用于检测大型网络中的小社区的相变
Phase transition for detecting a small community in a large network
摘要：
如何在大型网络中检测小社区是一个有趣的问题，包括集团检测作为一种特殊情况，其中基于朴素度的$χ^2$ -测试被证明是强大的，在存在Erdens-Renyi背景。使用Sinkhorn定理，我们表明，由$χ^2$测试捕获的信号可能是一个建模工件，它可能会消失，一旦我们用一个更广泛的网络模型取代Erdens-Renyi模型。我们表明，最近的SgnQ测试更适合这样的设置。该测试在检测规模与整个网络相当的社区时是最佳的，但从未在我们的设置中进行过研究，这是一个非常不同且更具挑战性的设置。使用度校正块模型（DCBM），我们建立了这个测试问题的相变的大小的小社区和边缘密度在小社区和大社区。当小社区的大小大于$\sqrt{n}$时，SgnQ测试是最佳的，因为它达到了计算下界（CLB），即允许多项式计算时间的方法的信息下界。当小社区的规模小于$\sqrt{n}$时，我们建立了SgnQ测试具有全功率的参数机制，并对CLB进行了一些说明。我们还研究了经典的信息下界（LB），并表明，总是有一个差距CLB和LB在我们感兴趣的范围。
英文摘要：
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 $χ^2$-test was shown to be powerful in the presence of an Erdős-Renyi background. Using Sinkhorn's theorem, we show that the signal captured by the $χ^2$-test may be a modeling artifact, and it may disappear once we replace the Erdős-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.
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