A Survey of Deep Graph Clustering: Taxonomy, Challenge, Application, and Open Resource

Graph clustering, which aims to divide nodes in the graph into several distinct clusters, is a fundamental yet challenging task. Benefiting from the powerful representation capability of deep learning, deep graph clustering methods have achieved great success in recent years. However, the corresponding survey paper is relatively scarce, and it is imminent to make a summary of this field. From this motivation, we conduct a comprehensive survey of deep graph clustering. Firstly, we introduce formulaic definition, evaluation, and development in this field. Secondly, the taxonomy of deep graph clustering methods is presented based on four different criteria, including graph type, network architecture, learning paradigm, and clustering method. Thirdly, we carefully analyze the existing methods via extensive experiments and summarize the challenges and opportunities from five perspectives, including graph data quality, stability, scalability, discriminative capability, and unknown cluster number. Besides, the applications of deep graph clustering methods in six domains, including computer vision, natural language processing, recommendation systems, social network analyses, bioinformatics, and medical science, are presented. Last but not least, this paper provides open resource supports, including 1) a collection (\url{https://github.com/yueliu1999/Awesome-Deep-Graph-Clustering}) of state-of-the-art deep graph clustering methods (papers, codes, and datasets) and 2) a unified framework (\url{https://github.com/Marigoldwu/A-Unified-Framework-for-Deep-Attribute-Graph-Clustering}) of deep graph clustering. We hope this work can serve as a quick guide and help researchers overcome challenges in this vibrant field.

[1]  Yue Liu,et al.  DealMVC: Dual Contrastive Calibration for Multi-view Clustering , 2023, ACM Multimedia.

[2]  Stan Z. Li,et al.  CONVERT: Contrastive Graph Clustering with Reliable Augmentation , 2023, ArXiv.

[3]  Stan Z. Li,et al.  Reinforcement Graph Clustering with Unknown Cluster Number , 2023, ArXiv.

[4]  Yue Liu,et al.  Learn from Relational Correlations and Periodic Events for Temporal Knowledge Graph Reasoning , 2023, SIGIR.

[5]  Siwei Wang,et al.  Unpaired Multi-View Graph Clustering with Cross-View Structure Matching , 2023, IEEE transactions on neural networks and learning systems.

[6]  Shifei Ding,et al.  Graph clustering network with structure embedding enhanced , 2023, Pattern Recognition.

[7]  Uday Singh Saini,et al.  CARL-G: Clustering-Accelerated Representation Learning on Graphs , 2023, KDD.

[8]  Yue Liu,et al.  arXiv4TGC: Large-Scale Datasets for Temporal Graph Clustering , 2023, ArXiv.

[9]  Stan Z. Li,et al.  Dink-Net: Neural Clustering on Large Graphs , 2023, ICML.

[10]  Jingbo Shang,et al.  ClusterLLM: Large Language Models as a Guide for Text Clustering , 2023, EMNLP.

[11]  Yue Liu,et al.  Message Intercommunication for Inductive Relation Reasoning , 2023, ArXiv.

[12]  Yue Liu,et al.  Deep Temporal Graph Clustering , 2023, ICLR.

[13]  Philip S. Yu,et al.  Contrastive Graph Clustering in Curvature Spaces , 2023, IJCAI.

[14]  Zhao Kang,et al.  Beyond Homophily: Reconstructing Structure for Graph-agnostic Clustering , 2023, ICML.

[15]  Xiaolei Huang,et al.  ABSLearn: a GNN-based framework for aliasing and buffer-size information retrieval , 2023, Pattern Analysis and Applications.

[16]  Yue Liu,et al.  Cluster-guided Contrastive Graph Clustering Network , 2023, AAAI.

[17]  Yue Liu,et al.  Hard Sample Aware Network for Contrastive Deep Graph Clustering , 2022, AAAI.

[18]  Yue Liu,et al.  Contrastive Deep Graph Clustering with Learnable Augmentation , 2022, ArXiv.

[19]  Rong Jiang,et al.  An access control model for medical big data based on clustering and risk , 2022, Inf. Sci..

[20]  Philip S. Yu,et al.  Deep Clustering: A Comprehensive Survey , 2022, IEEE transactions on neural networks and learning systems.

[21]  Neil Shah,et al.  Multi-task Self-supervised Graph Neural Networks Enable Stronger Task Generalization , 2022, ICLR.

[22]  Xifeng Guo,et al.  Deep graph clustering with multi-level subspace fusion , 2022, Pattern Recognit..

[23]  Siwei Wang,et al.  Multiple Kernel Clustering with Dual Noise Minimization , 2022, ACM Multimedia.

[24]  Kuan-Ching Li,et al.  Local Sample-weighted Multiple Kernel Clustering with Consensus Discriminative Graph , 2022, IEEE transactions on neural networks and learning systems.

[25]  K. Berger,et al.  Molecular Clustering Analysis of Blood Biomarkers in World Trade Center Exposed Community Members with Persistent Lower Respiratory Symptoms , 2022, International journal of environmental research and public health.

[26]  Sihang Zhou,et al.  Attributed Graph Clustering with Dual Redundancy Reduction , 2022, IJCAI.

[27]  Yue Liu,et al.  Initializing Then Refining: A Simple Graph Attribute Imputation Network , 2022, IJCAI.

[28]  M. Bouguessa,et al.  Escaping Feature Twist: A Variational Graph Auto-Encoder for Node Clustering , 2022, IJCAI.

[29]  Ka-chun Wong,et al.  ZINB-Based Graph Embedding Autoencoder for Single-Cell RNA-Seq Interpretations , 2022, AAAI.

[30]  Guanyu Yang,et al.  Neighborhood contrastive representation learning for attributed graph clustering , 2022, Neurocomputing.

[31]  Jiawei Chen,et al.  A Comprehensive Survey on Deep Clustering: Taxonomy, Challenges, and Future Directions , 2022, ArXiv.

[32]  Yue Liu,et al.  Mixed Graph Contrastive Network for Semi-Supervised Node Classification , 2022, ACM Transactions on Knowledge Discovery from Data.

[33]  Xinzhong Zhu,et al.  Highly-efficient Incomplete Largescale Multiview Clustering with Consensus Bipartite Graph , 2022, 2022 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR).

[34]  Q. Hu,et al.  Collaborative Decision-Reinforced Self-Supervision for Attributed Graph Clustering. , 2022, IEEE transactions on neural networks and learning systems.

[35]  Siwei Wang,et al.  Simple Contrastive Graph Clustering , 2022, IEEE transactions on neural networks and learning systems.

[36]  Di Jin,et al.  Exploring Temporal Community Structure via Network Embedding , 2022, IEEE Transactions on Cybernetics.

[37]  Gayan K. Kulatilleke,et al.  SCGC : Self-Supervised Contrastive Graph Clustering , 2022, ArXiv.

[38]  Nesreen K. Ahmed,et al.  CGC: Contrastive Graph Clustering forCommunity Detection and Tracking , 2022, WWW.

[39]  Peibo Li,et al.  Embedding Graph Auto-Encoder for Graph Clustering , 2022, IEEE Transactions on Neural Networks and Learning Systems.

[40]  Daniele Grattarola,et al.  Unsupervised Network Embedding Beyond Homophily , 2022, Trans. Mach. Learn. Res..

[41]  Yue Liu,et al.  Improved Dual Correlation Reduction Network , 2022, ArXiv.

[42]  En Zhu,et al.  Interpolation-based Contrastive Learning for Few-Label Semi-Supervised Learning , 2022, IEEE transactions on neural networks and learning systems.

[43]  Philip S. Yu,et al.  Graph Neural Networks for Graphs with Heterophily: A Survey , 2022, ArXiv.

[44]  M. Kleinsteuber,et al.  Cluster-Aware Heterogeneous Information Network Embedding , 2022, WSDM.

[45]  Senzhang Wang,et al.  Ada-NETS: Face Clustering via Adaptive Neighbour Discovery in the Structure Space , 2022, ICLR.

[46]  Stan Z. Li,et al.  SimGRACE: A Simple Framework for Graph Contrastive Learning without Data Augmentation , 2022, WWW.

[47]  Julian McAuley,et al.  Intent Contrastive Learning for Sequential Recommendation , 2022, WWW.

[48]  Ruiqi Hu,et al.  Deep neighbor-aware embedding for node clustering in attributed graphs , 2022, Pattern Recognit..

[49]  Qun Dai,et al.  Graph Clustering via Variational Graph Embedding , 2022, Pattern Recognit..

[50]  Hao Peng,et al.  Towards Unsupervised Deep Graph Structure Learning , 2022, WWW.

[51]  En Zhu,et al.  Deep Graph Clustering via Dual Correlation Reduction , 2021, AAAI.

[52]  Vijini Mallawaarachchi,et al.  RepBin: Constraint-based Graph Representation Learning for Metagenomic Binning , 2021, AAAI.

[53]  Chanyoung Park,et al.  Augmentation-Free Self-Supervised Learning on Graphs , 2021, AAAI.

[54]  Heli Sun,et al.  Graph Community Infomax , 2021, ACM Trans. Knowl. Discov. Data.

[55]  Zhihao Peng,et al.  Deep Attention-Guided Graph Clustering With Dual Self-Supervision , 2021, IEEE Transactions on Circuits and Systems for Video Technology.

[56]  Junbin Gao,et al.  Wasserstein Adversarially Regularized Graph Autoencoder , 2021, Neurocomputing.

[57]  Zhao Kang,et al.  Multi-view Contrastive Graph Clustering , 2021, NeurIPS.

[58]  Stan Z. Li,et al.  ProGCL: Rethinking Hard Negative Mining in Graph Contrastive Learning , 2021, ICML.

[59]  Shu Wu,et al.  An Empirical Study of Graph Contrastive Learning , 2021, NeurIPS Datasets and Benchmarks.

[60]  Youpeng Hu,et al.  Adaptive Hypergraph Auto-Encoder for Relational Data Clustering , 2021, IEEE Transactions on Knowledge and Data Engineering.

[61]  C. Ding,et al.  Tensorized Bipartite Graph Learning for Multi-View Clustering , 2021, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[62]  Hui Liu,et al.  Attention-driven Graph Clustering Network , 2021, ACM Multimedia.

[63]  Han Zhao,et al.  Graph Debiased Contrastive Learning with Joint Representation Clustering , 2021, IJCAI.

[64]  Riadh Ksantini,et al.  Rethinking Graph Auto-Encoder Models for Attributed Graph Clustering , 2021, IEEE Transactions on Knowledge and Data Engineering.

[65]  Xinbo Gao,et al.  Self-Supervised Graph Convolutional Network for Multi-View Clustering , 2021, IEEE Transactions on Multimedia.

[66]  Quan Z. Sheng,et al.  A Comprehensive Survey on Graph Anomaly Detection With Deep Learning , 2021, IEEE Transactions on Knowledge and Data Engineering.

[67]  Xiaorui Liu,et al.  Automated Self-Supervised Learning for Graphs , 2021, ICLR.

[68]  Yonina C. Eldar,et al.  Graph Signal Denoising Via Unrolling Networks , 2021, ICASSP 2021 - 2021 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).

[69]  Quan Z. Sheng,et al.  A Comprehensive Survey on Community Detection With Deep Learning , 2021, IEEE Transactions on Neural Networks and Learning Systems.

[70]  Chuan Shi,et al.  Self-supervised Heterogeneous Graph Neural Network with Co-contrastive Learning , 2021, KDD.

[71]  Xinbo Gao,et al.  Graph embedding clustering: Graph attention auto-encoder with cluster-specificity distribution , 2021, Neural Networks.

[72]  Zhouchen Lin,et al.  Graph Contrastive Clustering , 2021, 2021 IEEE/CVF International Conference on Computer Vision (ICCV).

[73]  Yann LeCun,et al.  Barlow Twins: Self-Supervised Learning via Redundancy Reduction , 2021, ICML.

[74]  Philip S. Yu,et al.  Graph Self-Supervised Learning: A Survey , 2021, IEEE Transactions on Knowledge and Data Engineering.

[75]  Jieren Cheng,et al.  Deep Fusion Clustering Network , 2020, AAAI.

[76]  Yu Ding,et al.  Graph Regularized Autoencoder and its Application in Unsupervised Anomaly Detection , 2020, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[77]  Zhangyang Wang,et al.  Graph Contrastive Learning with Augmentations , 2020, NeurIPS.

[78]  Jiawei Zhang,et al.  CommDGI: Community Detection Oriented Deep Graph Infomax , 2020, CIKM.

[79]  Honglei Zhang,et al.  Dirichlet Graph Variational Autoencoder , 2020, NeurIPS.

[80]  Siheng Chen,et al.  Learning on Attribute-Missing Graphs , 2020, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[81]  Xiangliang Zhang,et al.  SAIL: Self-Augmented Graph Contrastive Learning , 2020, AAAI.

[82]  Xiaochun Cao,et al.  JANE: Jointly Adversarial Network Embedding , 2020, IJCAI.

[83]  Jie Zhou,et al.  Adaptive Graph Encoder for Attributed Graph Embedding , 2020, KDD.

[84]  Qianqian Wang,et al.  Multi-View Attribute Graph Convolution Networks for Clustering , 2020, IJCAI.

[85]  Sunil Kumar Sahu,et al.  Autoencoding Keyword Correlation Graph for Document Clustering , 2020, ACL.

[86]  Emmanuel Müller,et al.  Graph Clustering with Graph Neural Networks , 2020, J. Mach. Learn. Res..

[87]  Pierre H. Richemond,et al.  Bootstrap Your Own Latent: A New Approach to Self-Supervised Learning , 2020, NeurIPS.

[88]  Liang Wang,et al.  Deep Graph Contrastive Representation Learning , 2020, ArXiv.

[89]  Kaveh Hassani,et al.  Contrastive Multi-View Representation Learning on Graphs , 2020, ICML.

[90]  J. Leskovec,et al.  Open Graph Benchmark: Datasets for Machine Learning on Graphs , 2020, NeurIPS.

[91]  Duc Minh Nguyen,et al.  Graph Auto-Encoder for Graph Signal Denoising , 2020, ICASSP 2020 - 2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).

[92]  Xiao Wang,et al.  One2Multi Graph Autoencoder for Multi-view Graph Clustering , 2020, WWW.

[93]  Qinghua Hu,et al.  Collaborative Graph Convolutional Networks: Unsupervised Learning Meets Semi-Supervised Learning , 2020, AAAI.

[94]  Liming Zhu,et al.  Going Deep: Graph Convolutional Ladder-Shape Networks , 2020, AAAI.

[95]  Manoj Diwakar,et al.  Clustering based Multi-modality Medical Image Fusion , 2020, Journal of Physics: Conference Series.

[96]  Tengpeng Li,et al.  Adaptive Graph Convolutional Network With Attention Graph Clustering for Co-Saliency Detection , 2020, 2020 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR).

[97]  Geoffrey E. Hinton,et al.  A Simple Framework for Contrastive Learning of Visual Representations , 2020, ICML.

[98]  Xiao Wang,et al.  Structural Deep Clustering Network , 2020, WWW.

[99]  Suhang Wang,et al.  Deep Multi-Graph Clustering via Attentive Cross-Graph Association , 2020, WSDM.

[100]  Bernard Ghanem,et al.  Self-Supervised Learning by Cross-Modal Audio-Video Clustering , 2019, NeurIPS.

[101]  Ross B. Girshick,et al.  Momentum Contrast for Unsupervised Visual Representation Learning , 2019, 2020 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR).

[102]  Robert Frederking,et al.  RWR-GAE: Random Walk Regularization for Graph Auto Encoders , 2019, ArXiv.

[103]  Xu Wang,et al.  An improved index for clustering validation based on Silhouette index and Calinski-Harabasz index , 2019, IOP Conference Series: Materials Science and Engineering.

[104]  Hyung Jin Chang,et al.  Symmetric Graph Convolutional Autoencoder for Unsupervised Graph Representation Learning , 2019, 2019 IEEE/CVF International Conference on Computer Vision (ICCV).

[105]  Weixiong Zhang,et al.  Network-Specific Variational Auto-Encoder for Embedding in Attribute Networks , 2019, IJCAI.

[106]  Yun Fu,et al.  Adversarial Graph Embedding for Ensemble Clustering , 2019, IJCAI.

[107]  Jian Pei,et al.  ProGAN: Network Embedding via Proximity Generative Adversarial Network , 2019, KDD.

[108]  Cesare Alippi,et al.  Spectral Clustering with Graph Neural Networks for Graph Pooling , 2019, ICML.

[109]  Xiaotong Zhang,et al.  Attributed Graph Clustering via Adaptive Graph Convolution , 2019, IJCAI.

[110]  Jing Jiang,et al.  Attributed Graph Clustering: A Deep Attentional Embedding Approach , 2019, IJCAI.

[111]  Mohamed El Halaby,et al.  The Application of Unsupervised Clustering Methods to Alzheimer’s Disease , 2019, Front. Comput. Neurosci..

[112]  Fernando Berzal Galiano,et al.  Evaluation Metrics for Unsupervised Learning Algorithms , 2019, ArXiv.

[113]  Jaewoo Kang,et al.  Self-Attention Graph Pooling , 2019, ICML.

[114]  Shengjin Wang,et al.  Linkage Based Face Clustering via Graph Convolution Network , 2019, 2019 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR).

[115]  Mourad Khayati,et al.  Accuracy Evaluation of Overlapping and Multi-Resolution Clustering Algorithms on Large Datasets , 2019, 2019 IEEE International Conference on Big Data and Smart Computing (BigComp).

[116]  Xinbing Wang,et al.  CommunityGAN: Community Detection with Generative Adversarial Nets , 2019, WWW.

[117]  Chengqi Zhang,et al.  Learning Graph Embedding With Adversarial Training Methods , 2019, IEEE Transactions on Cybernetics.

[118]  Pietro Liò,et al.  Deep Graph Infomax , 2018, ICLR.

[119]  R. Devon Hjelm,et al.  Learning deep representations by mutual information estimation and maximization , 2018, ICLR.

[120]  Qiang Liu,et al.  A Survey of Clustering With Deep Learning: From the Perspective of Network Architecture , 2018, IEEE Access.

[121]  Jure Leskovec,et al.  Hierarchical Graph Representation Learning with Differentiable Pooling , 2018, NeurIPS.

[122]  Charles A. Sutton,et al.  GEMSEC: Graph Embedding with Self Clustering , 2018, 2019 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining (ASONAM).

[123]  Daniel Cremers,et al.  Clustering with Deep Learning: Taxonomy and New Methods , 2018, ArXiv.

[124]  Liyuan Liu,et al.  Graph Clustering with Dynamic Embedding , 2017, ArXiv.

[125]  Marc Peter Deisenroth,et al.  Deep Reinforcement Learning: A Brief Survey , 2017, IEEE Signal Processing Magazine.

[126]  Chun Wang,et al.  MGAE: Marginalized Graph Autoencoder for Graph Clustering , 2017, CIKM.

[127]  Kevin Chen-Chuan Chang,et al.  Learning Community Embedding with Community Detection and Node Embedding on Graphs , 2017, CIKM.

[128]  Pietro Liò,et al.  Graph Attention Networks , 2017, ICLR.

[129]  Jianping Yin,et al.  Improved Deep Embedded Clustering with Local Structure Preservation , 2017, IJCAI.

[130]  Jure Leskovec,et al.  Inductive Representation Learning on Large Graphs , 2017, NIPS.

[131]  Terrance E. Boult,et al.  Towards Robust Deep Neural Networks with BANG , 2016, 2018 IEEE Winter Conference on Applications of Computer Vision (WACV).

[132]  Maosong Sun,et al.  A Unified Framework for Community Detection and Network Representation Learning , 2016, IEEE Transactions on Knowledge and Data Engineering.

[133]  Max Welling,et al.  Variational Graph Auto-Encoders , 2016, ArXiv.

[134]  Bo Yang,et al.  Towards K-means-friendly Spaces: Simultaneous Deep Learning and Clustering , 2016, ICML.

[135]  Max Welling,et al.  Semi-Supervised Classification with Graph Convolutional Networks , 2016, ICLR.

[136]  Jure Leskovec,et al.  node2vec: Scalable Feature Learning for Networks , 2016, KDD.

[137]  Yang Song,et al.  Improving the Robustness of Deep Neural Networks via Stability Training , 2016, 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR).

[138]  Wei Lu,et al.  Deep Neural Networks for Learning Graph Representations , 2016, AAAI.

[139]  Sean Hughes,et al.  Clustering by Fast Search and Find of Density Peaks , 2016 .

[140]  François Fleuret,et al.  Nested Mini-Batch K-Means , 2016, NIPS.

[141]  Ali Farhadi,et al.  Unsupervised Deep Embedding for Clustering Analysis , 2015, ICML.

[142]  Charu C. Aggarwal,et al.  Heterogeneous Network Embedding via Deep Architectures , 2015, KDD.

[143]  Jimmy Ba,et al.  Adam: A Method for Stochastic Optimization , 2014, ICLR.

[144]  Yoshua Bengio,et al.  Empirical Evaluation of Gated Recurrent Neural Networks on Sequence Modeling , 2014, ArXiv.

[145]  Alexander J. Smola,et al.  Efficient mini-batch training for stochastic optimization , 2014, KDD.

[146]  Enhong Chen,et al.  Learning Deep Representations for Graph Clustering , 2014, AAAI.

[147]  Derek Greene,et al.  Normalized Mutual Information to evaluate overlapping community finding algorithms , 2011, ArXiv.

[148]  Pierre Baldi,et al.  Autoencoders, Unsupervised Learning, and Deep Architectures , 2011, ICML Unsupervised and Transfer Learning.

[149]  Cosma Rohilla Shalizi,et al.  Homophily and Contagion Are Generically Confounded in Observational Social Network Studies , 2010, Sociological methods & research.

[150]  K. Thangavel,et al.  Clustering Categorical Data Using Silhouette Coefficient as a Relocating Measure , 2007, International Conference on Computational Intelligence and Multimedia Applications (ICCIMA 2007).

[151]  Ulrike von Luxburg,et al.  A tutorial on spectral clustering , 2007, Stat. Comput..

[152]  R. Nickerson.  Confirmation Bias: A Ubiquitous Phenomenon in Many Guises , 1998 .

[153]  Kenli Li,et al.  Multi-View Bipartite Graph Clustering With Coupled Noisy Feature Filter , 2023, IEEE Transactions on Knowledge and Data Engineering.

[154]  Shiping Wang,et al.  An Overview of Advanced Deep Graph Node Clustering , 2024, IEEE Transactions on Computational Social Systems.

[155]  Qianqian Wang,et al.  Self-consistent Contrastive Attributed Graph Clustering with Pseudo-label Prompt , 2022, IEEE Transactions on Multimedia.

[156]  Yue Liu,et al.  Relational Symmetry based Knowledge Graph Contrastive Learning , 2022, ArXiv.

[157]  Yanqiao Zhu,et al.  A Systematic Survey of Molecular Pre-trained Models , 2022, arXiv.org.

[158]  Prateek Jain,et al.  S3GC: Scalable Self-Supervised Graph Clustering , 2022, NeurIPS.

[159]  Michal Valko,et al.  Bootstrapped Representation Learning on Graphs , 2021, ArXiv.

[160]  Sheng Wu,et al.  Graph Convolution-Based Deep Clustering for Speech Separation , 2020, IEEE Access.

[161]  Jinsung Yoon,et al.  GENERATIVE ADVERSARIAL NETS , 2018 .

[162]  Guigang Zhang,et al.  Deep Learning , 2016, Int. J. Semantic Comput..

[163]  Jane Zundel.  MATCHING THEORY , 2011 .

[164]  Léon Bottou,et al.  Large-Scale Machine Learning with Stochastic Gradient Descent , 2010, COMPSTAT.

[165]  Douglas A. Reynolds,et al.  Gaussian Mixture Models , 2018, Encyclopedia of Biometrics.

[166]  Slobodan Petrovic,et al.  A Comparison Between the Silhouette Index and the Davies-Bouldin Index in Labelling IDS Clusters , 2006 .

[167]  J. Delvenne,et al.  Random walks on graphs , 2004 .

[168]  Ka Yee Yeung,et al.  Details of the Adjusted Rand index and Clustering algorithms Supplement to the paper “ An empirical study on Principal Component Analysis for clustering gene expression data ” ( to appear in Bioinformatics ) , 2001 .