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January 2018 An $L^{p}$ theory of sparse graph convergence II: LD convergence, quotients and right convergence
Christian Borgs, Jennifer T. Chayes, Henry Cohn, Yufei Zhao
Ann. Probab. 46(1): 337-396 (January 2018). DOI: 10.1214/17-AOP1187


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.


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Christian Borgs. Jennifer T. Chayes. Henry Cohn. Yufei Zhao. "An $L^{p}$ theory of sparse graph convergence II: LD convergence, quotients and right convergence." Ann. Probab. 46 (1) 337 - 396, January 2018.


Received: 1 February 2015; Revised: 1 March 2017; Published: January 2018
First available in Project Euclid: 5 February 2018

zbMATH: 06865125
MathSciNet: MR3758733
Digital Object Identifier: 10.1214/17-AOP1187

Primary: 05C80
Secondary: 82B99

Keywords: graph convergence , graph quotient , graphon , Sparse graph limit

Rights: Copyright © 2018 Institute of Mathematical Statistics


Vol.46 • No. 1 • January 2018
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