The scaling window for a random graph with a given degree sequence

We consider a random graph on a given degree sequence <i>D</i>, satisfying certain conditions. We focus on two parameters <i>Q</i> = <i>Q</i>(<i>D</i>),<i>R</i> = <i>R</i>(<i>D</i>). Molloy and Reed proved that <i>Q</i> = 0 is the threshold for the random graph to have a giant component. We prove that if |<i>Q</i>| = <i>O</i>(<i>n</i><sup>-1/3</sup><i>R</i><sup>2/3</sup>) then, with high probability, the size of the largest component of the random graph will be of order Θ(<i>n</i><sup>2/3</sup><i>R</i><sup>-1/3</sup>). If <i>Q</i> is asymptotically larger/smaller that <i>n</i><sup>-1/3</sup><i>R</i><sup>2/3</sup> then the size of the largest component is asymptotically larger/smaller than <i>n</i><sup>2/3</sup><i>R</i><sup>-1/3</sup>. In other words, we establish that |<i>Q</i>| = <i>O</i>(<i>n</i><sup>-1/3</sup><i>R</i><sup>2/3</sup>) is the scaling window.

[1]  M. Weigt,et al.  Phase transition and finite-size scaling in the vertex-cover problem , 2005, Comput. Phys. Commun..

[2]  Brendan D. McKay,et al.  Asymptotic enumeration by degree sequence of graphs with degreeso(n1/2) , 1991, Comb..

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

[4]  Johan van Leeuwaarden,et al.  Scaling limits for critical inhomogeneous random graphs with finite third moments , 2009, 0907.4279.

[5]  Riccardo Zecchina,et al.  Threshold values of random K‐SAT from the cavity method , 2003, Random Struct. Algorithms.

[6]  Mihyun Kang,et al.  The Critical Phase for Random Graphs with a Given Degree Sequence , 2008, Combinatorics, Probability and Computing.

[7]  Remco van der Hofstad,et al.  Novel scaling limits for critical inhomogeneous random graphs , 2009, 0909.1472.

[8]  Richard M. Karp,et al.  The Transitive Closure of a Random Digraph , 1990, Random Struct. Algorithms.

[9]  Béla Bollobás,et al.  The scaling window of the 2‐SAT transition , 1999, Random Struct. Algorithms.

[10]  A. Rbnyi.  ON THE EVOLUTION OF RANDOM GRAPHS , 2001 .

[11]  Svante Janson.  The Probability That a Random Multigraph is Simple , 2009, Comb. Probab. Comput..

[12]  Bruce A. Reed,et al.  Mick gets some (the odds are on his side) (satisfiability) , 1992, Proceedings., 33rd Annual Symposium on Foundations of Computer Science.

[13]  P. Erdos,et al.  On the evolution of random graphs , 1984 .

[14]  Svante Janson,et al.  A new approach to the giant component problem , 2007, Random Struct. Algorithms.

[15]  Béla Bollobás,et al.  A Probabilistic Proof of an Asymptotic Formula for the Number of Labelled Regular Graphs , 1980, Eur. J. Comb..

[16]  B. Pittel,et al.  The structure of a random graph at the point of the phase transition , 1994 .

[17]  Edward A. Bender,et al.  The Asymptotic Number of Labeled Graphs with Given Degree Sequences , 1978, J. Comb. Theory A.

[18]  Fan Chung Graham,et al.  Concentration Inequalities and Martingale Inequalities: A Survey , 2006, Internet Math..

[19]  Asaf Nachmias,et al.  The critical random graph, with martingales , 2005 .

[20]  Y. Peres,et al.  Component sizes of the random graph outside the scaling window , 2006, math/0610466.

[21]  Alan M. Frieze,et al.  The Size of the Largest Strongly Connected Component of a Random Digraph with a Given Degree Sequence , 2004, Combinatorics, Probability and Computing.

[22]  Christian Borgs,et al.  Sharp threshold and scaling window for the integer partitioning problem , 2001, STOC '01.

[23]  B. Bollobás,et al.  The phase transition in inhomogeneous random graphs , 2007 .

[24]  Tomasz Łuczak.  Component behavior near the critical point of the random graph process , 1990 .

[25]  M. Leone,et al.  Phase coexistence and finite size scaling in random combinatorial problems , 2001 .

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

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

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

[29]  Svante Janson,et al.  A new approach to the giant component problem , 2009 .

[30]  Béla Bollobás,et al.  The phase transition in inhomogeneous random graphs , 2007, Random Struct. Algorithms.

[31]  R. Arratia,et al.  Logarithmic Combinatorial Structures: A Probabilistic Approach , 2003 .

[32]  Elizabeth L. Wilmer,et al.  Markov Chains and Mixing Times , 2008 .

[33]  Andreas Goerdt,et al.  A Threshold for Unsatisfiability , 1992, MFCS.

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

[35]  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.

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