A Geometric Preferential Attachment Model of Networks

A detailed understanding of expansion in complex networks can greatly aid in the design and analysis of algorithms for a variety of important network tasks, including routing messages, ranking nodes, and compressing graphs. This has motivated several recent investigations of expansion properties in real-world graphs and also in random models of real-world graphs, like the preferential attachment graph. The results point to a gap between real-world observations and theoretical models. Some real-world graphs are expanders and others are not, but a graph generated by the preferential attachment model is an expander whp. 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 R3. After generating xt, we randomly connect it to m points from those points in x1, x2, ..., xt-1....

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

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

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

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

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

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

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

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

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

[10]  Andrei Z. Broder,et al.  Graph structure in the Web , 2000, Comput. Networks.

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

[12]  Albert-László Barabási,et al.  Internet: Diameter of the World-Wide Web , 1999, Nature.

[13]  Colin Macilwain.  Clinton flails congress over $1.8 billion civilian R&D cuts , 1999, Nature.

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

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

[16]  Christian Borgs,et al.  Competition-Induced Preferential Attachment , 2004, ICALP.

[17]  Deryk Osthus,et al.  Popularity based random graph models leading to a scale-free degree sequence , 2004, Discret. Math..

[18]  Abraham D. Flaxman.  Expansion and Lack Thereof in Randomly Perturbed Graphs , 2006, WAW.

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

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

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

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

[23]  Béla Bollobás,et al.  Degree Distribution of the FKP Network Model , 2003, ICALP.

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

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

[26]  F. Chung,et al.  Spectra of random graphs with given expected degrees , 2003, Proceedings of the National Academy of Sciences of the United States of America.

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

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

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

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

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

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

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

[34]  Duncan J. Watts,et al.  The Structure and Dynamics of Networks: (Princeton Studies in Complexity) , 2006 .

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

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

[37]  Alan M. Frieze,et al.  A general model of web graphs , 2003, Random Struct. Algorithms.

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

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

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

[41]  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.

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