稀疏图收敛的$L^p$理论II：LD收敛、逆收敛和右收敛
An $L^p$ theory of sparse graph convergence II: LD convergence, quotients, and right convergence
摘要：
我们扩展了$L^p$理论的稀疏图的限制，这是介绍了一个同伴的文件，通过分析不同的概念的收敛。在适当的节点权限制下，证明了度量收敛、商收敛、微正则基态能量收敛、微正则自由能收敛和大偏差收敛的等价性.我们的定理将稠密图收敛的广泛适用性扩展到所有具有无界平均度的稀疏图，而证明需要基于一致上正则性的新技术。我们的理论适用的例子包括随机块模型，幂律图，和稀疏版本的$W$ -随机图。
英文摘要：
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.
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