Analysis of higher-order organizations, usually small connected subgraphs called motifs, is a fundamental task on complex networks. This paper studies a new problem of testing higher-order clusterability: given query access to an undirected graph, can we judge whether this graph can be partitioned into a few clusters of highly-connected motifs? This problem is an extension of the former work proposed by Czumaj et al. (STOC' 15), who recognized cluster structure on graphs using the framework of property testing. In this paper, a good graph cluster on high dimensions is first defined for higher-order clustering. Then, query lower bound is given for testing whether this kind of good cluster exists. Finally, an optimal sublinear-time algorithm is developed for testing clusterability based on triangles.

翻译：对高阶组织（通常称为模体的小型连通子图）的分析是复杂网络中的一项基本任务。本文研究了一个新的高阶可聚类性测试问题：给定对无向图的查询访问，我们能否判断该图是否可以划分为几个高度连通的模体簇？该问题是Czumaj等人（STOC' 15）先前工作的扩展，他们利用性质测试框架识别图上的簇结构。本文首先针对高阶聚类定义了高维空间中的良好图簇，随后给出了测试此类良好簇存在性的查询下界，最后基于三角形结构开发了一种最优的亚线性时间算法来测试可聚类性。