图子图分支过程与分数同构
Graphon branching processes and fractional isomorphism
AI总结：
针对两类图子图参数化分支过程，证明其树结构分布相同的充要条件分别为图子图分数同构、相差标量倍数的分数同构，还给出图及图子图分数同构的连通分支刻画，结合相关结果揭示稠密图均匀生成树局部结构与缩放极限的关联。
摘要：
在研究非均匀随机图中的巨分支时，Bollobás、Janson和Riordan引入了一类由可能无界的图子图（graphon）参数化的分支过程。我们证明，两个此类分支过程的底层树结构具有相同分布，当且仅当对应的图子图是分数同构的——这一概念由Grebík和Rocha提出。\nHladký、Nachmias和Tran引入了另一类分支过程，用于研究逼近给定连通图子图的有限图中的均匀生成树（uniform spanning trees）。我们证明，两个此类分支过程的树结构具有相同分布，当且仅当对应的图子图是相差一个标量倍数的分数同构。结合Archer和Shalev的最新结果，这意味着若两个稠密图的均匀生成树具有相似局部结构，则它们具有相似的缩放极限。\n作为次要结果，我们给出了图及图子图的分数同构的刻画，该刻画基于它们的连通分支。
英文摘要：
In their study of the giant component in inhomogeneous random graphs, Bollobás, Janson, and Riordan introduced a class of branching processes parametrized by a possibly unbounded graphon. We prove that the tree structures underlying two such branching processes have the same distributions if and only if the corresponding graphons are fractionally isomorphic, a notion introduced by Grebík and Rocha. A different class of branching processes was introduced by Hladký, Nachmias, and Tran in relation to uniform spanning trees in finite graphs approximating a given connected graphon. We prove that that the tree structures of two such branching processes have the same distributions if and only if the corresponding graphons are fractionally isomorphic up to scalar multiple. Combined with a recent result of Archer and Shalev, this implies that if uniform spanning trees of two dense graphs have a similar local structure, they have a similar scaling limit. As a side result we give a characterization of fractional isomorphism for graphs as well as graphons in terms of their connected components.
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