stronger assumptions such as $C \geq K k^3$ (Gao et al., 2017). In the node-robust setting, the best known algorithm requires the substantially stronger condition $C \geq K k^{102}$ (Liu and Moitra, 2022). Our results close this gap by providing the first polynomial-time algorithm that achieves the optimal error rate near the {KS} threshold in both settings. Our work has two key technical contributions: (1) we robustify majority voting via the Sum-of-Squares framework, (2) we develop a novel graph bisectioning algorithm via robust majority voting, which allows us to significantly improve the misclassification rate to $1/\mathrm{poly}(k)$ for the initial estimation near the {KS} threshold.} }
Endnote
%0 Conference Paper %T Rate-optimal community detection near the KS threshold via node-robust algorithms %A Jingqiu Ding %A Yiding Hua %A Kasper Lindberg %A David Steurer %A Aleksandr Storozhenko %B Proceedings of Thirty Ninth Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2026 %E Steve Hanneke %E Tor Lattimore %F pmlr-v336-ding26b %I PMLR %P 1965--2037 %U https://proceedings.mlr.press/v336/ding26b.html %V 336 %X We study community detection in the \emph{symmetric $k$-stochastic block model}, where $n$ nodes are evenly partitioned into $k$ clusters with intra- and inter-cluster connection probabilities $p$ and $q$, respectively. Our main result is a polynomial-time algorithm that achieves the optimal misclassification rate $\exp(-(1 \pm o(1)) C/k)$, where $C = (\sqrt{pn} - \sqrt{qn})^2$, whenever $C \geq K k^2 \log k$ for some universal constant $K$, matching the Kesten–Stigum ({KS}) threshold up to a $\log k$ factor. Notably, this rate holds even when an adversary corrupts an $\eta \leq \exp(-(1 \pm o(1)) C/k)$ fraction of the nodes. To the best of our knowledge, this optimal error rate was previously only attainable either via computationally inefficient procedures (Zhang and Zhou, 2015) or via polynomial-time algorithms that require strictly stronger assumptions such as $C \geq K k^3$ (Gao et al., 2017). In the node-robust setting, the best known algorithm requires the substantially stronger condition $C \geq K k^{102}$ (Liu and Moitra, 2022). Our results close this gap by providing the first polynomial-time algorithm that achieves the optimal error rate near the {KS} threshold in both settings. Our work has two key technical contributions: (1) we robustify majority voting via the Sum-of-Squares framework, (2) we develop a novel graph bisectioning algorithm via robust majority voting, which allows us to significantly improve the misclassification rate to $1/\mathrm{poly}(k)$ for the initial estimation near the {KS} threshold.
APA
Ding, J., Hua, Y., Lindberg, K., Steurer, D. & Storozhenko, A.. (2026). Rate-optimal community detection near the KS threshold via node-robust algorithms. Proceedings of Thirty Ninth Conference on Learning Theory, in Proceedings of Machine Learning Research 336:1965-2037 Available from https://proceedings.mlr.press/v336/ding26b.html.

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