Testing Higher-order Clusterability on graphs

Abstract: 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.