## The Annals of Probability

- Ann. Probab.
- Volume 46, Number 1 (2018), 337-396.

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

Christian Borgs, Jennifer T. Chayes, Henry Cohn, and Yufei Zhao

#### 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.

#### Article information

**Source**

Ann. Probab., Volume 46, Number 1 (2018), 337-396.

**Dates**

Received: February 2015

Revised: March 2017

First available in Project Euclid: 5 February 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1517821225

**Digital Object Identifier**

doi:10.1214/17-AOP1187

**Mathematical Reviews number (MathSciNet)**

MR3758733

**Zentralblatt MATH identifier**

06865125

**Subjects**

Primary: 05C80: Random graphs [See also 60B20]

Secondary: 82B99: None of the above, but in this section

**Keywords**

Sparse graph limit graphon graph convergence graph quotient

#### Citation

Borgs, Christian; Chayes, Jennifer T.; Cohn, Henry; Zhao, Yufei. An $L^{p}$ theory of sparse graph convergence II: LD convergence, quotients and right convergence. Ann. Probab. 46 (2018), no. 1, 337--396. doi:10.1214/17-AOP1187. https://projecteuclid.org/euclid.aop/1517821225