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：我们考虑了检测电法不相容随机图是否包含几何界的问题, 我们将此设置为假设测试问题。 更确切地说, 我们假设我们从n 脊椎上的图形空间的未知分布中获得了样本。 在无效假设下, 样本来源于不相干随机图, 且具有重度序列。 在替代假设下, $k = o(n) o(n) o(n) o(n) o(n) o( o) o( o) o( o) o( o) o( o) o( o) o( o) o( o( o) o( o) o( o) o( o) o( o) o( o) o( o) ) o( o( o) o( o( o) o( o( o) o( o) o( o) o( o) o( o) o( o) o( o) o( o) o( o) o( o) o( t) ) ) o( ) ) o( o( ) o( t) o( t) ) ) ) ) o( o( o( o( ) ) o( o( o( o( o( o) o) o) ) ) o( o( o( ) ) o( ) o( o( ) ) o( o( ) ) ) ) o( o( o( ) ) ) ) ) ) o( ) o( o( o( o( o( o( o( o( o) o( o( o) ) o( o) o) o) ) o( o( o( o( o) o) o( o) o) </s>