Cliques in Regular Graphs and the Core-Periphery Problem in Social Networks

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BRANDES, Ulrik, Eugenia HOLM, Andreas KARRENBAUER, 2016. Cliques in Regular Graphs and the Core-Periphery Problem in Social Networks. COCOA 2016: Combinatorial Optimization and Applications. Hong Kong, 16. Dez 2016 - 18. Dez 2016. In: CHAN, T-H. Hubert, ed. and others. Combinatorial Optimization and Applications : 10th International Conference, COCOA 2016, Hong Kong, China, December 16-18, 2016, Proceedings. COCOA 2016: Combinatorial Optimization and Applications. Hong Kong, 16. Dez 2016 - 18. Dez 2016. Cham:Springer, pp. 175-186. ISSN 1611-3349. eISSN 1611-3349. ISBN 978-3-319-48748-9. Available under: doi: 10.1007/978-3-319-48749-6_13

@inproceedings{Brandes2016-10-31Cliqu-36733, title={Cliques in Regular Graphs and the Core-Periphery Problem in Social Networks}, year={2016}, doi={10.1007/978-3-319-48749-6_13}, number={10043}, isbn={978-3-319-48748-9}, issn={1611-3349}, address={Cham}, publisher={Springer}, series={Lecture Notes in Computer Science}, booktitle={Combinatorial Optimization and Applications : 10th International Conference, COCOA 2016, Hong Kong, China, December 16-18, 2016, Proceedings}, pages={175--186}, editor={Chan, T-H. Hubert}, author={Brandes, Ulrik and Holm, Eugenia and Karrenbauer, Andreas} }

Holm, Eugenia Karrenbauer, Andreas Holm, Eugenia eng 2017-01-17T12:39:48Z Brandes, Ulrik 2016-10-31 The existence of a densely knit core surrounded by a loosely connected periphery is a common macro-structural feature of social networks. Formally, the CorePeriphery problem is to partition the nodes of an undirected graph G=(V,E) such that a subset X⊂V, the core, induces a dense subgraph, and its complement V∖X , the periphery, induces a sparse subgraph. Split graphs represent the ideal case in which the core induces a clique and the periphery forms an independent set. The number of missing and superfluous edges in the core and the periphery, respectively, can be minimized in linear time via edit distance to the closest split graph. We show that the CorePeriphery becomes intractable for standard notions of density other than the absolute number of misclassified pairs. Our main tool is a regularization procedure that transforms a given graph with maximum degree d into a d-regular graph with the same clique number by adding at most d⋅n new nodes. This is of independent interest because it implies that finding a maximum clique in a regular graph is NP-hard to approximate to within a factor of n<sup>1/2−ε</sup> for all ε>0 . We gratefully acknowledge financial support from Deutsche Forschungsgemeinschaft (DFG) under grants Br 2158/6-1 and Ka 3042/3-1. This work is partially supported by the Zukunftskolleg of the University of Konstanz, and the Max Planck Center for Visual Computing and Communication (www.mpc-vcc.org). Brandes, Ulrik Karrenbauer, Andreas 2017-01-17T12:39:48Z Cliques in Regular Graphs and the Core-Periphery Problem in Social Networks

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