Logical convergence laws via stochastic approximation and Markov processes

TL;DR

This paper proves that uniform attachment random graphs with bounded degrees follow the first order convergence law by analyzing Markov chains and stochastic approximation processes.

Contribution

It introduces a novel method combining Markov chains and stochastic approximation to establish logical convergence laws for uniform attachment graphs.

Findings

Uniform attachment graphs with bounded degrees obey the first order convergence law.

The dynamics of first order equivalence classes can be modeled using Markov chains.

Convergence to a limit distribution is established via stochastic approximation.

Abstract

Since the paper of Kleinberg and Kleinberg, SODA'05, where it was proven that the preferential attachment random graph with degeneracy at least 3 does not obey the first order 0-1 law, no general methods were developed to study logical limit laws for recursive random graph models with arbitrary degeneracy. Even in the (possibly) simplest case of the uniform attachment, it is still not known whether the first order convergence law holds in this model. We prove that the uniform attachment random graph with bounded degrees obeys the first order convergence law. To prove the law, we describe dynamics of first order equivalence classes of the random graph using Markov chains. The convergence law follows from the existence of a limit distribution of the considered Markov chain. To show the latter convergence, we use stochastic approximation processes.