A Geometric Preferential Attachment Model of Networks II

We study a random graph Gn that combines certain aspects of geometric random graphs and preferential attachment graphs. This model yields a graph with power-law degree distribution where the expansion property depends on a tunable parameter of the model. The vertices of Gn are n sequentially generated points x1, x2, . . . , xn chosen uniformly at random from the unit sphere in R. After generating xt, we randomly connect it to m points from those points in x1, x2, . . . , xt−1.

[1]  Ravi Kumar,et al.  Trawling the Web for Emerging Cyber-Communities , 1999, Comput. Networks.

[2]  J. Dall,et al.  Random geometric graphs. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[3]  Alan M. Frieze,et al.  Random Deletion in a Scale-Free Random Graph Process , 2004, Internet Math..

[4]  Ernesto Estrada,et al.  Spectral scaling and good expansion properties in complex networks , 2006, Europhysics Letters (EPL).

[5]  Eli Upfal,et al.  The Web as a graph , 2000, PODS.

[6]  Mihaela Enachescu,et al.  Variations on Random Graph Models for the Web , 2001 .

[7]  Béla Bollobás,et al.  The degree sequence of a scale‐free random graph process , 2001, Random Struct. Algorithms.

[8]  Mark E. J. Newman,et al.  Structure and Dynamics of Networks , 2009 .

[9]  Alan M. Frieze,et al.  Random graphs , 2006, SODA '06.

[10]  Michalis Faloutsos,et al.  On power-law relationships of the Internet topology , 1999, SIGCOMM '99.

[11]  Fan Chung Graham,et al.  A random graph model for massive graphs , 2000, STOC '00.

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

[13]  Abraham D. Flaxman,et al.  Expansion and Lack Thereof in Randomly Perturbed Graphs , 2007, Internet Math..

[14]  Béla Bollobás,et al.  Coupling Scale-Free and Classical Random Graphs , 2004, Internet Math..

[15]  H. Simon,et al.  ON A CLASS OF SKEW DISTRIBUTION FUNCTIONS , 1955 .

[16]  N. Alon,et al.  The Probabilistic Method, Second Edition , 2000 .

[17]  Yamir Moreno,et al.  Local versus global knowledge in the Barabási-Albert scale-free network model. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[18]  Noga Alon,et al.  The Probabilistic Method , 2015, Fundamentals of Ramsey Theory.

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

[20]  Guy E. Blelloch,et al.  Compact representations of separable graphs , 2003, SODA '03.

[21]  Michael Mitzenmacher,et al.  A Brief History of Generative Models for Power Law and Lognormal Distributions , 2004, Internet Math..

[22]  Jie Wu,et al.  Small Worlds: The Dynamics of Networks between Order and Randomness , 2003 .

[23]  Alan M. Frieze,et al.  A General Model of Undirected Web Graphs , 2001, ESA.

[24]  Amin Saberi,et al.  On certain connectivity properties of the Internet topology , 2003, 44th Annual IEEE Symposium on Foundations of Computer Science, 2003. Proceedings..

[25]  F. Chung,et al.  Eigenvalues of Random Power law Graphs , 2003 .

[26]  Christos H. Papadimitriou,et al.  Heuristically Optimized Trade-Offs: A New Paradigm for Power Laws in the Internet , 2002, ICALP.

[27]  Fan Chung Graham,et al.  Random evolution in massive graphs , 2001, Proceedings 2001 IEEE International Conference on Cluster Computing.

[28]  B. Hayes.  Graph Theory in Practice: Part II , 2000, American Scientist.

[29]  Alan M. Frieze,et al.  A Geometric Preferential Attachment Model of Networks , 2006, Internet Math..

[30]  Walter Willinger,et al.  Towards a Theory of Scale-Free Graphs: Definition, Properties, and Implications , 2005, Internet Math..

[31]  Jon M. Kleinberg,et al.  The Web as a Graph: Measurements, Models, and Methods , 1999, COCOON.

[32]  Duncan J. Watts,et al.  Book Review: Small Worlds. The Dynamics of Networks Between Order and Randomness , 2000 .

[33]  G. Yule,et al.  A Mathematical Theory of Evolution, Based on the Conclusions of Dr. J. C. Willis, F.R.S. , 1925 .

[34]  Christos H. Papadimitriou,et al.  On the Eigenvalue Power Law , 2002, RANDOM.

[35]  Béla Bollobás,et al.  Degree distribution of the FKP network model , 2003, Theor. Comput. Sci..

[36]  Eli Upfal,et al.  Stochastic models for the Web graph , 2000, Proceedings 41st Annual Symposium on Foundations of Computer Science.

[37]  Béla Bollobás,et al.  The Diameter of a Scale-Free Random Graph , 2004, Comb..