# Logic in Computer Science (LICS 2003)

## Paper: Convergence Law for Random Graphs with Specified Degree Sequence (at LICS 2003)

Authors: James F. Lynch

### Abstract

The degree sequence of an n-vertex graph is d, ..., d, where each d is the number of vertices of degree i in the graph. A random graph with degree sequence d, ..., d is a randomly selected member of the set of graphs on {0, ..., n-1} with that degree sequence, all choices being equally likely. Let \lambda, \lambda, ... be a sequence of nonnegative reals summing to 1. A class of finite graphs has degree sequences approximated by \lambda, \lambda, ... if, for every i and n, the members of the class of size n have lambdan + o(n) vertices of degree i. Our main result is a convergence law for random graphs with degree sequences approximated by some sequence \lambda, \lambda, .... With certain conditions on the sequence \lambda, \lambda, ..., the probability of any first-order sentence on random graphs of size n converges to a limit as n grows.

### BibTeX

  @InProceedings{Lynch-ConvergenceLawforRa,
author = 	 {James F. Lynch},
title = 	 {Convergence Law for Random Graphs with Specified Degree Sequence},
booktitle =  {Proceedings of the Eighteenth Annual IEEE Symp. on Logic in Computer Science, {LICS} 2003},
year =	 2003,
editor =	 {Phokion G. Kolaitis},
month =	 {June},
pages =      {301--},