Clustering and the Hyperbolic Geometry of Complex Networks

Clustering is a fundamental property of complex networks and it is the mathematical expression of a ubiquitous phenomenon that arises in various types of self-organized networks such as biological networks, computer networks or social networks. In this paper, we consider what is called the global clustering coefficient of random graphs on the hyperbolic plane. This model of random graphs was proposed recently by Krioukov et al. [22] as a mathematical model of complex networks, implementing the assumption that hyperbolic geometry underlies the structure of these networks. We do a rigorous analysis of clustering and characterize the global clustering coefficient in terms of the parameters of the model. We show how the global clustering coefficient can be tuned by these parameters, giving an explicit formula.

[1]  Luca Gugelmann,et al.  Random Hyperbolic Graphs: Degree Sequence and Clustering , 2012, ArXiv.

[2]  Luca Gugelmann,et al.  Random Hyperbolic Graphs: Degree Sequence and Clustering - (Extended Abstract) , 2012, ICALP.

[3]  Michael Krivelevich,et al.  Submitted to the Annals of Applied Probability HAMILTON CYCLES IN RANDOM GEOMETRIC GRAPHS By , 2010 .

[4]  Svante Janson,et al.  Random graphs , 2000, Wiley-Interscience series in discrete mathematics and optimization.

[5]  F. Chung,et al.  Complex Graphs and Networks , 2006 .

[6]  Mindaugas Bloznelis,et al.  Degree and clustering coefficient in sparse random intersection graphs , 2013, 1303.3388.

[7]  M. Newman,et al.  Random graphs with arbitrary degree distributions and their applications. , 2000, Physical review. E, Statistical, nonlinear, and soft matter physics.

[8]  Béla Bollobás,et al.  Random Graphs , 1985 .

[9]  Amin Vahdat,et al.  Hyperbolic Geometry of Complex Networks , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[10]  M. Newman,et al.  Statistical mechanics of networks. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[11]  Albert-László Barabási,et al.  Statistical mechanics of complex networks , 2001, ArXiv.

[12]  Nicole Eggemann,et al.  The clustering coefficient of a scale-free random graph , 2008, Discret. Appl. Math..

[13]  Colin McDiarmid,et al.  On the chromatic number of random geometric graphs , 2011, Comb..

[14]  Nikolaos Fountoulakis,et al.  On the evolution of random graphs on spaces of negative curvature , 2012, ArXiv.

[15]  F. Chung,et al.  The average distances in random graphs with given expected degrees , 2002, Proceedings of the National Academy of Sciences of the United States of America.

[16]  S. Ross.  A random graph , 1981 .

[17]  Béla Bollobás,et al.  Sparse random graphs with clustering , 2008, Random Struct. Algorithms.

[18]  Béla Bollobás,et al.  Random Graphs: Notation , 2001 .

[19]  F. Chung,et al.  Connected Components in Random Graphs with Given Expected Degree Sequences , 2002 .

[20]  E. N. Gilbert,et al.  Random Plane Networks , 1961 .

[21]  Emilie Coupechoux,et al.  How Clustering Affects Epidemics in Random Networks , 2012, Advances in Applied Probability.

[22]  Sergey N. Dorogovtsev,et al.  Lectures on Complex Networks , 2010 .

[23]  Duncan J. Watts,et al.  Collective dynamics of ‘small-world’ networks , 1998, Nature.

[24]  Albert,et al.  Emergence of scaling in random networks , 1999, Science.

[25]  M. Weigt,et al.  On the properties of small-world network models , 1999, cond-mat/9903411.

[26]  R. Luce,et al.  A method of matrix analysis of group structure , 1949, Psychometrika.

[27]  Filippo Menczer,et al.  Growing and navigating the small world Web by local content , 2002, Proceedings of the National Academy of Sciences of the United States of America.

[28]  Robert Hafner,et al.  The asymptotic distribution of random clumps , 1972, Computing.

[29]  Béla Bollobás,et al.  Mathematical results on scale‐free random graphs , 2005 .

[30]  Robert L. Goldstone,et al.  The simultaneous evolution of author and paper networks , 2003, Proceedings of the National Academy of Sciences of the United States of America.

[31]  Mathew D. Penrose,et al.  Random Geometric Graphs , 2003 .