ABSTRACT
Attributed Graph Clustering (AGC) and Attributed Hypergraph Clustering (AHC) are important topics in graph mining with many applications. For AGC, amongst the unsupervised methods that combine the graph structure with node attributes, graph convolution has been shown to achieve impressive results. However, the effects of graph convolution on AGC have not yet been adequately studied. In this paper, we show that graph convolution attempts to find the best trade-off between node attribute distance and the number of inter-cluster edges. On the one hand, we show that compared to clustering node attributes directly, graph convolution produces a greater distance between node attributes in the same cluster and a smaller distance between node attributes in different clusters (which is detrimental for clustering). On the other hand, we show that graph convolution benefits clustering by considerably reducing the number of edges among different clusters. We then extend our result on AGC to AHC and leverage the hypergraph convolution to propose an unsupervised, fast, and memory-efficient algorithm (GRAC) for AHC, which achieves excellent performance on popular supervised clustering measures.
Supplemental Material
- Reid Andersen, Fan Chung, and Kevin Lang. 2007. Using PageRank to Locally Partition a Graph. Internet Mathematics 4 (01 2007), 35--64. https://doi.org/10.1080/15427951.2007.10129139Google Scholar
- Deng Cai, Xiaofei He, Jiawei Han, and Thomas S Huang. 2010. Graph regularized nonnegative matrix factorization for data representation. IEEE transactions on pattern analysis and machine intelligence 33, 8 (2010), 1548--1560. Google ScholarDigital Library
- T.-H. Hubert Chan and Zhibin Liang. 2020. Generalizing the hypergraph Laplacian via a diffusion process with mediators. Theor. Comput. Sci. 806 (2020), 416--428. https://doi.org/10.1016/j.tcs.2019.07.024Google ScholarCross Ref
- T.-H. Hubert Chan, Anand Louis, Zhihao Gavin Tang, and Chenzi Zhang. 2018. Spectral Properties of Hypergraph Laplacian and Approximation Algorithms. J. ACM 65, 3, Article 15 (March 2018), 48 pages. https://doi.org/10.1145/3178123 Google ScholarDigital Library
- Olivier Delalleau, Yoshua Bengio, and Nicolas Le Roux. 2005. Efficient NonParametric Function Induction in Semi-Supervised Learning. In Proceedings of the Tenth International Workshop on Artificial Intelligence and Statistics (Proceedings of Machine Learning Research, Vol. R5), Robert G. Cowell and Zoubin Ghahramani (Eds.). PMLR, 96--103. http://proceedings.mlr.press/r5/delalleau05a.html Reissued by PMLR on 30 March 2021.Google Scholar
- Inderjit Dhillon, Yuqiang Guan, and Brian Kulis. 2007. Weighted Graph Cuts without Eigenvectors A Multilevel Approach. IEEE transactions on pattern analysis and machine intelligence 29 (12 2007), 1944--57. https://doi.org/10.1109/TPAMI. 2007.1115 Google ScholarDigital Library
- Inderjit S. Dhillon, Yuqiang Guan, and Brian Kulis. 2004. Kernel k-means: spectral clustering and normalized cuts. In Proceedings of the Tenth ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, Seattle, Washington, USA, August 22--25, 2004, Won Kim, Ron Kohavi, Johannes Gehrke, and William DuMouchel (Eds.). ACM, 551--556. https://doi.org/10.1145/1014052.1014118 Google ScholarDigital Library
- Rundong Du, Barry L. Drake, and Haesun Park. 2017. Hybrid Clustering based on Content and Connection Structure using Joint Nonnegative Matrix Factorization. CoRR abs/1703.09646 (2017). arXiv:1703.09646 http://arxiv.org/abs/1703.09646Google Scholar
- Yifan Feng, Haoxuan You, Zizhao Zhang, Rongrong Ji, and Yue Gao. 2018. Hypergraph Neural Networks. AAAI 2019 (2018).Google Scholar
- Yifan Feng, Haoxuan You, Zizhao Zhang, Rongrong Ji, and Yue Gao. 2018. Hypergraph Neural Networks. AAAI 2019 (2018).Google Scholar
- Aditya Grover and Jure Leskovec. 2016. node2vec: Scalable Feature Learning for Networks. In Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining. Google ScholarDigital Library
- Ting Guo, Shirui Pan, Xingquan Zhu, and Chengqi Zhang. 2018. CFOND: consensus factorization for co-clustering networked data. IEEE Transactions on Knowledge and Data Engineering 31, 4 (2018), 706--719. Google ScholarDigital Library
- Will Hamilton, Zhitao Ying, and Jure Leskovec. 2017. Inductive representation learning on large graphs. In Advances in neural information processing systems. 1024--1034. Google ScholarDigital Library
- Po-Yao Huang, Robert Frederking, et al. 2019. RWR-GAE: Random Walk Regularization for Graph Auto Encoders. arXiv preprint arXiv:1908.04003 (2019).Google Scholar
- Jianbo Shi and J. Malik. 2000. Normalized cuts and image segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence 22, 8 (2000), 888--905. https://doi.org/10.1109/34.868688 Google ScholarDigital Library
- Jiang Jianwen, Wei Yuxuan, Feng Yifan, Cao Jingxuan, and Gao Yue. 2019. Dynamic Hypergraph Neural Networks. In Proceedings of International Joint Conferences on Artificial Intelligence. Google ScholarDigital Library
- Vassilis Kalofolias. 2016. How to Learn a Graph from Smooth Signals. In Proceedings of the 19th International Conference on Artificial Intelligence and Statistics (Proceedings of Machine Learning Research, Vol. 51), Arthur Gretton and Christian C. Robert (Eds.). PMLR, Cadiz, Spain, 920--929. http://proceedings.mlr.press/v51/kalofolias16.htmlGoogle Scholar
- Thomas N Kipf and Max Welling. 2016. Variational Graph Auto-Encoders. NIPS Workshop on Bayesian Deep Learning (2016).Google Scholar
- Thomas N. Kipf and Max Welling. 2017. Semi-Supervised Classification with Graph Convolutional Networks. In International Conference on Learning Representations (ICLR).Google Scholar
- Da Kuang, Chris Ding, and Haesun Park. 2012. Symmetric nonnegative matrix factorization for graph clustering. In Proceedings of the 2012 SIAM international conference on data mining. SIAM, 106--117.Google ScholarCross Ref
- Da Kuang, Chris Ding, and Haesun Park. 2012. Symmetric nonnegative matrix factorization for graph clustering. In Proceedings of the 2012 SIAM international conference on data mining. SIAM, 106--117.Google ScholarCross Ref
- Tarun Kumar, Sankaran Vaidyanathan, Harini Ananthapadmanabhan, Srinivasan Parthasarathy, and Balaraman Ravindran. 2018. Hypergraph Clustering: A Modularity Maximization Approach. CoRR abs/1812.10869 (2018). arXiv:1812.10869 http://arxiv.org/abs/1812.10869Google Scholar
- Marius Leordeanu and Cristian Sminchisescu. 2012. Efficient Hypergraph Clustering. AISTATS (01 2012).Google Scholar
- D. Li, Z. Xu, S. Li, and X. Sun. 2013. Link prediction in social networks based on hypergraph. WWW 2013 Companion - Proceedings of the 22nd International Conference on World Wide Web (01 2013), 41--42. Google ScholarDigital Library
- Pan Li and Olgica Milenkovic. 2017. Inhomogeneous Hypergraph Clustering with Applications. In Advances in Neural Information Processing Systems, I. Guyon, U. V. Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan, and R. Garnett (Eds.), Vol. 30. Curran Associates, Inc., 2308--2318. https://proceedings.neurips. cc/paper/2017/file/a50abba8132a77191791390c3eb19fe7-Paper.pdf Google ScholarDigital Library
- Q. Li, Z. Han, and X.-M. Wu. 2018. Deeper Insights into Graph Convolutional Networks for Semi-Supervised Learning. In The Thirty-Second AAAI Conference on Artificial Intelligence. AAAI.Google ScholarCross Ref
- Qimai Li, Xiao-Ming Wu, Han Liu, Xiaotong Zhang, and Zhichao Guan. 2019. Label efficient semi-supervised learning via graph filtering. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. 9582--9591.Google ScholarCross Ref
- Ulrike Luxburg. 2004. A Tutorial on Spectral Clustering. Statistics and Computing 17 (01 2004), 395--416. https://doi.org/10.1007/s11222-007--9033-z Google ScholarDigital Library
- James MacQueen et al. 1967. Some methods for classification and analysis of multivariate observations. In Proceedings of the fifth Berkeley symposium on mathematical statistics and probability, Vol. 1. Oakland, CA, USA, 281--297.Google Scholar
- James MacQueen et al. 1967. Some methods for classification and analysis of multivariate observations. In Proceedings of the fifth Berkeley symposium on mathematical statistics and probability, Vol. 1. Oakland, CA, USA, 281--297.Google Scholar
- Shirui Pan, Ruiqi Hu, Guodong Long, Jing Jiang, Lina Yao, and Chengqi Zhang. 2018. Adversarially Regularized Graph Autoencoder for Graph Embedding.. In IJCAI. 2609--2615. Google ScholarDigital Library
- P. Perona and W. Freeman. 1998. A Factorization Approach to Grouping. In in European Conference on Computer Vision. 655--670. Google ScholarDigital Library
- Bryan Perozzi, Rami Al-Rfou, and Steven Skiena. 2014. Deepwalk: Online learning of social representations. In Proceedings of the 20th ACM SIGKDD international conference on Knowledge discovery and data mining. 701--710. Google ScholarDigital Library
- J.A. Rodriguez. 2002. On the Laplacian Eigenvalues and Metric Parameters of Hypergraphs. Linear and Multilinear Algebra 50, 1 (2002), 1--14. https://doi.org/10.1080/03081080290011692 arXiv:https://doi.org/10.1080/03081080290011692Google ScholarCross Ref
- Shota Saito, Danilo P. Mandic, and Hideyuki Suzuki. 2017. Hypergraph p-Laplacian: A Differential Geometry View. CoRR abs/1711.08171 (2017). arXiv:1711.08171 http://arxiv.org/abs/1711.08171Google Scholar
- D. I. Shuman, S. K. Narang, P. Frossard, A. Ortega, and P. Vandergheynst. 2013. The emerging field of signal processing on graphs: Extending high-dimensional data analysis to networks and other irregular domains. IEEE Signal Processing Magazine 30, 3 (2013), 83--98. https://doi.org/10.1109/MSP.2012.2235192Google ScholarCross Ref
- Yuuki Takai, Atsushi Miyauchi, Masahiro Ikeda, and Yuichi Yoshida. 2020. Hypergraph Clustering Based on PageRank. 1970--1978. https://doi.org/10.1145/3394486.3403248Google Scholar
- Guillem Cucurull, Arantxa Casanova, Adriana Romero, Pietro Liò, and Yoshua Bengio. 2018. Graph Attention Networks. International Conference on Learning Representations (2018). https://openreview.net/forum?id= rJXMpikCZ accepted as poster.Google Scholar
- Chun Wang, Shirui Pan, Ruiqi Hu, Guodong Long, Jing Jiang, and Chengqi Zhang. 2019. Attributed Graph Clustering: a Deep Attentional Embedding approach. In Proceedings of the Twenty-Eighth International Joint Conference on Artificial Intelligence, Sarit Kraus (Ed.). Association for the Advancement of Artificial Intelligence (AAAI), United States of America, 3670--3676. https://doi.org/10.24963/ijcai.2019/509 Google ScholarDigital Library
- Chun Wang, Shirui Pan, Ruiqi Hu, Guodong Long, Jing Jiang, and Chengqi Zhang. 2019. Attributed Graph Clustering: a Deep Attentional Embedding approach. In Proceedings of the Twenty-Eighth International Joint Conference on Artificial Intelligence, Sarit Kraus (Ed.). Association for the Advancement of Artificial Intelligence (AAAI), United States of America, 3670--3676. https://doi.org/10.24963/ijcai.2019/509 Google ScholarDigital Library
- Chun Wang, Shirui Pan, Guodong Long, Xingquan Zhu, and Jing Jiang. 2017. Mgae: Marginalized graph autoencoder for graph clustering. In Proceedings of the 2017 ACM on Conference on Information and Knowledge Management. 889--898. Google ScholarDigital Library
- Joyce Whang, Rundong Du, Sangwon Jung, Geon Lee, Barry Drake, Qingqing Liu, Seonggoo Kang, and Haesun Park. 2020. MEGA: multi-view semi-supervised clustering of hypergraphs. Proceedings of the VLDB Endowment 13 (01 2020), 698--711. https://doi.org/10.14778/3377369.3377378 Google ScholarDigital Library
- Felix Wu, Amauri Souza, Tianyi Zhang, Christopher Fifty, Tao Yu, and Kilian Weinberger. 2019. Simplifying Graph Convolutional Networks. In Proceedings of the 36th International Conference on Machine Learning. PMLR, 6861--6871.Google Scholar
- Felix Wu, Amauri Souza, Tianyi Zhang, Christopher Fifty, Tao Yu, and Kilian Weinberger. 2019. Simplifying Graph Convolutional Networks. In Proceedings of the 36th International Conference on Machine Learning. PMLR, 6861--6871.Google Scholar
- Rongkai Xia, Y. Pan, Lei Du, and J. Yin. 2014. Robust Multi-View Spectral Clustering via Low-Rank and Sparse Decomposition. In AAAI. Google ScholarDigital Library
- Keyulu Xu, Weihua Hu, Jure Leskovec, and Stefanie Jegelka. 2019. How Powerful are Graph Neural Networks?. In International Conference on Learning Representations. https://openreview.net/forum?id=ryGs6iA5KmGoogle Scholar
- Zhiqiang Xu, Yiping Ke, Yi Wang, Hong Cheng, and James Cheng. 2012. A model-based approach to attributed graph clustering. Proceedings of the 2012 ACM SIGMOD International Conference on Management of Data, SIGMOD '12 (05 2012). https://doi.org/10.1145/2213836.2213894 Google ScholarDigital Library
- Zhiqiang Xu, Yiping Ke, Yi Wang, Hong Cheng, and James Cheng. 2014. GBAGC: A General Bayesian Framework for Attributed Graph Clustering. ACM Trans. Knowl. Discov. Data 9 (2014), 5:1--5:43. http://dblp.uni-trier.de/db/journals/tkdd/tkdd9.html#XuK0CC14 Google ScholarDigital Library
- Naganand Yadati, Madhav Nimishakavi, Prateek Yadav, Vikram Nitin, Anand Louis, and Partha Talukdar. 2019. HyperGCN: A New Method For Training Graph Convolutional Networks on Hypergraphs. In Advances in Neural Information Processing Systems (NeurIPS) 32. Curran Associates, Inc., 1509--1520. Google ScholarDigital Library
- Tianbao Yang, Rong Jin, Yun Chi, and Shenghuo Zhu. 2009. Combining Link and Content for Community Detection: A Discriminative Approach. In Proceedings of the 15th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (Paris, France) (KDD '09). Association for Computing Machinery, New York, NY, USA, 927--936. https://doi.org/10.1145/1557019.1557120 Google ScholarDigital Library
- Xiaotong Zhang, Han Liu, Qimai Li, and Xiao-Ming Wu. 2019. Attributed Graph Clustering via Adaptive Graph Convolution. 4327--4333. https://doi.org/10.24963/ijcai.2019/601 Google ScholarDigital Library
- Lingxiao Zhao and Leman Akoglu. 2020. PairNorm: Tackling Oversmoothing in GNNs. In International Conference on Learning Representations. https://openreview.net/forum?id=rkecl1rtwBGoogle Scholar
- Dengyong Zhou, Jiayuan Huang, and Bernhard Schölkopf. 2006. Learning with Hypergraphs: Clustering, Classification, and Embedding. Advances in Neural Information Processing Systems 19: Proceedings of the 2006 Conference, 1601--1608 (2007) 19, 1601--1608. Google ScholarDigital Library
Index Terms
- HyperGraph Convolution Based Attributed HyperGraph Clustering
Recommendations
Hypergraph Based Berge Hypergraphs
AbstractFix a hypergraph . A hypergraph is called a Berge copy of or Berge- if we can choose a subset of each hyperedge of to obtain a copy of . A hypergraph is Berge--free if it does not contain a subhypergraph which is Berge copy of . This ...
Acyclic Hypergraph Projections
Hypergraphs can be partitioned into two classes: tree (acyclic) hypergraphs and cyclic hypergraphs. This paper analyzes a new class of cyclic hypergraphs called Xrings. HypergraphHis an Xring if the hyperedges ofHcan be circularly ordered so that for ...
Efficient and Effective Attributed Hypergraph Clustering via K-Nearest Neighbor Augmentation
PACMMODHypergraphs are an omnipresent data structure used to represent high-order interactions among entities. Given a hypergraph H wherein nodes are associated with attributes, attributed hypergraph clustering (AHC) aims to partition the nodes in H into k ...
Comments