Fanchen Bu
ABSTRACT
Graphs are a powerful mathematical model, and they are used to represent real-world structures in various fields. In many applications, real-world structures with high connectivity and robustness are preferable. For enhancing the connectivity and robustness of graphs, two operations, adding edges and anchoring nodes, have been extensively studied. However, merging nodes, which is a realistic operation in many scenarios (e.g., bus station reorganization, multiple team formation), has been overlooked. In this work, we study the problem of improving graph cohesiveness by merging nodes. First, we formulate the problem mathematically using the size of the k-truss, for a given k, as the objective. Then, we prove the NP-hardness and non-modularity of the problem. After that, we develop BATMAN, a fast and effective algorithm for choosing sets of nodes to be merged, based on our theoretical findings and empirical observations. Lastly, we demonstrate the superiority of BATMAN over several baselines, in terms of speed and effectiveness, through extensive experiments on fourteen real-world graphs.
Supplemental Material
- 2023. Online supplementary material. https://github.com/bokveizen/cohesive-truss-mergeGoogle Scholar
- Ramesh Bobby Addanki et al. 2020. Multi-team Formation using Community Based Approach in Real-World Networks. arXiv preprint arXiv:2008.11191 (2020).Google Scholar
- Esra Akbas and Peixiang Zhao. 2017. Truss-based community search: a truss-equivalence based indexing approach. Proceedings of the VLDB Endowment 10, 11 (2017), 1298--1309.Google ScholarDigital Library
- Vivek Singh Baghel and S Durga Bhavani. 2018. Multiple team formation using an evolutionary approach. In IC3.Google Scholar
- Richard Barrett, Michael Berry, Tony F Chan, James Demmel, June Donato, Jack Dongarra, Victor Eijkhout, Roldan Pozo, Charles Romine, and Henk Van der Vorst. 1994. Templates for the solution of linear systems: building blocks for iterative methods. SIAM.Google Scholar
- Alina Beygelzimer, Geoffrey Grinstein, Ralph Linsker, and Irina Rish. 2005. Im-proving network robustness by edge modification. Physica A: Statistical Mechanics and its Applications 357, 3-4 (2005), 593--612.Google Scholar
- Kshipra Bhawalkar, Jon Kleinberg, Kevin Lewi, Tim Roughgarden, and Aneesh Sharma. 2015. Preventing unraveling in social networks: the anchored k-core problem. SIAM Journal on Discrete Mathematics 29, 3 (2015), 1452--1475.Google ScholarDigital Library
- Edward C Brewer and Terence L Holmes. 2016. Better communication= better teams: A communication exercise to improve team performance. IEEE Transactions on Professional Communication 59, 3 (2016), 288--298.Google ScholarCross Ref
- Sylvain Brohee and Jacques Van Helden. 2006. Evaluation of clustering algorithms for protein-protein interaction networks. BMC bioinformatics 7, 1 (2006), 1--19.Google Scholar
- Guo-Ray Cai and Yu-Geng Sun. 1989. The minimum augmentation of any graph to a K-edge-connected graph. Networks 19, 1 (1989), 151--172.Google ScholarCross Ref
- Hau Chan, Leman Akoglu, and Hanghang Tong. 2014. Make it or break it: Manipulating robustness in large networks. In SDM.Google Scholar
- Chen Chen, Mengqi Zhang, Renjie Sun, Xiaoyang Wang, Weijie Zhu, and Xun Wang. 2022. Locating pivotal connections: The K-Truss minimization and maximization problems. World Wide Web 25, 2 (2022), 899--926.Google ScholarDigital Library
- Huiping Chen, Alessio Conte, Roberto Grossi, Grigorios Loukides, Solon P Pissis, and Michelle Sweering. 2021. On breaking truss-based communities. In KDD.Google Scholar
- Zi Chen, Long Yuan, Li Han, and Zhengping Qian. 2021. Higher-Order Truss Decomposition in Graphs. IEEE Transactions on Knowledge and Data Engineering (2021).Google Scholar
- Meenal Chhabra, Sanmay Das, and Boleslaw Szymanski. 2013. Team formation in social networks. Computer and information sciences III (2013), 291--299.Google Scholar
- Eunjoon Cho, Seth A Myers, and Jure Leskovec. 2011. Friendship and mobility: user movement in location-based social networks. In KDD.Google Scholar
- Jonathan Cohen. 2008. Trusses: Cohesive subgraphs for social network analysis. National security agency technical report (2008).Google Scholar
- CTtransit. 2010. https://www.cttransit.com/sites/default/files/PDF_files/Bus% 20Stop%20Conslidation_NH.pdfGoogle Scholar
- Sybil Derrible and Christopher Kennedy. 2010. The complexity and robustness of metro networks. Physica A: Statistical Mechanics and its Applications 389, 17 (2010), 3678--3691.Google Scholar
- Issa Moussa Diop, Chantal Cherifi, Cherif Diallo, and Hocine Cherifi. 2020. On local and global components of the air transportation network. In Conference on Complex Systems (CCS).Google Scholar
- Nurcan Durak, Ali Pinar, Tamara G Kolda, and C Seshadhri. 2012. Degree relations of triangles in real-world networks and graph models. In CIKM.Google Scholar
- Wendy Ellens and Robert E Kooij. 2013. Graph measures and network robustness. arXiv preprint arXiv:1311.5064 (2013).Google Scholar
- Wendy Ellens, Floske M Spieksma, Piet Van Mieghem, Almerima Jamakovic, and Robert E Kooij. 2011. Effective graph resistance. Linear algebra and its applications 435, 10 (2011), 2491--2506.Google Scholar
- Paul Erd's, Alfréd Rényi, et al. 1960. On the evolution of random graphs. Pub-lication of the Mathematical Institute of the Hungarian Academy of Sciences 5, 1 (1960), 17--60.Google Scholar
- Scott Freitas, Diyi Yang, Srijan Kumar, Hanghang Tong, and Duen Horng Chau. 2021. Evaluating graph vulnerability and robustness using tiger. In CIKM.Google Scholar
- Gene. 2013. Efficient algorithm for finding all maximal subsets. https:// stackoverflow.com/questions/14106121Google Scholar
- Zakariya Ghalmane, Mohammed El Hassouni, Chantal Cherifi, and Hocine Cherifi. 2018. K-truss decomposition for modular centrality. In International Symposium on Signal, Image, Video and Communications (ISIVC).Google ScholarCross Ref
- Arpita Ghosh, Stephen Boyd, and Amin Saberi. 2008. Minimizing effective resistance of a graph. SIAM review 50, 1 (2008), 37--66.Google Scholar
- Jimmy H Gutiérrez, César A Astudillo, Pablo Ballesteros-Pérez, Daniel Mora-Melià, and Alfredo Candia-Véjar. 2016. The multiple team formation problem using sociometry. Computers & Operations Research 75 (2016), 150--162.Google ScholarDigital Library
- Xin Huang, Hong Cheng, Lu Qin, Wentao Tian, and Jeffrey Xu Yu. 2014. Querying k-truss community in large and dynamic graphs. In SIGMOD.Google Scholar
- Xin Huang, Wei Lu, and Laks VS Lakshmanan. 2016. Truss decomposition of probabilistic graphs: Semantics and algorithms. In SIGMOD.Google Scholar
- Jian Gang Jin, Loon Ching Tang, Lijun Sun, and Der-Horng Lee. 2014. Enhancing metro network resilience via localized integration with bus services. Transportation Research Part E: Logistics and Transportation Review 63 (2014), 17--30.Google ScholarCross Ref
- Camille Jordan. 1869. Sur les assemblages de lignes. Journal für die reine und angewandte Mathematik 71 (1869), 185--190.Google Scholar
- Russell Jurney. 2013. Efficient algorithm for finding all maximal subsets. https: //data.world/datasyndrome/relato-business-graph-databaseGoogle Scholar
- Steve WJ Kozlowski and Bradford S Bell. 2013. Work groups and teams in organizations. (2013).Google Scholar
- Ricky Laishram, Ahmet Erdem Sar, Tina Eliassi-Rad, Ali Pinar, and Sucheta Soundarajan. 2020. Residual core maximization: An efficient algorithm for maximizing the size of the k-core. In SDM.Google Scholar
- Naomi R Lamoreaux. 1988. The great merger movement in American business, 1895--1904. Cambridge University Press.Google Scholar
- Jure Leskovec, Daniel Huttenlocher, and Jon Kleinberg. 2010. Predicting positive and negative links in online social networks. In TheWebConf (fka. WWW).Google Scholar
- Jure Leskovec, Daniel Huttenlocher, and Jon Kleinberg. 2010. Signed networks in social media. In CHI.Google Scholar
- Jure Leskovec, Jon Kleinberg, and Christos Faloutsos. 2005. Graphs over time: densification laws, shrinking diameters and possible explanations. In KDD.Google Scholar
- Jure Leskovec, Jon Kleinberg, and Christos Faloutsos. 2007. Graph evolution: Densification and shrinking diameters. The ACM Transactions on Knowledge Discovery from Data 1, 1 (2007), 2--es.Google ScholarDigital Library
- Jure Leskovec, Kevin J Lang, Anirban Dasgupta, and Michael W Mahoney. 2009. Community structure in large networks: Natural cluster sizes and the absence of large well-defined clusters. Internet Mathematics 6, 1 (2009), 29--123.Google ScholarCross Ref
- Jure Leskovec and Julian Mcauley. 2012. Learning to discover social circles in ego networks. In NeurIPS (fka. NIPS).Google Scholar
- Qingyuan Linghu, Fan Zhang, Xuemin Lin, Wenjie Zhang, and Ying Zhang. 2020. Global reinforcement of social networks: The anchored coreness problem. In SIGMOD.Google Scholar
- Qingyuan Linghu, Fan Zhang, Xuemin Lin, Wenjie Zhang, and Ying Zhang. 2022. Anchored coreness: efficient reinforcement of social networks. The VLDB Journal 31, 2 (2022), 227--252.Google ScholarDigital Library
- Kaixin Liu, Sibo Wang, Yong Zhang, and Chunxiao Xing. 2021. An Efficient Algorithm for the Anchored k-Core Budget Minimization Problem. In ICDE.Google Scholar
- Qing Liu, Minjun Zhao, Xin Huang, Jianliang Xu, and Yunjun Gao. 2020. Truss-based community search over large directed graphs. In SIGMOD.Google Scholar
- R Duncan Luce. 1950. Connectivity and generalized cliques in sociometric group structure. Psychometrika 15, 2 (1950), 169--190.Google ScholarCross Ref
- Qi Luo, Dongxiao Yu, Xiuzhen Cheng, Zhipeng Cai, Jiguo Yu, and Weifeng Lv. 2020. Batch processing for truss maintenance in large dynamic graphs. IEEE Transactions on Computational Social Systems 7, 6 (2020), 1435--1446.Google ScholarCross Ref
- Fragkiskos D Malliaros, Vasileios Megalooikonomou, and Christos Faloutsos. 2012. Fast robustness estimation in large social graphs: Communities and anomaly detection. In SDM.Google Scholar
- David E Manolopoulos and Patrick W Fowler. 1992. Molecular graphs, point groups, and fullerenes. The Journal of chemical physics 96, 10 (1992), 7603--7614.Google ScholarCross Ref
- Sourav Medya, Tianyi Ma, Arlei Silva, and Ambuj Singh. 2020. A Game Theoretic Approach For Core Resilience. In IJCAI.Google Scholar
- Robert J Mokken et al. 1979. Cliques, clubs and clans. Quality & Quantity 13, 2 (1979), 161--173.Google ScholarCross Ref
- Richard L Moreland, Linda Argote, and Ranjani Krishnan. 2002. Training people to work in groups. In Theory and research on small groups. Springer, 37--60.Google Scholar
- James G Oxley. 2006. Matroid theory.Google Scholar
- Giulia Preti, Gianmarco De Francisci Morales, and Francesco Bonchi. 2021. STruD: Truss Decomposition of Simplicial Complexes. In TheWebConf (fka. WWW).Google Scholar
- Matthew Richardson, Rakesh Agrawal, and Pedro Domingos. 2003. Trust management for the semantic web. In ISWC.Google Scholar
- Ryan A. Rossi and Nesreen K. Ahmed. 2015. The Network Data Repository with Interactive Graph Analytics and Visualization. In AAAI. https: //networkrepository.comGoogle ScholarDigital Library
- John Scott. 1988. Social network analysis. Sociology 22, 1 (1988), 109--127.Google ScholarCross Ref
- Stephen B Seidman. 1983. Network structure and minimum degree. Social networks 5, 3 (1983), 269--287.Google Scholar
- Stephen B Seidman and Brian L Foster. 1978. A graph-theoretic generalization of the clique concept. Journal of Mathematical sociology 6, 1 (1978), 139--154.Google ScholarCross Ref
- Nitai B Silva, Ren Tsang, George DC Cavalcanti, and Jyh Tsang. 2010. A graph-based friend recommendation system using genetic algorithm. In CEC.Google Scholar
- Xin Sun, Xin Huang, and Di Jin. 2022. Fast algorithms for core maximization on large graphs. Proceedings of the VLDB Endowment 15, 7 (2022), 1350--1362.Google ScholarDigital Library
- Xin Sun, Xin Huang, Zitan Sun, and Di Jin. 2021. Budget-constrained Truss Maximization over Large Graphs: A Component-based Approach. In CIKM.Google Scholar
- Zitan Sun, Xin Huang, Jianliang Xu, and Francesco Bonchi. 2021. Efficient probabilistic truss indexing on uncertain graphs. In TheWebConf (fka. WWW).Google Scholar
- Jia Wang and James Cheng. 2012. Truss decomposition in massive networks. Proceedings of the VLDB Endowment 5, 9 (2012), 812--823.Google ScholarDigital Library
- Sheng Wei, Lei Wang, Xiongwu Fu, and Tao Jia. 2020. Using open big data to build and analyze urban bus network models within and across administrations Complexity (2020).Google Scholar
- Jaewon Yang and Jure Leskovec. 2015. Defining and evaluating network communities based on ground-truth. Knowledge and Information Systems 42, 1 (2015), 181--213.Google ScholarDigital Library
- Zhibang Yang, Xiaoxue Li, Xu Zhang, Wensheng Luo, and Kenli Li. 2022. K-truss community most favorites query based on top-t. World Wide Web 25, 2 (2022), 949--969.Google ScholarDigital Library
- Daniel M Yellin. 1992. Algorithms for subset testing and finding maximal sets. In SODA.Google Scholar
- Hao Yin, Austin R Benson, Jure Leskovec, and David F Gleich. 2017. Local higher-order graph clustering. In KDD.Google Scholar
- Fan Zhang, Conggai Li, Ying Zhang, Lu Qin, and Wenjie Zhang. 2018. Finding critical users in social communities: The collapsed core and truss problems. IEEE Transactions on Knowledge and Data Engineering 32, 1 (2018), 78--91.Google ScholarCross Ref
- Fan Zhang, Conggai Li, Ying Zhang, Lu Qin, and Wenjie Zhang. 2020. Finding Critical Users in Social Communities: The Collapsed Core and Truss Problems. IEEE Transactions on Knowledge and Data Engineering 32, 1 (2020), 78--91.Google ScholarCross Ref
- Fan Zhang, Wenjie Zhang, Ying Zhang, Lu Qin, and Xuemin Lin. 2017. OLAK: an efficient algorithm to prevent unraveling in social networks. Proceedings of the VLDB Endowment 10, 6 (2017), 649--660.Google ScholarDigital Library
- Fan Zhang, Ying Zhang, Lu Qin, Wenjie Zhang, and Xuemin Lin. 2017. Finding critical users for social network engagement: The collapsed k-core problem. In AAAI.Google Scholar
- Fan Zhang, Ying Zhang, Lu Qin, Wenjie Zhang, and Xuemin Lin. 2018. Efficiently reinforcing social networks over user engagement and tie strength. In ICDE.Google Scholar
- Yikai Zhang and Jeffrey Xu Yu. 2019. Unboundedness and efficiency of truss maintenance in evolving graphs. In SIGMOD.Google Scholar
- Jun Zhao, Renjie Sun, Qiuyu Zhu, Xiaoyang Wang, and Chen Chen. 2020. Community identification in signed networks: a k-truss based model. In CIKM.Google Scholar
- Kangfei Zhao, Zhiwei Zhang, Yu Rong, Jeffrey Xu Yu, and Junzhou Huang. 2021. Finding critical users in social communities via graph convolutions. IEEE Transactions on Knowledge and Data Engineering 35, 1 (2021), 456--468.Google Scholar
- Zibin Zheng, Fanghua Ye, Rong-Hua Li, Guohui Ling, and Tan Jin. 2017. Finding weighted k-truss communities in large networks. Information Sciences 417 (2017), 344--360.Google ScholarDigital Library
- Yaoming Zhou, Junwei Wang, and Hai Yang. 2019. Resilience of transportation systems: concepts and comprehensive review. IEEE Transactions on Intelligent Transportation Systems] 20, 12 (2019), 4262--4276.Google ScholarDigital Library
- Zhongxin Zhou, Wenchao Zhang, Fan Zhang, Deming Chu, and Binghao Li. 2021. VEK: a vertex-oriented approach for edge k-core problem. In TheWebConf (fka. WWW).Google Scholar
- Difeng Zhu, Guojiang Shen, Jingjing Chen, Wenfeng Zhou, and Xiangjie Kong. 2022. A higher-order motif-based spatiotemporal graph imputation approach for transportation networks. Wireless Communications and Mobile Computing 2022 (2022), 1--16.Google Scholar
- Weijie Zhu, Chen Chen, Xiaoyang Wang, and Xuemin Lin. 2018. K-core minimization: An edge manipulation approach. In CIKM.Google Scholar
- Weijie Zhu, Mengqi Zhang, Chen Chen, Xiaoyang Wang, Fan Zhang, and Xuemin Lin. 2019. Pivotal Relationship Identification: The K-Truss Minimization Problem.. In IJCAI.Google Scholar
Index Terms
- On Improving the Cohesiveness of Graphs by Merging Nodes: Formulation, Analysis, and Algorithms
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