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

**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 Symposium on Logic in Computer Science (LICS 2003)}, year = {2003}, month = {June}, pages = {301--310}, location = {Ottawa, Canada}, publisher = {IEEE Computer Society Press} }