A logical approach to concentration

TL;DR

This paper introduces a logical framework with aggregate operators for graphs and proves that all terms in this logic are concentrated as random variables in Erdős-Rényi models, extending existing zero-one laws.

Contribution

It develops a new real-valued logic with aggregate operators and proves concentration results for all terms within this logic on Erdős-Rényi graphs, both dense and sparse.

Findings

All terms in the logic are concentrated on Erdős-Rényi graphs.

Extends zero-one laws to a broader class of graph properties.

Provides a meta-theorem for deriving concentration results in random graphs.

Abstract

Concentration results say that a sequence of random variables becomes progressively concentrated around the mean. Such results are common in the study of functions of random graphs. We introduce a real-valued logic with various aggregate operators on graphs, including summation, and prove that every term in the language, seen as a random variable on random graphs within the classical Erd\H{o}s-R\'{e}nyi random graph model, is concentrated. We prove this for dense and sparse variants of Erd\H{o}s-R\'{e}nyi graphs. On the one hand, our results extend the line of work originating with Fagin and Glebskii et al. on zero-one laws for dense random graphs, as well as the zero-one law of Shelah and Spencer for sparse random graphs. On the other hand, they can be seen as a meta-theorem for inferring concentration results on random graphs, and we give examples of such applications.