ABSTRACT
The problem of selecting a group of vertices under certain constraints that maximize their joint centrality arises in many practical scenarios. In this paper, we extend the notion of current flow closeness centrality (CFCC) to a set of vertices in a graph, and investigate the problem of selecting a subset S to maximizes its CFCC C(S), with the cardinality constraint |S| = k. We show the NP-hardness of the problem, but propose two greedy algorithms to minimize the reciprocal of C(S). We prove the approximation ratios by showing the monotonicity and supermodularity. A proposed deterministic greedy algorithm has an approximation factor and cubic running time. To compare with, a proposed randomized algorithm gives -approximation in nearly-linear time, for any ? > 0. Extensive experiments on model and real networks demonstrate the effectiveness and efficiency of the proposed algorithms, with the randomized algorithm being applied to massive networks with more than a million vertices.
- Haim Avron and Sivan Toledo. 2011. Randomized algorithms for estimating the trace of an implicit symmetric positive semi-definite matrix. J. ACM 58, 2 (2011), 8:1-8:34. Google ScholarDigital Library
- Albert-László Barabási and Re´ka Albert. 1999. Emergence of scaling in random networks. Science 286, 5439 (1999), 509-512.Google Scholar
- Alex Bavelas. 1948. A mathematical model for group structures. Hum. Oran. 7, 3 (1948), 16-30.Google ScholarCross Ref
- Alex Bavelas. 1950. Communication patterns in task-oriented groups. J. Acoust. Soc. Am. 22, 6 (1950), 725-730.Google ScholarCross Ref
- Michele Benzi and Christine Klymko. 2015. On the limiting behavior of parameter-dependent network centrality measures. SIAM J. Matrix Anal. Appl. 36, 2 (2015), 686-706.Google ScholarDigital Library
- Elisabetta Bergamini, Tanya Gonser, and Henning Meyerhenke. 2018. Scaling up Group Closeness Maximization. In Proceedings of the Twentieth Workshop on Algorithm Engineering and Experiments (ALENEX '18). SIAM, 209-222.Google ScholarCross Ref
- Elisabetta Bergamini, Michael Wegner, Dimitar Lukarski, and Henning Meyerhenke. 2016. Estimating Current-Flow Closeness Centrality with a Multigrid Laplacian Solver. In Proceedings of the 7th SIAM Workshop on Combinatorial Scientific Computing (CSC '16). SIAM, 1-12.Google ScholarCross Ref
- Paolo Boldi and Sebastiano Vigna. 2014. Axioms for centrality. Internet Math. 10, 3-4 (2014), 222-262.Google ScholarCross Ref
- Francesco Bonchi, Gianmarco De Francisci Morales, and Matteo Riondato. 2016. Centrality measures on big graphs: Exact, approximated, and distributed algorithms. In Proceedings of the 25th International Conference on World Wide Web (WWW '16). ACM, 1017-1020. Google ScholarDigital Library
- Enrico Bozzo and Massimo Franceschet. 2013. Resistance distance, closeness, and betweenness. Soc. Networks 35, 3 (2013), 460-469.Google ScholarCross Ref
- Ulrik Brandes and Daniel Fleischer. 2005. Centrality Measures Based on Current Flow. In 22nd Annual Symposium on Theoretical Aspects of Computer Science, Proceedings (STACS '05). Springer, 533-544. Google ScholarDigital Library
- Chen Chen, Hanghang Tong, B Aditya Prakash, Charalampos E Tsourakakis, Tina Eliassi-Rad, Christos Faloutsos, and Duen Horng Chau. 2016. Node immunization on large graphs: Theory and algorithms. IEEE Trans. Knowl. Data Eng. 28, 1(2016), 113-126. Google ScholarDigital Library
- Chen Chen, Wei Wang, and Xiaoyang Wang. 2016. Efficient maximum closeness centrality group identification. In Databases Theory and Applications - 27th Australasian Database Conference, Proceedings (ADC '16). Springer, 43-55.Google Scholar
- Andrew Clark, Qiqiang Hou, Linda Bushnell, and Radha Poovendran. 2017. A submodular optimization approach to leader-follower consensus in networks with negative edges. In Proceedings of American Control Conference (ACC '17). IEEE, 1346-1352.Google ScholarCross Ref
- Andrew Clark and Radha Poovendran. 2011. A submodular optimization framework for leader selection in linear multi-agent systems. In Proceedings of the 50th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC '11). IEEE, 3614-3621.Google ScholarCross Ref
- Michael B. Cohen, Rasmus Kyng, Gary L. Miller, Jakub W. Pachocki, Richard Peng, Anup Rao, and Shen Chen Xu. 2014. Solving SDD linear systems in nearly mlog?1/2n time. In Proceedings of the 46th annual ACM Symposium on Theory of Computing (STOC '14). ACM, 343-352. Google ScholarDigital Library
- Samuel I. Daitch and Daniel A. Spielman. 2008. Faster approximate lossy generalized flow via interior point algorithms. In Proceedings of the 40th Annual ACM Symposium on Theory of Computing (STOC '08). ACM, 451-460. Google ScholarDigital Library
- Shlomi Dolev, Yuval Elovici, Rami Puzis, and Polina Zilberman. 2009. Incremental deployment of network monitors based on group betweenness centrality. Inform. Process. Lett. 109, 20 (2009), 1172-1176. Google ScholarDigital Library
- Peter G Doyle and J Laurie Snell. 1984. Random Walks and Electric Networks. Mathematical Association of America.Google Scholar
- Paul Erdös and Alfre´d Re´nyi. 1959. On random graphs, I. Publ. Math. Debrecen 6 (1959), 290-297.Google ScholarCross Ref
- Martin G Everett and Stephen P Borgatti. 1999. The centrality of groups and classes. J. Math. Sociol. 23, 3 (1999), 181-201.Google ScholarCross Ref
- Martin Fink and Joachim Spoerhase. 2011. Maximum betweenness centrality: approximability and tractable cases. In Algorithms and Computation - 5th International Workshop, Proceedings (WALCOM '11). Springer, 9-20. Google ScholarDigital Library
- Gerd Fricke, Stephen T. Hedetniemi, and David Pokrass Jacobs. 1998. Independence and Irredundance in k-Regular Graphs. Ars Comb. 49(1998).Google Scholar
- Rumi Ghosh, Shang-hua Teng, Kristina Lerman, and Xiaoran Yan. 2014. The interplay between dynamics and networks: centrality, communities, and cheeger inequality. In Proceedings of the 20th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (KDD '14). ACM, 1406-1415. Google ScholarDigital Library
- Christos Gkantsidis, Milena Mihail, and Amin Saberi. 2006. Random walks in peer-to-peer networks: algorithms and evaluation. Perform. Evaluation 63, 3 (2006), 241-263. Google ScholarDigital Library
- MF Hutchinson. 1989. A stochastic estimator of the trace of the influence matrix for Laplacian smoothing splines. Commun. Stat. Simul. Comput. 18, 3(1989), 1059-1076.Google ScholarCross Ref
- N Sh Izmailian, R Kenna, and FY Wu. 2013. The two-point resistance of a resistor network: a new formulation and application to the cobweb network. J. Phys. A 47, 3(2013), 035003.Google ScholarCross Ref
- Donald B. Johnson. 1977. Efficient algorithms for shortest paths in sparse networks. J. ACM (1977). Google ScholarDigital Library
- William Johnson and Joram Lindenstrauss. 1984. Extensions of Lipschitz mappings into a Hilbert space. Contemp. Math. 26, 189-206 (1984), 1.Google ScholarCross Ref
- Jonathan A. Kelner, Gary L. Miller, and Richard Peng. 2012. Faster approximate multicommodity flow using quadratically coupled flows. In Proceedings of the 44th Symposium on Theory of Computing Conference (STOC 2012). ACM, 1-18. Google ScholarDigital Library
- David Kempe, Jon Kleinberg, and E&Acute;va Tardos. 2003. Maximizing the spread of influence through a social network. In Proceedings of the 9th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (KDD '03). ACM, 137-146. Google ScholarDigital Library
- Douglas J Klein and Milan Randic. 1993. Resistance distance. J. Math. Chem. 12, 1 (1993), 81-95.Google ScholarCross Ref
- Ioannis Koutis, Alex Levin, and Richard Peng. 2016. Faster Spectral Sparsification and Numerical Algorithms for SDD Matrices. ACM Trans. Algorithms 12, 2 (2016), 17:1-17:16. Google ScholarDigital Library
- Je´r⊚me Kunegis. 2013. Konect: the koblenz network collection. In 22nd International World Wide Web Conference, Companion Volume (WWW '13). ACM, 1343-1350. Google ScholarDigital Library
- Amy N Langville and Carl D Meyer. 2012. Who's# 1?: the science of rating and ranking. Princeton University Press. Google ScholarDigital Library
- Jure Leskovec and Andrej Krevl. 2014. SNAP Datasets: Stanford Large Network Dataset Collection. http://snap.stanford.edu/data. (June 2014).Google Scholar
- Rong-Hua Li, Jeffrey Xu Yu, Xin Huang, and Hong Cheng. 2014. Random-walk domination in large graphs. In Proceedings of IEEE 30th International Conference on Data Engineering (ICDE '14). IEEE, 736-747.Google ScholarCross Ref
- Linyuan Lü, Duanbing Chen, Xiao-Long Ren, Qian-Ming Zhang, Yi-Cheng Zhang, and Tao Zhou. 2016. Vital nodes identification in complex networks. Phys. Rep. 650(2016), 1-63.Google ScholarCross Ref
- Ahmad Mahmoody, Charalampos E Tsourakakis, and Eli Upfal. 2016. Scalable betweenness centrality maximization via sampling. In Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (KDD '16). ACM, 1765-1773. Google ScholarDigital Library
- Charalampos Mavroforakis, Michael Mathioudakis, and Aristides Gionis. 2015. Absorbing random-walk centrality: Theory and algorithms. In Proceedings of IEEE International Conference on Data Mining (ICDM '15). IEEE, 901-906. Google ScholarDigital Library
- Gary L. Miller and Richard Peng. 2013. Approximate Maximum Flow on Separable Undirected Graphs. In Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA '13). SIAM, 1151-1170. Google ScholarDigital Library
- George L. Nemhauser, Laurence A. Wolsey, and Marshall L. Fisher. 1978. An analysis of approximations for maximizing submodular set functions - I. Math. Program. 14, 1 (1978), 265-294. Google ScholarDigital Library
- Mark E. J. Newman. 2005. A measure of betweenness centrality based on random walks. Soc. Networks 27, 1 (2005), 39-54.Google ScholarCross Ref
- Mark E. J. Newman. 2010. Networks: An Introduction. Oxford University Press. Google ScholarCross Ref
- Stacy Patterson and Bassam Bamieh. 2010. Leader selection for optimal network coherence. In Proceedings of 49th IEEE Conference on Decision and Control (CDC '10). IEEE, 2692-2697.Google ScholarCross Ref
- Mohammad Pirani and Shreyas Sundaram. 2014. Spectral properties of the grounded Laplacian matrix with applications to consensus in the presence of stubborn agents. In Proceedings of American Control Conference (ACC '14). IEEE, 2160-2165.Google ScholarCross Ref
- Mojtaba Rezvani, Weifa Liang, Wenzheng Xu, and Chengfei Liu. 2015. Identifying top-k structural hole spanners in large-scale social networks. In Proceedings of the 24th ACM International on Conference on Information and Knowledge Management (CIKM '15). ACM, 263-272. Google ScholarDigital Library
- Daniel A. Spielman and Shanghua Teng. 2014. Nearly Linear Time Algorithms for Preconditioning and Solving Symmetric, Diagonally Dominant Linear Systems. SIAM J. Matrix Anal. Appl. 35, 3 (2014), 835-885.Google ScholarDigital Library
- Daniel A. Spielman and Nikhil Srivastava. 2011. Graph Sparsification by Effective Resistances. SIAM J. Comput. 40, 6 (2011), 1913-1926. Google ScholarDigital Library
- Karen Stephenson and Marvin Zelen. 1989. Rethinking centrality: Methods and examples. Soc. Networks 11, 1 (1989), 1-37.Google ScholarCross Ref
- Hanghang Tong, B Aditya Prakash, Charalampos Tsourakakis, Tina Eliassi-Rad, Christos Faloutsos, and Duen Horng Chau. 2010. On the vulnerability of large graphs. In Proceedings of IEEE 10th International Conference on Data Mining (ICDM '10). IEEE, 1091-1096. Google ScholarDigital Library
- Duncan J Watts and Steven H Strogatz. 1998. Collective dynamics of small-world networks. Nature 393, 6684 (1998), 440.Google Scholar
- Scott White and Padhraic Smyth. 2003. Algorithms for estimating relative importance in networks. In Proceedings of the 20th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (KDD '03). ACM, 266-275. Google ScholarDigital Library
- Wenzheng Xu, Mojtaba Rezvani, Weifa Liang, Jeffrey Xu Yu, and Chengfei Liu. 2017. Efficient Algorithms for the Identification of Top-k Structural Hole Spanners in Large Social Networks. IEEE Trans. Knowl. Data Eng. 29, 5(2017), 1017-1030. Google ScholarDigital Library
- Yuichi Yoshida. 2014. Almost linear-time algorithms for adaptive betweenness centrality using hypergraph sketches. In Proceedings of the 20th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (KDD '14). ACM, 1416-1425. Google ScholarDigital Library
- Junzhou Zhao, John Lui, Don Towsley, and Xiaohong Guan. 2014. Measuring and maximizing group closeness centrality over disk-resident graphs. In Proceedings of the 23rd International Conference on World Wide Web (WWW '14). ACM, 689-694. Google ScholarDigital Library
- Pengpeng Zhao, Yongkun Li, Hong Xie, Zhiyong Wu, Yinlong Xu, and John CS Lui. 2017. Measuring and Maximizing Influence via Random Walk in Social Activity Networks. In Database Systems for Advanced Applications - 22nd International Conference, Proceedings, Part II (DASFAA '17). Springer, 323-338.Google Scholar
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