单模随机图的永恒家族树与动力学
Eternal Family Trees and Dynamics on Unimodular Random Graphs
摘要：
本文主要研究单模随机图和随机网络上的协变动力学，即图或网络同构所保持的顶点集到自身的映射。这种动态在这里被称为顶点移位。本文的第一个结果是对单模随机网络上的顶点移位进行了分类。每个这样的顶点移位将顶点划分成连接的组件和箔的集合。后者是离散的类似动力学的稳定流形。分类是基于连接的组件和箔的基数。在零概率事件之前，连通分支中有三类叶理：F/F（具有无穷多个有限叶）、I/F（无限多个有限叶）和I/I（无限多个无限叶）。随机网络上顶点移动图的无限连通分支形成一棵具有一个选定端点的无限树，称为永恒家谱。这样的树可以看作是分支过程的随机扩展。幺模永恒家谱可以看作是临界分支过程的扩展。在本文中引入的后代不变的永恒家族树类，允许人们分析不一定是幺模的网络上的动力学。这些可以被看作是非必要临界分支过程的扩展。提出了几种永恒家谱的构建技术，如树木的连接或将根移动到远后代。永恒的高尔顿-沃森树和永恒的多型高尔顿-沃森树也介绍了满足额外的独立性的永恒的家庭树的特殊情况。这些例子可以表明，结果永恒的家谱统一和延伸到依赖的情况下，几个著名的定理的文献分支过程。
英文摘要：
This paper is centered on covariant dynamics on unimodular random graphs and random networks, namely maps from the set of vertices to itself which are preserved by graph or network isomorphisms. Such dynamics are referred to as vertex-shifts here. The first result of the paper is a classification of vertex-shifts on unimodular random networks. Each such vertex-shift partitions the vertices into a collection of connected components and foils. The latter are discrete analogues the stable manifold of the dynamics. The classification is based on the cardinality of the connected components and foils. Up to an event of zero probability, there are three classes of foliations in a connected component: F/F (with finitely many finite foils), I/F (infinitely many finite foils), and I/I (infinitely many infinite foils). An infinite connected component of the graph of a vertex-shift on a random network forms an infinite tree with one selected end which is referred to as an Eternal Family Tree. Such trees can be seen as stochastic extensions of branching processes. Unimodular Eternal Family Trees can be seen as extensions of critical branching processes. The class of offspring-invariant Eternal Family Trees, which is introduced in the paper, allows one to analyze dynamics on networks which are not necessarily unimodular. These can be seen as extensions of non-necessarily critical branching processes. Several construction techniques of Eternal Family Trees are proposed, like the joining of trees or moving the root to a far descendant. Eternal Galton-Watson Trees and Eternal Multi-type Galton-Watson Trees are also introduced as special cases of Eternal Family Trees satisfying additional independence properties. These examples allow one to show that the results on Eternal Family Trees unify and extend to the dependent case several well known theorems of the literature on branching processes.
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