We consider the problem of community detection from the joint observation of a high-dimensional covariate matrix and $L$ sparse networks, all encoding noisy, partial information about the latent community labels of $n$ subjects. In the asymptotic regime where the networks have constant average degree and the number of features $p$ grows proportionally with $n$, we derive a sharp threshold under which detecting and estimating the subject labels is possible. Our results extend the work of \cite{MN23} to the constant-degree regime with noisy measurements, and also resolve a conjecture in \cite{YLS24+} when the number of networks is a constant. Our information-theoretic lower bound is obtained via a novel comparison inequality between Bernoulli and Gaussian moments, as well as a statistical variant of the ``recovery to chi-square divergence reduction'' argument inspired by \cite{DHSS25}. On the algorithmic side, we design efficient algorithms based on counting decorated cycles and decorated paths and prove that they achieve the sharp threshold for both detection and weak recovery. In particular, our results show that there is no statistical-computational gap in this setting.

翻译：我们考虑从高维协变量矩阵和$L$个稀疏网络的联合观测中进行社区检测的问题，这些数据均编码了关于$n$个主体潜在社区标签的噪声部分信息。在网络具有恒定平均度且特征数量$p$与$n$成比例增长的渐近机制下，我们推导出一个尖锐阈值，在此阈值之下检测和估计主体标签成为可能。我们的结果将\cite{MN23}的工作推广至具有噪声测量的恒定度机制，并解决了\cite{YLS24+}中当网络数量为常数时的一个猜想。我们的信息论下界通过伯努利与高斯矩之间的新型比较不等式，以及受\cite{DHSS25}启发的"恢复至卡方散度约简"论证的统计变体获得。在算法方面，我们设计了基于计数装饰环和装饰路径的高效算法，并证明它们在检测和弱恢复任务上均达到了该尖锐阈值。特别地，我们的结果表明在此设定中不存在统计-计算间隙。