15. Graph limits II: regularity and counting

TL;DR

Graph limits and convergence are concepts in graph theory that involve the analysis of sequences of graphons and their convergence properties.

Transcript

[SQUEAKING] PROFESSOR: Last time, we started discussing graph limits. And let me remind you some of the notions and definitions that were involved. One of the main objects in graph limits is that of a graphon, which are symmetric, measurable functions from the unit squared to the unit interval. So here, symmetric means that w of x, comma, y equals ... Read More

Key Insights

• 📈 Graph limits involve the study of graphons and their convergence properties.
• 💇 The cut distance and cut norm are used to measure the difference between graphons.
• ❓ The weak regularity lemma provides a partitioning technique to approximate graphons.
• 👍 The martingale convergence theorem proves the convergence of bounded martingales.

Questions & Answers

Q: What is a graphon?

A graphon is a symmetric, measurable function from the unit square to the unit interval.

Q: How is convergence defined in graph limits?

Convergence in graph limits is defined as the convergence of the homomorphism densities of a sequence of graphons.

Q: What is the cut distance and cut norm in graph limits?

The cut distance is a measure of difference between two graphons, while the cut norm is a measure of maximum deviation of a graphon on subsets of the unit interval.

Q: What is the weak regularity lemma in graph limits?

The weak regularity lemma provides a partitioning technique to approximate graphons by dividing each part of a given partition into at most four parts.

Q: What is the martingale convergence theorem?

The martingale convergence theorem states that every bounded martingale converges almost surely.

Summary & Key Takeaways

• Graph limits involve the study of graphons, which are symmetric, measurable functions from the unit square to the unit interval.

• Convergence in graph limits is defined as the convergence of the homomorphism densities of a sequence of graphons.

• The cut distance is a measure of difference between two graphons, while the cut norm is a measure of maximum deviation of a graphon on subsets of the unit interval.

• The weak regularity lemma provides a partitioning technique to approximate graphons, while the martingale convergence theorem proves the convergence of bounded martingales.