Qualitative graph limit theory. Cantor Dynamical Systems and Constant-Time Distributed Algorithms

Abstract

The goal of the paper is to lay the foundation for the qualitative analogue of the classical, quantitative sparse graph limit theory. In the first part of the paper we introduce the qualitative analogues of the Benjamini-Schramm and local-global graph limit theories for sparse graphs. The natural limit objects are continuous actions of finitely generated groups on totally disconnected compact metric spaces. We prove that the space of weak equivalent classes of free Cantor actions is compact and contains a smallest element, as in the measurable case. We will introduce and study various notions of almost finiteness, the qualitative analogue of hyperfiniteness, for classes of bounded degree graphs. We prove the almost finiteness of a new class of étale groupoids associated to Cantor actions and construct an example of a nonamenable, almost finite totally disconnected étale groupoid, answering a query of Suzuki. Motivated by the notions and results on qualitative graph limits, in the second part of our paper we give a precise definition of constant-time distributed algorithms on sparse graphs. We construct such constant-time algorithms for various approximation problems for hyperfinite and almost finite graph classes. We also prove the Hausdorff convergence of the spectra of convergent graph sequences in the strongly almost finite category.