We study the two inference problems of detecting and recovering an isolated community of \emph{general} structure planted in a random graph. The detection problem is formalized as a hypothesis testing problem, where under the null hypothesis, the graph is a realization of an Erd\H{o}s-R\'{e}nyi random graph $\mathcal{G}(n,q)$ with edge density $q\in(0,1)$; under the alternative, there is an unknown structure $\Gamma_k$ on $k$ nodes, planted in $\mathcal{G}(n,q)$, such that it appears as an \emph{induced subgraph}. In case of a successful detection, we are concerned with the task of recovering the corresponding structure. For these problems, we investigate the fundamental limits from both the statistical and computational perspectives. Specifically, we derive lower bounds for detecting/recovering the structure $\Gamma_k$ in terms of the parameters $(n,k,q)$, as well as certain properties of $\Gamma_k$, and exhibit computationally unbounded optimal algorithms that achieve these lower bounds. We also consider the problem of testing in polynomial-time. As is customary in many similar structured high-dimensional problems, our model undergoes an "easy-hard-impossible" phase transition and computational constraints can severely penalize the statistical performance. To provide an evidence for this phenomenon, we show that the class of low-degree polynomials algorithms match the statistical performance of the polynomial-time algorithms we develop.

翻译：我们研究了在随机图中植入的孤立的分子群的探测和回收 emph{gener} 结构的两种推论问题。 检测问题被正式确定为假设测试问题, 在无效假设下, 图表是Erd\H{ o}s- R\\\ e} 随机图 $\ mathcal{G}( n, q) 美元, 边际密度 $Q( 01) 美元 ; 在替代方案下, 一个未知的结构 $\ gamma_ k$, 以美元( gn, q), 是一个假设测试问题。 在成功检测的情况下, 我们担心的是恢复相应结构的任务。 对于这些问题, 我们从统计和计算角度都调查了基本限度。 具体地说, 我们的模型结构结构 $\ gamma_k], 以美元, 以( rc) 美元, 以美元, 以美元,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, 。,,,,,,, 。,,,,, 。,,,,,,,, 。,,,,,,,, 。,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,