Differential calculus on the space of countable labelled graphs

TL;DR

This paper develops a mathematical framework for differentiating functions on the space of countable labeled graphs, connecting graph limits with p-adic integers and the Cantor set, and establishing calculus analogues.

Contribution

It introduces a model for calculus on graph space, including derivatives and automorphisms, extending graph limit theory with differential structures.

Findings

Established analogues of convergence and homomorphism densities

Developed calculus tools like derivatives and tests for graph functions

Classified automorphisms preserving the graph space structure

Abstract

The study of very large graphs is a prominent theme in modern-day mathematics. In this paper we develop a rigorous foundation for studying the space of finite labelled graphs and their limits. These limiting objects are naturally countable graphs, and the completed graph space $\mathscr{G}(V)$ is identified with the 2-adic integers as well as the Cantor set. The goal of this paper is to develop a model for differentiation on graph space in the spirit of the Newton-Leibnitz calculus. To this end, we first study the space of all finite labelled graphs and their limiting objects, and establish analogues of left-convergence, homomorphism densities, a Counting Lemma, and a large family of topologically equivalent metrics on labelled graph space. We then establish results akin to the First and Second Derivative Tests for real-valued functions on countable graphs, and completely classify the permutation automorphisms of graph space that preserve its topological and differential structures.