How to detect a small community in a large network is an interesting problem, including clique detection as a special case, where a naive degree-based $\chi^2$-test was shown to be powerful in the presence of an Erd\H{o}s-Renyi background. Using Sinkhorn's theorem, we show that the signal captured by the $\chi^2$-test may be a modeling artifact, and it may disappear once we replace the Erd\H{o}s-Renyi model by a broader network model. We show that the recent SgnQ test is more appropriate for such a setting. The test is optimal in detecting communities with sizes comparable to the whole network, but has never been studied for our setting, which is substantially different and more challenging. Using a degree-corrected block model (DCBM), we establish phase transitions of this testing problem concerning the size of the small community and the edge densities in small and large communities. When the size of the small community is larger than $\sqrt{n}$, the SgnQ test is optimal for it attains the computational lower bound (CLB), the information lower bound for methods allowing polynomial computation time. When the size of the small community is smaller than $\sqrt{n}$, we establish the parameter regime where the SgnQ test has full power and make some conjectures of the CLB. We also study the classical information lower bound (LB) and show that there is always a gap between the CLB and LB in our range of interest.

翻译：如何在大规模网络中检测小型社区是一个有趣的问题，其中包含作为特例的团检测问题。研究表明，在Erdős-Rényi背景模型下，基于度数的朴素χ²检验具有显著功效。通过Sinkhorn定理，我们证明χ²检验捕获的信号可能是建模伪影，且一旦将Erdős-Rényi模型替换为更广泛的网络模型，该信号可能消失。我们证明近期提出的SgnQ检验更适用于此类场景。该检验在检测规模与整个网络相当的社区时具有最优性，但从未在本文所研究这一显著不同且更具挑战性的场景中得到验证。采用度修正块模型（DCBM），我们建立了关于小型社区规模及大小社区中边密度这一检验问题的相变。当小型社区规模大于√n时，SgnQ检验达到计算下界（CLB）——即允许多项式计算时间方法的信息下界，因而具有最优性。当小型社区规模小于√n时，我们刻画了SgnQ检验具有完全功效的参数区间，并对计算下界提出若干猜想。我们还研究了经典信息下界（LB），证明在我们关注的参数范围内，计算下界与信息下界始终存在差距。