Lack of Gromov-hyperbolicity in small-world networks

Barahona M., Pecora L.M., Synchronization in small-world systems, Phys. Rev. Lett., 2002, 89(5), #054101

Bermudo S., Rodríguez J.M., Sigarreta J.M., Vilaire J.-M., Mathematical properties of Gromov hyperbolic graphs, In: International Conference of Numerical Analysis and Applied Mathematics, Rhodes, September 19–25, 2010, AIP Conf. Proc., 1281, American Institute of Physics, Melville, 2010, 575–578

Bollobás B., Random Graphs, Cambridge Stud. Adv. Math., 73, Cambridge University Press, Cambridge, 2001

Brisdon M.R., Haefliger A., Metric Spaces of Non-positive Curvature, Grundlehren Math. Wiss., 319, Springer, Berlin, 1999

Clauset A., Moore C., Newman M.E.J., Hierarchical structure and the prediction of missing links in networks, Nature, 2008, 453, 98–101

Dress A., Holland B., Huber K.T., Koolen J.H., Moulton V., Weyer-Menkhoff J., δ additive and δ ultra-additive maps, Gromov’s trees, and the Farris transform, Discrete Appl. Math., 2005, 146(1), 51–73

Dress A., Huber K.T., Moulton V., Some uses of the Farris transform in mathematics and phylogenetics — a review, Ann. Comb., 2007, 11(1), 1–37

Gromov M., Hyperbolic groups, In: Essays in Group Theory, Math. Sci. Res. Inst. Publ., 8, Springer, New York, 1987, 75–263

Gromov M., Metric Structures for Riemannian and Non-Riemannian Spaces, Progr. Math., 152, Birkhäuser, Boston, 1999

Jonckheere E., Lohsoonthorn P., Geometry of network security, In: American Control Conference, vol. 2, Boston, June 30–July 2, 2004, 976–981, available at http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=1386698

Jonckheere E.A., Lohsoonthorn P., A hyperbolic geometry approach to multipath routing, In: 10th Mediterranean Conference on Control and Automation, Lisbon, July 9–12, 2002, available at http://med.ee.nd.edu/MED10/pdf/373.pdf

Jonckheere E.A., Lou M., Hespanha J., Barooah P., Effective resistance of Gromov-hyperbolic graphs: application to asymptotic sensor network problems, In: 46th IEEE Conference on Decision and Control, New Orleans, December 12–14, 2007, 1453–1458, DOI: 10.1109/CDC.2007.4434878

Kleinberg J.M., Navigation in a small world, Nature, 2000, 406, 845

Krioukov D., Papadopoulos F., Kitsak M., Vahdat A., Boguñá M., Hyperbolic geometry of complex networks, Phys. Rev. E, 2010, 82(3), #036106

Maldacena J., The large-N limit of superconformal field theories and supergravity, Internat. J. Theoret. Phys., 1999, 38(4), 1113–1133

Narayan O., Saniee I., Large-scale curvature of networks, Phys. Rev. E, 2011, 84, #066108

Narayan O., Saniee I., Tucci G.H., Lack of spectral gap and hyperbolicity in asymptotic Erdős-Rényi random graphs, preprint available at http://arxiv.org/abs/1009.5700

Newman M.E.J., Models of the small world, J. Stat. Phys., 2000, 101(3–4), 819–841

Newman M.E.J., Moore C., Watts D.J., Mean-field solution of the small-world network model, Phys. Rev. Lett., 2000, 84(14), 3201–3204

Shang Y., Lack of Gromov-hyperbolicity in colored random networks, Panamer. Math. J., 2011, 21(1), 27–36

Shang Y., Multi-type directed scale-free percolation, Commun. Theor. Phys. (in press)

Watts D.J., Strogatz S.H., Collective dynamics of ’small-world’ networks, Nature, 1998, 393, 440–442