Three-query PCPs with perfect completeness over non-Boolean domains

In this paper we study a random graph with N nodes, where node j has degree Dj and lDjr j=1N are i.i.d. with P(Dj ≤ x) = F(x). We assume that 1 - F(x) ≤ cx-t+1 for some t > 3 and some constant c > 0. This graph model is a variant of the so-called configuration model, and includes heavy tail degrees with finite variance. The minimal number of edges between two arbitrary connected nodes, also known as the graph distance or the hopcount, is investigated when N → ∞. We prove that the graph distance grows like logn N, when the base of the logarithm equals n = E[Dj(Dj - 1)]/E[Dj] > 1. This confirms the heuristic argument of Newman, Strogatz, and Watts [Phys Rev E 64 (2002), 026118, 1–17]. In addition, the random fluctuations around this asymptotic mean logn N are characterized and shown to be uniformly bounded. In particular, we show convergence in distribution of the centered graph distance along exponentially growing subsequences. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2005

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

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

[3]  Bruce A. Reed,et al.  The Size of the Giant Component of a Random Graph with a Given Degree Sequence , 1998, Combinatorics, Probability and Computing.

[4]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1951 .

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

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

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

[8]  Remco van der Hofstad,et al.  Distances in random graphs with infinite mean degrees , 2004, math/0407091.

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

[10]  Fan Chung Graham,et al.  A Random Graph Model for Power Law Graphs , 2001, Exp. Math..

[11]  H. Thorisson.  Coupling, stationarity, and regeneration , 2000 .

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

[13]  Walter Willinger,et al.  Network topology generators: degree-based vs. structural , 2002, SIGCOMM 2002.

[14]  Per Bak,et al.  Small Worlds: The Dynamics of Networks between Order and Randomness, by Duncan J. Watts , 2000 .

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

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

[17]  I. Glynn,et al.  A hundred years of sodium pumping. , 2002, Annual review of physiology.

[18]  Albert-László Barabási,et al.  Linked: The New Science of Networks , 2002 .

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

[20]  L. Rogers,et al.  Diffusions, Markov processes, and martingales , 1979 .

[21]  Bruce A. Reed,et al.  A Critical Point for Random Graphs with a Given Degree Sequence , 1995, Random Struct. Algorithms.

[22]  Remco van der Hofstad,et al.  Distances in Random Graphs with Finite Mean and Infinite Variance Degrees , 2005, math/0502581.

[23]  Mark E. J. Newman,et al.  The Structure and Function of Complex Networks , 2003, SIAM Rev..

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

[25]  Gesine Reinert,et al.  Small worlds , 2001, Random Struct. Algorithms.

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

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

[28]  Béla Bollobás,et al.  Directed scale-free graphs , 2003, SODA '03.

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

[30]  S. N. Dorogovtsev,et al.  Pseudofractal scale-free web. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[31]  S. N. Dorogovtsev,et al.  Metric structure of random networks , 2002, cond-mat/0210085.

[32]  Ilkka Norros,et al.  On the power-law random graph model of massive data networks , 2004, Perform. Evaluation.

[33]  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.

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

[35]  P. V. Mieghem,et al.  On the Mean Distance in Scale Free Graphs , 2005 .

[36]  Vern Paxson,et al.  End-to-end routing behavior in the Internet , 1996, TNET.

[37]  Svante Janson,et al.  Random graphs , 2000, ZOR Methods Model. Oper. Res..

[38]  Gerard Hooghiemstra,et al.  A scaling law for the hopcount , 2000 .

[39]  Noga Alon,et al.  The Probabilistic Method, Second Edition , 2004 .

[40]  Linyuan Lu,et al.  Random evolution in massive graphs , 2001 .

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

[42]  S. Strogatz.  Exploring complex networks , 2001, Nature.

[43]  Béla Bollobás,et al.  The diameter of random regular graphs , 1982, Comb..

[44]  D. Watts,et al.  Small Worlds: The Dynamics of Networks between Order and Randomness , 2001 .

[45]  M. Markus,et al.  Oscillations and turbulence induced by an activating agent in an active medium. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

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