An Analytic Approach to Stability

TL;DR

This paper introduces an analytic approach to graph stability using graph limits, providing new proofs and insights into stability properties and edit distances in extremal graph theory.

Contribution

It applies graph limit theory to stability, offering a new proof of the Erdős–Simonovits Stability Theorem and analyzing properties of edit distances.

Findings

New proof of Erdős–Simonovits Stability Theorem

Fractional and combinatorial edit distances are within a constant factor

Graph limit approach simplifies stability analysis

Abstract

The stability method is very useful for obtaining exact solutions of many extremal graph problems. Its key step is to establish the stability property which, roughly speaking, states that any two almost optimal graphs of the same order $n$ can be made isomorphic by changing o(n^2) edges. Here we show how the recently developed theory of graph limits can be used to give an analytic approach to stability. As an application, we present a new proof of the Erdos-Simonovits Stability Theorem. Also, we investigate various properties of the edit distance. In particular, we show that the combinatorial and fractional versions are within a constant factor from each other, thus answering a question of Goldreich, Krivelevich, Newman, and Rozenberg.