First-order limits, an analytical perspective

TL;DR

This paper introduces a new analytical framework for graph and structural limits using model theory and analysis, unifying various convergence notions and proposing new conjectures.

Contribution

It develops a general theory of structural limits based on dualities, covering all known examples and introducing new intermediate cases and conjectures.

Findings

Unified framework for structural convergence

New intermediate examples of graph limits

Proposed a grand conjecture for sparse graphs

Abstract

In this paper we present a novel approach to graph (and structural) limits based on model theory and analysis. The role of Stone and Gelfand dualities is displayed prominently and leads to a general theory, which we believe is naturally emerging. This approach covers all the particular examples of structural convergence and it put the whole in new context. As an application, it leads to new intermediate examples of structural convergence and to a "grand conjecture" dealing with sparse graphs. We survey the recent developments.