Every property is testable on a natural class of scale-free multigraphs

In this paper, we introduce a natural class of multigraphs called hierarchical-scale-free (HSF) multigraphs, and consider constant-time testability on the class. We show that a very wide subclass, specifically, that in which the power-law exponent is greater than two, of HSF is hyperfinite. Based on this result, an algorithm for a deterministic partitioning oracle can be constructed. We conclude by showing that every property is constant-time testable on the above subclass of HSF. This algorithm utilizes findings by Newman and Sohler of STOC'11. However, their algorithm is based on the bounded-degree model, while it is known that actual scale-free networks usually include hubs, which have a very large degree. HSF is based on scale-free properties and includes such hubs. This is the first universal result of constant-time testability on the general graph model, and it has the potential to be applicable on a very wide range of scale-free networks.Comment: 13 pages, one figure. Difference from ver. 1: Definitions of HSF and SF become more general. Typos were fixe

Orbit Equivalence and Measured Group Theory

We give a survey of various recent developments in orbit equivalence and measured group theory. This subject aims at studying infinite countable groups through their measure preserving actions.Comment: 2010 Hyderabad ICM proceeding; Dans Proceedings of the International Congress of Mathematicians, Hyderabad, India - International Congress of Mathematicians (ICM), Hyderabad : India (2010

Ergodicity and indistinguishability in percolation theory

This paper explores the link between the ergodicity of the clus-ter equivalence relation restricted to its infinite locus and the indis-tinguishability of infinite clusters. It is an important element of the dictionary connecting orbit equivalence and percolation theory. This note starts with a short exposition of some standard material of these theories. Then, the classic correspondence between ergodicity and in-distinguishability is presented. Finally, we introduce a notion of strong indistinguishability that corresponds to strong ergodicity, and obtain that this strong indistinguishability holds in the Bernoulli case. We also define an invariant percolation that is not insertion-tolerant, sat-isfies the Indistinguishability Property and does not satisfy the Strong Indistinguishability Property

Local Problems on Grids from the Perspective of Distributed Algorithms, Finitary Factors, and Descriptive Combinatorics

We present an intimate connection among the following fields: (a) distributed local algorithms: coming from the area of computer science, (b) finitary factors of iid processes: coming from the area of analysis of randomized processes, (c) descriptive combinatorics: coming from the area of combinatorics and measure theory. In particular, we study locally checkable labellings in grid graphs from all three perspectives. Most of our results are for the perspective (b) where we prove time hierarchy theorems akin to those known in the field (a) [Chang, Pettie FOCS 2017]. This approach that borrows techniques from the fields (a) and (c) implies a number of results about possible complexities of finitary factor solutions. Among others, it answers three open questions of [Holroyd et al. Annals of Prob. 2017] or the more general question of [Brandt et al. PODC 2017] who asked for a formal connection between the fields (a) and (b). In general, we hope that our treatment will help to view all three perspectives as a part of a common theory of locality, in which we follow the insightful paper of [Bernshteyn 2020+]