An Lp theory of sparse graph convergence II: LD convergence, quotients and right convergence

Christian Borgs ^[2] ; Jennifer T. Chayes ^[2] ; Henry Cohn ^[2] ; Yufei Zhao ^[1]
1. [1] Massachusetts Institute of Technology
  
  Massachusetts Institute of Technology
  
  City of Cambridge, Estados Unidos
2. [2] Microsoft Research (USA)
Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 46, Nº. 1, 2018, págs. 337-396
Idioma: inglés
DOI: 10.1214/17-AOP1187
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Resumen
- We extend the LpLp 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 WW-random graphs.