On computing the Gromov hyperbolicity

ACM Journal of Experimental Algorithmics

Volumn 20, Issue 1, 2015, Pages

  Cohen, Nathann ^a   Coudert, David ^b,c   Lancin, Aurélien ^b,c  

^a UNIVERSITÉ PARIS SACLAY   (France)

^b INRIA   (France)

^c UNIVERSITÉ CÔTE D'AZUR   (France)

Author keywords

Algorithms; Gromov hyperbolicity; Networks

Indexed keywords

EID: 84938839773     PISSN: 10846654     EISSN: None     Source Type: Journal    
DOI: 10.1145/2780652     Document Type: Article

References (47)

1. Aaron B. Adcock, Blair D. Sullivan, Oscar R. Hernandez, and Michael W. Mahoney. 2013. Evaluating OpenMP tasking at scale for the computation of graph hyperbolicity. In 9th International Workshop on OpenMP (IWOMP)(Lecture Notes in Computer Science), Vol. 8122. Springer, 71-83. DOI:http://dx.doi.org/10.1007/978-3-642-40698-0-6
2. Hans-Jürgen Bandelt and Victor Chepoi. 2003. 1-hyperbolic graphs. SIAM Journal on Discrete Mathematics 16, 2, 323-334. DOI:http://dx.doi.org/10.1137/S0895480100380902
3. Anne Berry, Romain Pogorelcnik, and Geneviève Simonet. 2010. An introduction to clique minimal separator decomposition. Algorithms 3, 2, 197-215. DOI:http://dx.doi.org/10.3390/a3020197
4. Marián Boguñá, Fragkiskos Papadopoulos, and Dmitri V. Krioukov. 2010. Sustaining the internet with hyperbolic mapping. Nature Communications 1, 62 (Oct. 2010), 1-18. DOI:http://dx.doi.org/10.1038/ncomms1063
5. Gunnar Brinkmann, Jack H. Koolen, and Vincent Moulton. 2001. On the hyperbolicity of chordal graphs. Annals of Combinatorics 5, 1, 61-69. DOI:http://dx.doi.org/10.1007/s00026-001-8007-7
6. John Chakerian and Susan Holmes. 2010. distory: Distance Between Phylogenetic Histories. Retrieved from http://cran.r-project.org/web/packages/distory/.
7. John Chakerian and Susan Holmes. 2012. Computational tools for evaluating phylogenetic and hierarchical clustering trees. Journal of Computational and Graphical Statistics 21, 3, 581-599. DOI:http://dx.doi.org/10.1080/10618600.2012.640901
8. Pierre Charbit, Fabien de Montgolfier, and Mathieu Raffinot. 2012. Linear time split decomposition revisited. SIAM Journal on Discrete Mathematics 26, 2, 499-514. DOI:http://dx.doi.org/10.1137/10080052X
9. Victor Chepoi, Feodor F. Dragan, Bertrand Estellon, Michel Habib, and Yann Vaxès. 2008. Notes on diameters, centers, and approximating trees of δ-hyperbolic geodesic spaces and graphs. Electronic Notes in Discrete Mathematics 31, 231-234. DOI:http://dx.doi.org/10.1016/j.endm.2008.06.046
10. Victor Chepoi, Feodor F. Dragan, Bertrand Estellon, Michel Habib, Yann Vaxès, and Yang Xiang. 2012. Additive spanners and distance and routing labeling schemes for hyperbolic graphs. Algorithmica 62, 3-4, 713-732. DOI:http://dx.doi.org/10.1007/s00453-010-9478-x
11. Eunjoon Cho, Seth A. Myers, and Jure Leskovec. 2011. Friendship and mobility: User movement in locationbased social networks. In ACM SIGKDD International Conference on Knowledge Discovery and Data Mining. ACM, 1082-1090. DOI:http://dx.doi.org/10.1145/2020408.2020579
12. Nathann Cohen, David Coudert, Guillaume Ducoffe, and Aurélien Lancin. 2014. Applying cliquedecomposition for computing Gromov hyperbolicity. Research Report RR-8535. Inria. 30 pages. Retrieved from http://hal.inria.fr/hal-00989024.
13. David Coudert. 2014. Gromov Hyperbolicity of Graphs: C Source Code. Retrieved from http://www-sop. inria.fr/members/David.Coudert/code/hyperbolicity.shtml.
14. 4-free and 1/2-hyperbolic graphs. SIAM Journal on Discrete Mathematics 28, 3 (Sept. 2014), 1601-1617. DOI:http://dx.doi.org/10.1137/140954787
15. William H. Cunningham. 1982. Decomposition of directed graphs. SIAM Journal on Algebraic Discrete Methods 3, 2, 214-228. DOI:http://dx.doi.org/10.1137/0603021
16. William H. Cunningham and Jack Edmonds. 1980. A combinatorial decomposition theory. Canadian Journal of Mathematics 32, 3, 734-765. DOI:http://dx.doi.org/10.4153/CJM-1980-057-7
17. Fabien de Montgolfier, Mauricio Soto, and Laurent Viennot. 2011. Treewidth and hyperbolicity of the internet. In 10th IEEE International Symposium on Network Computing and Applications (NCA). IEEE, 25-32. DOI:http://dx.doi.org/10.1109/NCA.2011.11
18. Andreas Dress, Katharina Huber, Jacobus Koolen, Vincent Moulton, and Andreas Spillner. 2011. Basic Phylogenetic Combinatorics. Cambridge University Press, Cambridge, UK. I-XII,1-264 pages.
19. Ran Duan. 2014. Approximation algorithms for the gromov hyperbolicity of discrete metric spaces. In 11th Latin American Theoretical Informatics Symposium (LATIN) (Lecture Notes in Computer Science), Vol. 8392. Springer, 285-293. DOI:http://dx.doi.org/10.1007/978-3-642-54423-1-25
20. Ran Duan and Seth Pettie. 2009. Fast algorithms for (max, min)-matrix multiplication and bottleneck shortest paths. In 20th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA). SIAM, New York, NY, 384-391. DOI:http://dx.doi.org/10.1145/1496770.1496813
21. Hervé Fournier, Anas Ismail, and Antoine Vigneron. 2015. Computing the Gromov hyperbolicity of a discrete metric space. Inform. Process. Lett. 115, 6-8, 576-579. DOI:http://dx.doi.org/10.1016/j.ipl.2015.02.002
22. Tibor Gallai. 1967. Transitiv orientierbare graphen. Acta Mathematica Hungarica 18, 1, 25-66. DOI:http://dx.doi.org/10.1007/BF02020961
23. Micha Gromov. 1987. Hyperbolic groups. In Essays in Group Theory, S.M. Gersten (Ed.). Mathematical Sciences Research Institute Publications, Vol. 8. Springer, New York, 75-263. DOI:http://dx. doi.org/10.1007/978-1-4613-9586-7-3
24. Michel Habib and Christophe Paul. 2010. A survey of the algorithmic aspects of modular decomposition. Computer Science Review 4, 1, 41-59. DOI:http://dx.doi.org/10.1016/j.cosrev.2010.01.001
25. Luc Hogie and others. 2014. Grph, the high performance graph library for Java (Version 1.4.17). The Grph Development Team. Retrieved from http://grph.inria.fr.
26. Edmond A. Jonckheere and Poonsuk Lohsoonthorn. 2004. Geometry of network security. In American Control Conference, Vol. 2. IEEE, 976-981.
27. W. Sean Kennedy, Onuttom Narayan, and Iraj Saniee. 2013. On the Hyperbolicity of Large-Scale Networks. Technical Report arXiv:1307.0031. ArXiv. http://arxiv.org/abs/1307.0031
28. Jack H. Koolen and Vincent Moulton. 2002. Hyperbolic bridged graphs. European Journal of Combinatorics 23, 6, 683-699. DOI:http://dx.doi.org/10.1006/eujc.2002.0591
29. Dmitri V. Krioukov, Fragkiskos Papadopoulos, Maksim Kitsak, Amin Vahdat, and Marián Boguñá. 2010. Hyperbolic geometry of complex networks. Physical Review E 82, 3 (Sept. 2010), 036106. DOI:http://dx.doi.org/10.1103/PhysRevE.82.036106
30. Jure Leskovec, Jon Kleinberg, and Christos Faloutsos. 2007. Graph evolution: Densification and shrinking diameters. ACM Transactions on Knowledge Discovery from Data 1, 1 (March 2007), 1-41. DOI:http://dx.doi.org/10.1145/1217299.1217301
31. Daniel Lokshtanov. 2009. Finding the longest isometric cycle in a graph. Discrete Applied Mathematics 157, 12, 2670-2674. DOI:http://dx.doi.org/10.1016/j.dam.2008.08.008
32. Clémence Magnien, Matthieu Latapy, and Michel Habib. 2009. Fast computation of empirically tight bounds for the diameter of massive graphs. ACM Journal of Experimental Algorithmics 13, Article 10 (Feb. 2009), 1.8 pages. DOI:http://dx.doi.org/10.1145/1412228.1455266
33. Onuttom Narayan and Iraj Saniee. 2011. The large scale curvature of networks. Physical Review E 84 (Dec. 2011), 066108. DOI:http://dx.doi.org/10.1103/PhysRevE.84.066108
34. Onuttom Narayan, Iraj Saniee, and Gabriel H. Tucci. 2015. Lack of hyperbolicity in asymptotic Erdös-Renyi sparse random graphs. Internet Mathematics 11, 3, 277-288. DOI:http://dx.doi.org/10.1080/15427951.2014.1002640
35. Damien Noguès. 2009. δ-hyperbolicité et graphes. Master's thesis. MPRI, Université Paris 7.
36. OpenMP Architecture Review Board. 2008. OpenMP API Version 3.0. Retrieved from http://www.openmp.org/.
37. Fragkiskos Papadopoulos, Dmitri V. Krioukov, Marián Boguñá, and Amin Vahdat. 2009. Greedy forwarding in scale-free networks embedded in hyperbolic metric spaces. ACMSIGMETRICS Performance Evaluation Review 37, 2 (Sept. 2009), 15-17. DOI:http://dx.doi.org/10.1145/1639562.1639568
38. R Core Team. 2014. R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. Retrieved from http://www.r-project.org.
39. Venugopalan Ramasubramanian, Dahlia Malkhi, Fabian Kuhn, Mahesh Balakrishnan, Archit Gupta, and Aditya Akella. 2009. On the treeness of internet latency and bandwidth. ACM SIGMETRICS Performance Evaluation Review 37, 1 (June 2009), 61-72. DOI:http://dx.doi.org/10.1145/2492101.1555357
40. Yilun Shang. 2011. Lack of Gromov-hyperbolicity in colored random networks. Pan American Mathematical Journal 21, 1, 27-36.
41. Yuval Shavitt and Eran Shir. 2005. DIMES: Let the internet measure itself. ACM SIGCOMM Computer Communication Review 35, 5 (Oct. 2005), 71-74. DOI:http://dx.doi.org/10.1145/1096536.1096546
42. Mauricio A. Soto Gómez. 2011. Quelques propriétés topologiques des graphes et applications à internet et aux réseaux. Ph.D. Dissertation. Univ. Paris Diderot (Paris 7). Retrieved from http://www.dim.uchile.cl/~mausoto/memoires/these.pdf.
43. William A. Stein and others. 2014. Sage Mathematics Software (Version 6.1). The Sage Development Team. http://www.sagemath.org.
44. Robert Endre Tarjan. 1972. Depth-first search and linear graph algorithms. SIAM Journal of Computing 1, 2, 146-160. DOI:http://dx.doi.org/10.1137/0201010
45. Robert Endre Tarjan. 1985. Decomposition by clique separators. Discrete Mathematics 55, 2, 221-232. DOI:http://dx.doi.org/10.1016/0012-365X(85)90051-2
46. The Cooperative Association for Internet Data Analysis (CAIDA). 2013. The CAIDA AS Relationships Dataset. Retrieved from http://www.caida.org/data/active/as-relationships/.
47. Yaokun Wu and Chengpeng Zhang. 2011. Hyperbolicity and chordality of a graph. The Electron Journal of Combinatorics 18, 1, P43.