Clustering is an important topic in algorithms, and has a number of applications in machine learning, computer vision, statistics, and several other research disciplines. Traditional objectives of graph clustering are to find clusters with low conductance. Not only are these objectives just applicable for undirected graphs, they are also incapable to take the relationships between clusters into account, which could be crucial for many applications. To overcome these downsides, we study directed graphs (digraphs) whose clusters exhibit further "structural" information amongst each other. Based on the Hermitian matrix representation of digraphs, we present a nearly-linear time algorithm for digraph clustering, and further show that our proposed algorithm can be implemented in sublinear time under reasonable assumptions. The significance of our theoretical work is demonstrated by extensive experimental results on the UN Comtrade Dataset: the output clustering of our algorithm exhibits not only how the clusters (sets of countries) relate to each other with respect to their import and export records, but also how these clusters evolve over time, in accordance with known facts in international trade.

Summary and Contributions: The paper considers a graph clustering on directed graphs. The authors introduce a new notion of clustering objective denoted by flow ratio. For any ordered partition of vertex set V into k pairwise disjoint subset (S0, ..., Sk-1), the flow ratio of the partition is sum of the average flow (i.e. The optimal clustering is the partitioning of V that maximizes the flow ratio among all possible partitions. The authors represent the directed graph using the Hermitian adjacency matrix.

Clustering is an important topic in algorithms, and has a number of applications in machine learning, computer vision, statistics, and several other research disciplines. Traditional objectives of graph clustering are to find clusters with low conductance. Not only are these objectives just applicable for undirected graphs, they are also incapable to take the relationships between clusters into account, which could be crucial for many applications. To overcome these downsides, we study directed graphs (digraphs) whose clusters exhibit further "structural" information amongst each other. Based on the Hermitian matrix representation of digraphs, we present a nearly-linear time algorithm for digraph clustering, and further show that our proposed algorithm can be implemented in sublinear time under reasonable assumptions.

The present paper is devoted to clustering geometric graphs. While the standard spectral clustering is often not effective for geometric graphs, we present an effective generalization, which we call higher-order spectral clustering. It resembles in concept the classical spectral clustering method but uses for partitioning the eigenvector associated with a higher-order eigenvalue. We establish the weak consistency of this algorithm for a wide class of geometric graphs which we call Soft Geometric Block Model. A small adjustment of the algorithm provides strong consistency. We also show that our method is effective in numerical experiments even for graphs of modest size.