The Annals of Probability

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

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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

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

Received: February 2015
Revised: March 2017
First available in Project Euclid: 5 February 2018

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05C80: Random graphs [See also 60B20]
Secondary: 82B99: None of the above, but in this section

Sparse graph limit graphon graph convergence graph quotient


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.

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