An $L^p$ theory of sparse graph convergence II: LD convergence, quotients, and right convergence

TL;DR

This paper extends the $L^p$ theory of sparse graph limits by establishing equivalences among various convergence notions for unbounded degree graphs, broadening the scope of graph limit theory.

Contribution

It introduces new equivalence results among different convergence notions for sparse graphs with unbounded degrees, using novel techniques based on uniform upper regularity.

Findings

Proves equivalence of metric, quotient, and energy convergence under certain conditions.

Extends dense graph convergence concepts to sparse graphs with unbounded degrees.

Applies to models like stochastic block, power law, and sparse $W$-random graphs.

Abstract

We extend the $L^p$ theory of sparse graph limits, which was introduced in a companion paper, by analyzing different notions of convergence. Under suitable restrictions on node weights, we prove the equivalence of metric convergence, quotient convergence, microcanonical ground state energy convergence, microcanonical free energy convergence, and large deviation convergence. Our theorems extend the broad applicability of dense graph convergence to all sparse graphs with unbounded average degree, while the proofs require new techniques based on uniform upper regularity. Examples to which our theory applies include stochastic block models, power law graphs, and sparse versions of $W$-random graphs.