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Algorithms for graph partitioning on the planted partition model
Anne Condon
Corresponding Author
E-mail address: condon@cs.wisc.edu
Computer Sciences Department, University of Wisconsin, 1210 West Dayton St., Madison, WI 53706
The Department of Computer Science, University of British Colombia, 201‐2366 Main Mall, Vancouver, V6T 1Z4.Search for more papers by this authorRichard M. Karp
E-mail address: karp@cs.washington.edu
Department of Computer Science and Engineering, University of Washington, Seattle, WA 98195
Search for more papers by this authorAnne Condon
Corresponding Author
E-mail address: condon@cs.wisc.edu
Computer Sciences Department, University of Wisconsin, 1210 West Dayton St., Madison, WI 53706
The Department of Computer Science, University of British Colombia, 201‐2366 Main Mall, Vancouver, V6T 1Z4.Search for more papers by this authorRichard M. Karp
E-mail address: karp@cs.washington.edu
Department of Computer Science and Engineering, University of Washington, Seattle, WA 98195
Search for more papers by this authorFirst published: 26 January 2001
Citations: 167
Abstract
The NP‐hard graph bisection problem is to partition the nodes of an undirected graph into two equal‐sized groups so as to minimize the number of edges that cross the partition. The more general graph l ‐partition problem is to partition the nodes of an undirected graph into l equal‐sized groups so as to minimize the total number of edges that cross between groups. We present a simple, linear‐time algorithm for the graph l ‐partition problem and we analyze it on a random “planted l ‐partition” model. In this model, the n nodes of a graph are partitioned into l groups, each of size n /l ; two nodes in the same group are connected by an edge with some probability p , and two nodes in different groups are connected by an edge with some probability r <p . We show that if p −r ≥n −1/2+ϵ for some constant ϵ, then the algorithm finds the optimal partition with probability 1− exp(−n Θ(ε)). © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 18: 116–140, 2001
Citing Literature
Number of times cited according to CrossRef: 167
- Danica Vukadinović Greetham, Nathaniel Charlton, Anush Poghosyan, Total Positive Influence Domination on Weighted Networks, Complex Networks and Their Applications VIII, 10.1007/978-3-030-36687-2_27, (325-336), (2020).
- Giulio Rossetti, Exorcising the Demon: Angel, Efficient Node-Centric Community Discovery, Complex Networks and Their Applications VIII, 10.1007/978-3-030-36687-2_13, (152-163), (2020).
- Mohamed Bouguessa, A Model-Based Approach for Mining Anomalous Nodes in Networks, Putting Social Media and Networking Data in Practice for Education, Planning, Prediction and Recommendation, 10.1007/978-3-030-33698-1_12, (213-237), (2020).
- Mostafa Rahmani, Andre Beckus, Adel Karimian, George Atia, Scalable and Robust Community Detection with Randomized Sketching, IEEE Transactions on Signal Processing, 10.1109/TSP.2020.2965818, (1-1), (2020).
- Hadrien Van Lierde, Tommy W. S. Chow, Guanrong Chen, Scalable Spectral Clustering for Overlapping Community Detection in Large-Scale Networks, IEEE Transactions on Knowledge and Data Engineering, 10.1109/TKDE.2019.2892096, 32, 4, (754-767), (2020).
- Jesse Laeuchli, Fast Community Detection with Graph Sparsification, Advances in Knowledge Discovery and Data Mining, 10.1007/978-3-030-47426-3_23, (291-304), (2020).
- Giulio Rossetti, ANGEL: efficient, and effective, node-centric community discovery in static and dynamic networks, Applied Network Science, 10.1007/s41109-020-00270-6, 5, 1, (2020).
- Vivek Bagaria, Jian Ding, David Tse, Yihong Wu, Jiaming Xu, Hidden Hamiltonian Cycle Recovery via Linear Programming, Operations Research, 10.1287/opre.2019.1886, (2020).
- Zengyou He, Hao Liang, Zheng Chen, Can Zhao, Yan Liu, Computing exact P-values for community detection, Data Mining and Knowledge Discovery, 10.1007/s10618-020-00681-0, (2020).
- Mohammad Esmaeili, Hussein Saad, Aria Nosratinia, undefined, ICASSP 2019 - 2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 10.1109/ICASSP.2019.8682223, (3477-3481), (2019).
- Amani Chouchane, Oualid Boutemine, Mohamed Bouguessa, On Detecting Multidimensional Communities, The Human Factor in a Mission to Mars, 10.1007/978-3-030-11286-8_3, (45-78), (2019).
- Qian Chen, Hongyi Su, Jiamou Liu, Bo Yan, Hong Zheng, He Zhao, In Pursuit of Social Capital: Upgrading Social Circle Through Edge Rewiring, Web and Big Data, 10.1007/978-3-030-26072-9_15, (207-222), (2019).
- Haijiao Liu, Huifang Ma, Yang Chang, Zhixin Li, Wenjuan Wu, Target Community Detection With User’s Preference and Attribute Subspace, IEEE Access, 10.1109/ACCESS.2019.2909736, 7, (46583-46594), (2019).
- Nandinee Fariah Haq, Mehdi Moradi, Z. Jane Wang, Community Structure Detection from Networks with Weighted Modularity, Pattern Recognition Letters, 10.1016/j.patrec.2019.02.005, (2019).
- Antonela Tommasel, Daniela Godoy, On the Evaluation of Community Detection Algorithms on Heterogeneous Social Media Data, Linking and Mining Heterogeneous and Multi-view Data, 10.1007/978-3-030-01872-6_12, (295-333), (2019).
- Madhurima Nath, Yihui Ren, Stephen Eubank, An Approach to Structural Analysis Using Moore-Shannon Network Reliability, Complex Networks and Their Applications VII, 10.1007/978-3-030-05411-3_44, (537-549), (2019).
- Tiago P. Peixoto, Bayesian Stochastic Blockmodeling, Advances in Network Clustering and Blockmodeling, 10.1002/9781119483298, (289-332), (2019).
- Eliav Buchnik, Edith Cohen, Bootstrapped Graph Diffusions, ACM SIGMETRICS Performance Evaluation Review, 10.1145/3308809.3308815, 46, 1, (8-10), (2019).
- A. Roxana Pamfil, Sam D. Howison, Renaud Lambiotte, Mason A. Porter, Relating Modularity Maximization and Stochastic Block Models in Multilayer Networks, SIAM Journal on Mathematics of Data Science, 10.1137/18M1231304, 1, 4, (667-698), (2019).
- I Eli Chien, Chung-Yi Lin, I-Hsiang Wang, On the Minimax Misclassification Ratio of Hypergraph Community Detection, IEEE Transactions on Information Theory, 10.1109/TIT.2019.2928301, 65, 12, (8095-8118), (2019).
- Reza Fathi, Anisur Rahaman Molla, Gopal Pandurangan, undefined, 2019 IEEE 39th International Conference on Distributed Computing Systems (ICDCS), 10.1109/ICDCS.2019.00048, (409-419), (2019).
- Andre Beckus, George K. Atia, undefined, 2019 57th Annual Allerton Conference on Communication, Control, and Computing (Allerton), 10.1109/ALLERTON.2019.8919954, (298-302), (2019).
- Hocine Cherifi, Gergely Palla, Boleslaw K. Szymanski, Xiaoyan Lu, On community structure in complex networks: challenges and opportunities, Applied Network Science, 10.1007/s41109-019-0238-9, 4, 1, (2019).
- John Matta, Gunes Ercal, Koushik Sinha, Comparing the speed and accuracy of approaches to betweenness centrality approximation, Computational Social Networks, 10.1186/s40649-019-0062-5, 6, 1, (2019).
- Bo Yan, Fanku Meng, Jiamou Liu, Yiping Liu, Hongyi Su, undefined, 2019 15th International Conference on Mobile Ad-Hoc and Sensor Networks (MSN), 10.1109/MSN48538.2019.00048, (206-211), (2019).
- Éder Mílton Schneider, Sebastián Gonçalves, José Roberto Iglesias, Bruno Requião da Cunha, Dynamic Modular Networks Model Mediated by Confinement, Applied Network Science, 10.1007/s41109-019-0143-2, 4, 1, (2019).
- Nasrin Mazaheri Soudani, Afsaneh Fatemi, Mohammadali Nematbakhsh, PPR-partitioning: a distributed graph partitioning algorithm based on the personalized PageRank vectors in vertex-centric systems, Knowledge and Information Systems, 10.1007/s10115-019-01328-3, (2019).
- Nasrin Mazaheri Soudani, Afsaneh Fatemi, Mohammadali Nematbakhsh, An investigation of big graph partitioning methods for distribution of graphs in vertex-centric systems, Distributed and Parallel Databases, 10.1007/s10619-019-07256-z, (2019).
- Zachary M. Boyd, Mason A. Porter, Andrea L. Bertozzi, Stochastic Block Models are a Discrete Surface Tension, Journal of Nonlinear Science, 10.1007/s00332-019-09541-8, (2019).
- Eliav Buchnik, Edith Cohen, Bootstrapped Graph Diffusions, ACM SIGMETRICS Performance Evaluation Review, 10.1145/3292040.3219621, 46, 1, (8-10), (2018).
- Haiving Wang, Huiru Zheng, undefined, 2018 14th International Conference on Natural Computation, Fuzzy Systems and Knowledge Discovery (ICNC-FSKD), 10.1109/FSKD.2018.8687269, (758-763), (2018).
- Kristen M. Altenburger, Johan Ugander, Monophily in social networks introduces similarity among friends-of-friends, Nature Human Behaviour, 10.1038/s41562-018-0321-8, 2, 4, (284-290), (2018).
- Thomas H. McCoy, Mapping the delirium literature through probabilistic topic modeling and network analysis: a computational scoping review, Psychosomatics, 10.1016/j.psym.2018.12.003, (2018).
- Israel Rocha, Jeannette Janssen, Nauzer Kalyaniwalla, Recovering the structure of random linear graphs, Linear Algebra and its Applications, 10.1016/j.laa.2018.07.029, 557, (234-264), (2018).
- Viviana Amati, Alessandro Lomi, Antonietta Mira, Social Network Modeling, Annual Review of Statistics and Its Application, 10.1146/annurev-statistics-031017-100746, 5, 1, (343-369), (2018).
- undefined, Abstracts of the 2018 ACM International Conference on Measurement and Modeling of Computer Systems - SIGMETRICS '18, 10.1145/3219617.3219621, (8-10), (2018).
- Giulio Rossetti, Rémy Cazabet, Community Discovery in Dynamic Networks, ACM Computing Surveys, 10.1145/3172867, 51, 2, (1-37), (2018).
- Avik Ray, Sujay Sanghavi, Sanjay Shakkottai, Searching for a Single Community in a Graph, ACM Transactions on Modeling and Performance Evaluation of Computing Systems, 10.1145/3200863, 3, 3, (1-17), (2018).
- Colin Cooper, Ngoc Vu, An Experimental Study of the k-MXT Algorithm with Applications to Clustering Geo-Tagged Data, Algorithms and Models for the Web Graph, 10.1007/978-3-319-92871-5_10, (145-169), (2018).
- Yudong Chen, Yuejie Chi, Harnessing Structures in Big Data via Guaranteed Low-Rank Matrix Estimation: Recent Theory and Fast Algorithms via Convex and Nonconvex Optimization, IEEE Signal Processing Magazine, 10.1109/MSP.2018.2821706, 35, 4, (14-31), (2018).
- Jun Jin Choong, Xin Liu, Tsuyoshi Murata, Variational Approach for Learning Community Structures, Complexity, 10.1155/2018/4867304, 2018, (1-13), (2018).
- Buchnik Eliav, Edith Cohen, Bootstrapped Graph Diffusions, Proceedings of the ACM on Measurement and Analysis of Computing Systems, 10.1145/3179413, 2, 1, (1-19), (2018).
- Zhenhai Chang, Xianjun Yin, Caiyan Jia, Xiaoyang Wang, Mixture models with entropy regularization for community detection in networks, Physica A: Statistical Mechanics and its Applications, 10.1016/j.physa.2018.01.002, 496, (339-350), (2018).
- Han Zhang, Chang-Dong Wang, Jian-Huang Lai, Philip S. Yu, Community detection using multilayer edge mixture model, Knowledge and Information Systems, 10.1007/s10115-018-1228-4, (2018).
- VAN VU, A Simple SVD Algorithm for Finding Hidden Partitions, Combinatorics, Probability and Computing, 10.1017/S0963548317000463, 27, 1, (124-140), (2017).
- ELVIS H. W. XU, PAK MING HUI, Efficient detection of communities with significant overlaps in networks: Partial community merger algorithm, Network Science, 10.1017/nws.2017.32, 6, 1, (71-96), (2017).
- Divya Pandove, Shivani Goel, Rinkle Rani, Correlation clustering methodologies and their fundamental results, Expert Systems, 10.1111/exsy.12229, 35, 1, (2017).
- Seung-Hee Bae, Daniel Halperin, Jevin D. West, Martin Rosvall, Bill Howe, Scalable and Efficient Flow-Based Community Detection for Large-Scale Graph Analysis, ACM Transactions on Knowledge Discovery from Data, 10.1145/2992785, 11, 3, (1-30), (2017).
- Naman Agarwal, Afonso S. Bandeira, Konstantinos Koiliaris, Alexandra Kolla, Multisection in the Stochastic Block Model Using Semidefinite Programming, Compressed Sensing and its Applications, 10.1007/978-3-319-69802-1_4, (125-162), (2017).
- Amit Saxena, Mukesh Prasad, Akshansh Gupta, Neha Bharill, Om Prakash Patel, Aruna Tiwari, Meng Joo Er, Weiping Ding, Chin-Teng Lin, A review of clustering techniques and developments, Neurocomputing, 10.1016/j.neucom.2017.06.053, 267, (664-681), (2017).
- Zheng Ran, Hua Yan, Huimin Zhang, Yun Li, Approximate optimal AUTOSAR software components deploying approach for automotive E/E system, International Journal of Automotive Technology, 10.1007/s12239-017-0108-3, 18, 6, (1109-1119), (2017).
- Bo Yang, He He, Xiaoming Hu, Detecting community structure in networks via consensus dynamics and spatial transformation, Physica A: Statistical Mechanics and its Applications, 10.1016/j.physa.2017.04.098, 483, (156-170), (2017).
- undefined, Proceedings of the 26th International Conference on World Wide Web Companion - WWW '17 Companion, 10.1145/3041021.3055139, (983-992), (2017).
- Jia-Rong Xie, Bing-Hong Wang, Modularity-like objective function in annotated networks, Frontiers of Physics, 10.1007/s11467-017-0657-y, 12, 6, (2017).
- Oualid Boutemine, Mohamed Bouguessa, undefined, Proceedings of the 2017 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining 2017 - ASONAM '17, 10.1145/3110025.3110052, (291-296), (2017).
- undefined, Proceedings of the 23rd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining - KDD '17, 10.1145/3097983.3098156, (737-746), (2017).
- Guangquan Cheng, Yang Ma, Jincai Huang, Kuihua Huang, undefined, 2017 IEEE 29th International Conference on Tools with Artificial Intelligence (ICTAI), 10.1109/ICTAI.2017.00095, (591-595), (2017).
- Lian Duan, Yanchi Liu, W. Nick Street, Haibing Lu, Utilizing advances in correlation analysis for community structure detection, Expert Systems with Applications, 10.1016/j.eswa.2017.05.010, 84, (74-91), (2017).
- Oualid Boutemine, Mohamed Bouguessa, Mining Community Structures in Multidimensional Networks, ACM Transactions on Knowledge Discovery from Data, 10.1145/3080574, 11, 4, (1-36), (2017).
- Elchanan Mossel, Joe Neeman, Allan Sly, A proof of the block model threshold conjecture, Combinatorica, 10.1007/s00493-016-3238-8, (2017).
- Filippo Palombi, Simona Toti, Topological Aspects of the Multi-Language Phases of the Naming Game on Community-Based Networks, Games, 10.3390/g8010012, 8, 1, (12), (2017).
- Jiaojiao Jiang, Sheng Wen, Shui Yu, Yang Xiang, Wanlei Zhou, Houcine Hassan, The structure of communities in scale‐free networks, Concurrency and Computation: Practice and Experience, 10.1002/cpe.4040, 29, 14, (2016).
- Isabel M. Kloumann, Johan Ugander, Jon Kleinberg, Block models and personalized PageRank, Proceedings of the National Academy of Sciences, 10.1073/pnas.1611275114, 114, 1, (33-38), (2016).
- Jihui Han, Wei Li, Zhu Su, Longfeng Zhao, Weibing Deng, Community detection by label propagation with compression of flow, The European Physical Journal B, 10.1140/epjb/e2016-70264-6, 89, 12, (2016).
- Zhao Yang, René Algesheimer, Claudio J. Tessone, A Comparative Analysis of Community Detection Algorithms on Artificial Networks, Scientific Reports, 10.1038/srep30750, 6, 1, (2016).
- Zhao Yang, Renn Algesheimer, A Comparative Analysis of Community Detection Algorithms on Artificial Networks, SSRN Electronic Journal, 10.2139/ssrn.2937843, (2016).
- Wei-Feng Guo, Shao-Wu Zhang, A general method of community detection by identifying community centers with affinity propagation, Physica A: Statistical Mechanics and its Applications, 10.1016/j.physa.2015.12.037, 447, (508-519), (2016).
- Avik Ray, Sujay Sanghavi, Sanjay Shakkottai, Searching For A Single Community in a Graph, ACM SIGMETRICS Performance Evaluation Review, 10.1145/2964791.2901494, 44, 1, (399-400), (2016).
- undefined, Proceedings of the 2016 ACM SIGMETRICS International Conference on Measurement and Modeling of Computer Science - SIGMETRICS '16, 10.1145/2896377.2901494, (399-400), (2016).
- Liangxun Shuo, Bianfang Chai, Discussion of the community detection algorithm based on statistical inference, Perspectives in Science, 10.1016/j.pisc.2015.11.020, 7, (122-125), (2016).
- Cristina G. Fernandes, Marcos Kiwi, Repetition-free longest common subsequence of random sequences, Discrete Applied Mathematics, 10.1016/j.dam.2015.07.005, 210, (75-87), (2016).
- Santo Fortunato, Darko Hric, Community detection in networks: A user guide, Physics Reports, 10.1016/j.physrep.2016.09.002, 659, (1-44), (2016).
- Tanja Hartmann, Andrea Kappes, Dorothea Wagner, Clustering Evolving Networks, Algorithm Engineering, 10.1007/978-3-319-49487-6_9, (280-329), (2016).
- Subu Surendran, D. Chithraprasad, M. Ramachandra Kaimal, A scalable geometric algorithm for community detection from social networks with incremental update, Social Network Analysis and Mining, 10.1007/s13278-016-0399-9, 6, 1, (2016).
- undefined, Proceedings of the 2016 ACM Conference on Innovations in Theoretical Computer Science - ITCS '16, 10.1145/2840728.2840749, (71-80), (2016).
- R. Franke, CHIMERA: Top-down model for hierarchical, overlapping and directed cluster structures in directed and weighted complex networks, Physica A: Statistical Mechanics and its Applications, 10.1016/j.physa.2016.05.063, 461, (384-408), (2016).
- Vladimir Estivill-Castro, Mahdi Parsa, undefined Unknown, undefined, Proceedings of the Australasian Computer Science Week Multiconference on - ACSW '16, 10.1145/2843043.2843053, (1-6), (2016).
- Atsushi Miyauchi, Yasushi Kawase, Z-Score-Based Modularity for Community Detection in Networks, PLOS ONE, 10.1371/journal.pone.0147805, 11, 1, (e0147805), (2016).
- Chi‐Hyon Lee, Manuela N. Hoehn‐Weiss, Samina Karim, Grouping interdependent tasks: Using spectral graph partitioning to study complex systems, Strategic Management Journal, 10.1002/smj.2455, 37, 1, (177-191), (2015).
- Abhinav Nellore, Rachel Ward, Recovery guarantees for exemplar-based clustering, Information and Computation, 10.1016/j.ic.2015.09.002, 245, (165-180), (2015).
- Matteo Pontecorvi, Vijaya Ramachandran, Fully Dynamic Betweenness Centrality, Algorithms and Computation, 10.1007/978-3-662-48971-0_29, (331-342), (2015).
- Andrea Clementi, Miriam Di Ianni, Giorgio Gambosi, Emanuele Natale, Riccardo Silvestri, Distributed community detection in dynamic graphs, Theoretical Computer Science, 10.1016/j.tcs.2014.11.026, 584, (19-41), (2015).
- Yashen Wang, Heyan Huang, Chong Feng, Zhirun Liu, Community Detection Based on Minimum-Cut Graph Partitioning, Web-Age Information Management, 10.1007/978-3-319-21042-1_5, (57-69), (2015).
- Madalina Olteanu, Nathalie Villa-Vialaneix, On-line relational and multiple relational SOM, Neurocomputing, 10.1016/j.neucom.2013.11.047, 147, (15-30), (2015).
- undefined, Proceedings of the 2015 ACM on Conference on Online Social Networks - COSN '15, 10.1145/2817946.2817950, (27-35), (2015).
- M. Sh. Levin, Combinatorial clustering: Literature review, methods, examples, Journal of Communications Technology and Electronics, 10.1134/S1064226915120177, 60, 12, (1403-1428), (2015).
- undefined, Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing - STOC '15, 10.1145/2746539.2746603, (69-75), (2015).
- Angsheng Li, Jiankou Li, Yicheng Pan, Discovering natural communities in networks, Physica A: Statistical Mechanics and its Applications, 10.1016/j.physa.2015.05.039, 436, (878-896), (2015).
- Faraz Zaidi, Muhammad Qasim Pasta, Arnaud Sallaberry, Guy Melançon, Social ties, homophily and extraversion--introversion to generate complex networks, Social Network Analysis and Mining, 10.1007/s13278-015-0270-4, 5, 1, (2015).
- undefined, Proceedings of the 24th ACM International on Conference on Information and Knowledge Management - CIKM '15, 10.1145/2806416.2806555, (1471-1480), (2015).
- undefined, Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis on - SC '15, 10.1145/2807591.2807668, (1-12), (2015).
- Elchanan Mossel, Joe Neeman, Allan Sly, Reconstruction and estimation in the planted partition model, Probability Theory and Related Fields, 10.1007/s00440-014-0576-6, 162, 3-4, (431-461), (2014).
- Tiago P. Peixoto, Efficient Monte Carlo and greedy heuristic for the inference of stochastic block models, Physical Review E, 10.1103/PhysRevE.89.012804, 89, 1, (2014).
- Jiamou Liu, Ziheng Wei, From a Local to a Global Perspective of Community Detection in Networks, PRICAI 2014: Trends in Artificial Intelligence, 10.1007/978-3-319-13560-1_90, (1036-1049), (2014).
- Leman Akoglu, Rohit Khandekar, Vibhore Kumar, Srinivasan Parthasarathy, Deepak Rajan, Kun-Lung Wu, Fast Nearest Neighbor Search on Large Time-Evolving Graphs, Machine Learning and Knowledge Discovery in Databases, 10.1007/978-3-662-44848-9_2, (17-33), (2014).
- Renana Peres, Christophe Van den Bulte, When to Take or Forgo New Product Exclusivity: Balancing Protection from Competition against Word-of-Mouth Spillover, Journal of Marketing, 10.1509/jm.12.0344, 78, 2, (83-100), (2014).
- Mariá C.V. Nascimento, Community detection in networks via a spectral heuristic based on the clustering coefficient, Discrete Applied Mathematics, 10.1016/j.dam.2013.09.017, 176, (89-99), (2014).
- V. K. Sihag, Abhineet Anand, Ravi Tomar, Jagdish Chandra, Rajeev Tiwari, Ankur Dumka, A. S. Poonia, undefined, 2014 IEEE Students' Conference on Electrical, Electronics and Computer Science, 10.1109/SCEECS.2014.6804502, (1-9), (2014).
- Tiago P. Peixoto, Hierarchical Block Structures and High-Resolution Model Selection in Large Networks, Physical Review X, 10.1103/PhysRevX.4.011047, 4, 1, (2014).
- Elizabeth Santiago, Jorge X. Velasco-Hernández, Manuel Romero-Salcedo, A methodology for the characterization of flow conductivity through the identification of communities in samples of fractured rocks, Expert Systems with Applications, 10.1016/j.eswa.2013.08.011, 41, 3, (811-820), (2014).
Metrics
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Details
Copyright © 2001 John Wiley & Sons, Inc.
Funding Information
- NSF. Grant Numbers: HRD‐627241, CCR‐9257241, DBI‐9601046
Publication History
- 26 January 2001
- 26 January 2001
- 17 October 2000
- 28 April 2000
- 16 March 1999
