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# Hyperbolic Random Graphs: Separators and Treewidth

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LIPIcs.ESA.2016.15.pdf
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## Cite As

Thomas Bläsius, Tobias Friedrich, and Anton Krohmer. Hyperbolic Random Graphs: Separators and Treewidth. In 24th Annual European Symposium on Algorithms (ESA 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 57, pp. 15:1-15:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)
https://doi.org/10.4230/LIPIcs.ESA.2016.15

## Abstract

Hyperbolic random graphs share many common properties with complex real-world networks; e.g., small diameter and average distance, large clustering coefficient, and a power-law degree sequence with adjustable exponent beta. Thus, when analyzing algorithms for large networks, potentially more realistic results can be achieved by assuming the input to be a hyperbolic random graph of size n. The worst-case run-time is then replaced by the expected run-time or by bounds that hold with high probability (whp), i.e., with probability 1-O(1/n). Though many structural properties of hyperbolic random graphs have been studied, almost no algorithmic results are known. Divide-and-conquer is an important algorithmic design principle that works particularly well if the instance admits small separators. We show that hyperbolic random graphs in fact have comparatively small separators. More precisely, we show that they can be expected to have balanced separator hierarchies with separators of size O(n^{3/2-beta/2}), O(log n), and O(1) if 2 < beta < 3, beta = 3, and 3 < beta, respectively. We infer that these graphs have whp a treewidth of O(n^{3/2-beta/2}), O(log^2 n), and O(log n), respectively. For 2 < \beta < 3, this matches a known lower bound. To demonstrate the usefulness of our results, we give several algorithmic applications.
##### Keywords
• hyperbolic random graphs
• scale-free networks
• power-law graphs
• separators
• treewidth

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## References

1. Mohammed Amin Abdullah, Michel Bode, and Nikolaos Fountoulakis. Typical distances in a geometric model for complex networks. CoRR, abs/1506.07811:1-33, 2015.
2. Albert-László Barabási and Réka Albert. Emergence of scaling in random networks. Science, 286(5439):509-512, 1999.
3. Michael Bode, Nikolaos Fountoulakis, and Tobias Müller. On the largest component of a hyperbolic model of complex networks. The Electronic Journal of Combinatorics, 22(3):1-46, 2015.
4. Marián Boguñá, Fragkiskos Papadopoulos, and Dmitri Krioukov. Sustaining the internet with hyperbolic mapping. Nature Communications, 1(62), 2010.
5. Béla Bollobás and Oliver M. Riordan. Mathematical Results on Scale-Free Random Graphs, chapter 1, pages 1-34. Wiley, 2005.
6. Karl Bringmann, Ralph Keusch, and Johannes Lengler. Geometric inhomogeneous random graphs. CoRR, abs/1511.00576:1-42, 2015.
7. Fan Chung and Linyuan Lu. The average distance in a random graph with given expected degrees. Internet Mathematics, 1(1):91-113, 2003.
8. Aaron Clauset, Cosma Rohilla Shalizi, and M. E. J. Newman. Power-law distributions in empirical data. SIAM Review, 51(4):661-703, 2009.
9. Marek Cygan, Fedor V. Fomin, Łukasz Kowalik, Daniel Lokshtanov, Daniel Marx, Marcin Pilipczuk, Michał Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer International Publishing, 2015.
10. Rodney G. Downey and Michael R. Fellows. Parameterized Complexity. Springer-Verlag New York, 1999.
11. Devdatt P. Dubhashi and Alessandro Panconesi. Concentration of Measure for the Analysis of Randomized Algorithms. Cambridge University Press, 2012.
12. P. Erdős and A. Rényi. On random graphs. I. Publicationes Mathematicae, 6:290-297, 1959.
13. Fedor V. Fomin, Daniel Lokshtanov, Michał Pilipczuk, Saket Saurabh, and Marcin Wrochna. Fully polynomial-time parameterized computations for graphs and matrices of low treewidth. CoRR, abs/1511.01379:1-44, 2015.
14. Tobias Friedrich and Anton Krohmer. Cliques in hyperbolic random graphs. In Proceedings of the IEEE Conference on Computer Communications (INFOCOM'15), pages 1544-1552, 2015.
15. Tobias Friedrich and Anton Krohmer. On the diameter of hyperbolic random graphs. In Proceedings of the 42nd International Colloquium on Automata, Languages, and Programming (ICALP'15), pages 614-625, 2015.
16. Yong Gao. On the threshold of having a linear treewidth in random graphs. In Proceedings of the 12th Annual International Conference on Computing and Combinatorics (COCOON'06), pages 226-234, 2006.
17. Yong Gao. Treewidth of Erdős-Rényi random graphs, random intersection graphs, and scale-free random graphs. Discrete Applied Mathematics, 160(4-5):566-578, 2012.
18. Luca Gugelmann, Konstantinos Panagiotou, and Ueli Peter. Random hyperbolic graphs: Degree sequence and clustering. In Proceedings of the 39th International Colloquium on Automata, Languages, and Programming (ICALP'12), pages 573-585, 2012.
19. Michał Karoński, Edward R. Scheinerman, and Karen B. Singer-Cohen. On random intersection graphs: The subgraph problem. Combinatorics, Probability and Computing, 8(1-2):131-159, 1999.
20. Marcos Kiwi and Dieter Mitsche. A bound for the diameter of random hyperbolic graphs. In Proceedings of the 12th Workshop on Analytic Algorithmics and Combinatorics (ANALCO'15), pages 26-39, 2015.
21. Dmitri Krioukov, Fragkiskos Papadopoulos, Maksim Kitsak, Amin Vahdat, and Marián Boguñá. Hyperbolic geometry of complex networks. Physical Review E, 82:036106, 2010.
22. Choongbum Lee, Joonkyung Lee, and Sang il Oum. Rank-width of random graphs. Journal of Graph Theory, 70(3):339-347, 2012.
23. Anshui Li and Tobias Müller. On the treewidth of random geometric graphs and percolated grids. Manuscript, 2015.
24. Richard J. Lipton and Robert E. Tarjan. A separator theorem for planar graphs. SIAM Journal on Applied Mathematics, 36(2):177-189, 1979.
25. Richard J. Lipton and Robert E. Tarjan. Applications of a planar separator theorem. SIAM Journal on Computing, 9(3):615-627, 1980.
26. Silvio Micali and Vijay V. Vazirani. An o(√|V| |e|) algorithm for finding maximum matching in general graphs. In Proceedings of the 21st Annual Symposium on Foundations of Computer Science (FOCS'80), pages 17-27, 1980.
27. Dieter Mitsche and Guillem Perarnau. On the treewidth and related parameters of random geometric graphs. In Proceedings of the 29th International Symposium on Theoretical Aspects of Computer Science (STACS'12), pages 408-419, 2012.
28. Fragkiskos Papadopoulos, Dmitri Krioukov, Marián Boguñá, and Amin Vahdat. Greedy forwarding in dynamic scale-free networks embedded in hyperbolic metric spaces. In Proceedings of the 29th Conference on Information Communications (INFOCOM'10), pages 2973-2981, 2010.
29. Remco van der Hofstad. Random graphs and complex networks. Vol. II. http://www.win.tue.nl/~rhofstad/NotesRGCNII.pdf, 2014.
30. David Zuckerman. Linear degree extractors and the inapproximability of max clique and chromatic number. In Proceedings of the 38th Annual ACM Symposium on Theory of Computing (STOC'06), pages 681-690, 2006.
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