We consider the problem of detecting whether a power-law inhomogeneous random graph contains a geometric community, and we frame this as an hypothesis testing problem. More precisely, we assume that we are given a sample from an unknown distribution on the space of graphs on n vertices. Under the null hypothesis, the sample originates from the inhomogeneous random graph with a heavy-tailed degree sequence. Under the alternative hypothesis, $k = o(n)$ vertices are given spatial locations and connect between each other following the geometric inhomogeneous random graph connection rule. The remaining $n-k$ vertices follow the inhomogeneous random graph connection rule. We propose a simple and efficient test, which is based on counting normalized triangles, to differentiate between the two hypotheses. We prove that our test correctly detects the presence of the community with high probability as $n \to \infty$, and identifies large-degree vertices of the community with high probability.

翻译：我们考虑检测幂律非均匀随机图是否包含几何群落的问题，并将此构建为假设检验问题。更精确地说，假设我们从顶点数为n的图空间上的未知分布中获取一个样本。在原假设下，样本源自具有重尾度序列的非均匀随机图。在备择假设下，$k = o(n)$个顶点被赋予空间位置，并遵循几何非均匀随机图连接规则相互连接。其余$n-k$个顶点则遵循非均匀随机图连接规则。我们提出一种简单高效的检验方法（基于归一化三角形计数）来区分两种假设。我们证明，当$n \to \infty$时，该检验能以高概率正确检测群落的存在，并以高概率识别群落中度数大的顶点。