A logical limit law for the sequential model of preferential attachment graphs

TL;DR

This paper establishes a logical limit law for preferential attachment graphs, extending the understanding of their asymptotic properties and logical expressibility, similar to known results for Erdős-Rényi graphs.

Contribution

It proves a limit law for preferential attachment graphs with Pólya urn representation, covering key models like uniform attachment and Barabási-Albert, highlighting differences in cycle distributions.

Findings

Limit law established for preferential attachment graphs.

Different behaviors observed in cycle distributions for key models.

Extends logical limit law results beyond Erdős-Rényi graphs.

Abstract

For a sequence of random graphs, the limit law we refer to is the existence of a limiting probability of any graph property that can be expressed in terms of predicate logic. A zero-one limit law is shown by Shelah and Spencer for Erd\"{o}s-Renyi graphs given that the connection rate has an irrational exponent. We show a limit law for preferential attachment graphs which admit a P\'{o}lya urn representation. The two extreme cases of the parametric model, the uniform attachment graph and the sequential Barab\'{a}si-Albert model, are covered separately as they exhibit qualitative differences regarding the distribution of cycles of bounded length in the graph.