Differential calculus on the space of countable labelled graphs

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 $\mathcal{G}(V)$ 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.

21 pages. Final version, to appear in Advances in Operator Theory (the journal version is longer)

References in corpus (5)

• An $L^p$ theory of sparse graph convergence I: limits, sparse random graph models, and power law distributions
• An $L^p$ theory of sparse graph convergence II: LD convergence, quotients, and right convergence
• Differential Calculus on Graphon Space
• The Hoffmann-Jorgensen inequality in metric semigroups
• Probability inequalities and tail estimates for metric semigroups

Cited by in corpus (2)

• The Hoffmann-Jorgensen inequality in metric semigroups
• The Khinchin-Kahane and Levy inequalities for abelian metric groups, and transfer from normed (abelian semi)groups to Banach spaces