Local graph clustering is an important machine learning task that aims to find a well-connected cluster near a set of seed nodes. Recent results have revealed that incorporating higher order information significantly enhances the results of graph clustering techniques. The majority of existing research in this area focuses on spectral graph theory-based techniques. However, an alternative perspective on local graph clustering arises from using max-flow and min-cut on the objectives, which offer distinctly different guarantees. For instance, a new method called capacity releasing diffusion (CRD) was recently proposed and shown to preserve local structure around the seeds better than spectral methods. The method was also the first local clustering technique that is not subject to the quadratic Cheeger inequality by assuming a good cluster near the seed nodes. In this paper, we propose a local hypergraph clustering technique called hypergraph CRD (HG-CRD) by extending the CRD process to cluster based on higher order patterns, encoded as hyperedges of a hypergraph. Moreover, we theoretically show that HG-CRD gives results about a quantity called motif conductance, rather than a biased version used in previous experiments. Experimental results on synthetic datasets and real world graphs show that HG-CRD enhances the clustering quality.
more » « less- NSF-PAR ID:
- 10542636
- Editor(s):
- Sendiña-Nadal, Irene
- Publisher / Repository:
- Public Library of Science (PLoS)
- Date Published:
- Journal Name:
- PLOS ONE
- Volume:
- 15
- Issue:
- 12
- ISSN:
- 1932-6203
- Page Range / eLocation ID:
- e0243485
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
We study p -Laplacians and spectral clustering for a recently proposed hypergraph model that incorporates edge-dependent vertex weights (EDVW). These weights can reflect different importance of vertices within a hyperedge, thus conferring the hypergraph model higher expressivity and flexibility. By constructing submodular EDVW-based splitting functions, we convert hypergraphs with EDVW into submodular hypergraphs for which the spectral theory is better developed. In this way, existing concepts and theorems such as p -Laplacians and Cheeger inequalities proposed under the submodular hypergraph setting can be directly extended to hypergraphs with EDVW. For submodular hypergraphs with EDVW-based splitting functions, we propose an efficient algorithm to compute the eigenvector associated with the second smallest eigenvalue of the hypergraph 1-Laplacian. We then utilize this eigenvector to cluster the vertices, achieving higher clustering accuracy than traditional spectral clustering based on the 2-Laplacian. More broadly, the proposed algorithm works for all submodular hypergraphs that are graph reducible. Numerical experiments using real-world data demonstrate the effectiveness of combining spectral clustering based on the 1-Laplacian and EDVW.more » « less
-
This paper proposes and analyzes a novel clustering algorithm, called \emph{learning by unsupervised nonlinear diffusion (LUND)}, that combines graph-based diffusion geometry with techniques based on density and mode estimation. LUND is suitable for data generated from mixtures of distributions with densities that are both multimodal and supported near nonlinear sets. A crucial aspect of this algorithm is the use of time of a data-adapted diffusion process, and associated diffusion distances, as a scale parameter that is different from the local spatial scale parameter used in many clustering algorithms. We prove estimates for the behavior of diffusion distances with respect to this time parameter under a flexible nonparametric data model, identifying a range of times in which the mesoscopic equilibria of the underlying process are revealed, corresponding to a gap between within-cluster and between-cluster diffusion distances. These structures may be missed by the top eigenvectors of the graph Laplacian, commonly used in spectral clustering. This analysis is leveraged to prove sufficient conditions guaranteeing the accuracy of LUND. We implement LUND and confirm its theoretical properties on illustrative data sets, demonstrating its theoretical and empirical advantages over both spectral and density-based clustering.more » « less
-
Path-based solutions have been shown to be useful for various graph analysis tasks, such as link prediction and graph clustering. However, they are no longer adequate for handling complex and gigantic graphs. Recently, motif-based analysis has attracted a lot of attention. A motif, or a small graph with a few nodes, is often considered as a fundamental unit of a graph. Motif-based analysis captures high-order structure between nodes, and performs better than traditional "edge-based" solutions. In this paper, we study motif-path , which is conceptually a concatenation of one or more motif instances. We examine how motif-paths can be used in three path-based mining tasks, namely link prediction, local graph clustering and node ranking. We further address the situation when two graph nodes are not connected through a motif-path, and develop a novel defragmentation method to enhance it. Experimental results on real graph datasets demonstrate the use of motif-paths and defragmentation techniques improves graph analysis effectiveness.more » « less
-
Multi-graph clustering aims to improve clustering accuracy by leveraging information from different domains, which has been shown to be extremely effective for achieving better clustering results than single graph based clustering algorithms. Despite the previous success, existing multi-graph clustering methods mostly use shallow models, which are incapable to capture the highly non-linear structures and the complex cluster associations in multigraph, thus result in sub-optimal results. Inspired by the powerful representation learning capability of neural networks, in this paper, we propose an end-to-end deep learning model to simultaneously infer cluster assignments and cluster associations in multi-graph. Specifically, we use autoencoding networks to learn node embeddings. Meanwhile, we propose a minimum-entropy based clustering strategy to cluster nodes in the embedding space for each graph. We introduce two regularizers to leverage both within-graph and cross-graph dependencies. An attentive mechanism is further developed to learn cross-graph cluster associations. Through extensive experiments on a variety of datasets, we observe that our method outperforms state-of-the-art baselines by a large margin.more » « less
-
null (Ed.)Hypergraph spectral analysis has emerged as an effective tool processing complex data structures in data analysis. The surface of a three-dimensional (3D) point cloud and the multilateral relationship among their points can be naturally captured by the high-dimensional hyperedges. This work investigates the power of hypergraph spectral analysis in unsupervised segmentation of 3D point clouds. We estimate and order the hypergraph spectrum from observed point cloud coordinates. By trimming the redundancy from the estimated hypergraph spectral space based on spectral component strengths, we develop a clustering-based segmentation method. We apply the proposed method to various point clouds, and analyze their respective spectral properties. Our experimental results demonstrate the effectiveness and efficiency of the proposed segmentation method.more » « less