AN Lp THEORY OF SPARSE GRAPH CONVERGENCE II: LD CONVERGENCE, QUOTIENTS AND RIGHT CONVERGENCE

• 成果类型: Article
• 署名作者: Borgs, Christian; Chayes, Jennifer T.; Cohn, Henry; Zhao, Yufei
• 署名单位: Microsoft; Massachusetts Institute of Technology (MIT)
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/17-AOP1187
• 发表日期: 2018
• 页码: 337-396
• 关键词: sequences

摘要

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.