从翻转过程到图幂上的动力系统
From flip processes to dynamical systems on graphons
AI总结：
本文引入翻转过程这一类随机图过程，在图幂空间构造其时间索引轨迹并证明高概率接近性，还从动力系统视角研究轨迹性质并给出周期轨迹示例。
摘要：
我们引入一类随机图过程，称之为翻转过程。每个这类过程由一个规则$\mathcal{R}:\mathcal{H}_k\rightarrow\mathcal{H}_k$给出，该规则是从所有标记$k$-顶点图到自身的函数（$k$固定）。过程以给定的$n$-顶点图$G_0$开始。每一步中，图$G_i$通过采样$G_{i-1}$的$k$个随机顶点$v_1,\ldots,v_k$，并将导出子图$F:=G_{i-1}[v_1,\ldots,v_k]$替换为$\mathcal{R}(F)$得到。这类过程包含若干先前研究过的过程，如Erdős--Rényi随机图过程和三角形移除过程。实际上，我们对翻转过程的定义更具一般性：$\mathcal{R}(F)$是$\mathcal{H}_k$上的概率分布，因此允许随机化替换。给定具有规则$\mathcal{R}$的翻转过程，我们在图幂空间中构造时间索引轨迹$\Phi:W_0\times[0,\infty)\rightarrow W_0$。我们证明，对于任意$T>0$，若以接近图幂$W_0$（在割范数下）的大型有限图$G_0$开始，以高概率，翻转过程在将时间按图的阶数平方缩放后，将停留在轨迹$(\Phi(W_0,t))_{t=0}^T$周围的细“香肠”区域内。随后，我们从动力系统的视角研究这些图幂轨迹。在诸多主题中，我们研究轨迹关于时间和初始图幂的连续性、不动点的存在性与稳定性，以及收敛速度（当无限时间极限存在时）。我们给出一个具有周期轨迹的翻转过程示例。
英文摘要：
We introduce a class of random graph processes, which we call flip processes. Each such process is given by a rule which is a function $\mathcal{R}:\mathcal{H}_k\rightarrow \mathcal{H}_k$ from all labeled $k$-vertex graphs into itself ($k$ is fixed). The process starts with a given $n$-vertex graph $G_0$. In each step, the graph $G_i$ is obtained by sampling $k$ random vertices $v_1,\ldots,v_k$ of $G_{i-1}$ and replacing the induced graph $F:=G_{i-1}[v_1,\ldots,v_k]$ by $\mathcal{R}(F)$. This class contains several previously studied processes including the Erdős--Rényi random graph process and the triangle removal process. Actually, our definition of flip processes is more general, in that $\mathcal{R}(F)$ is a probability distribution on $\mathcal{H}_k$, thus allowing randomised replacements. Given a flip process with a rule $\mathcal{R}$, we construct time-indexed trajectories $Φ:\mathcal{W}_0\times [0,\infty)\rightarrow\mathcal{W}_0$ in the space of graphons. We prove that for any $T > 0$ starting with a large finite graph $G_0$ which is close to a graphon $W_0$ in the cut norm, with high probability the flip process will stay in a thin sausage around the trajectory $(Φ(W_0,t))_{t=0}^T$ (after rescaling the time by the square of the order of the graph). These graphon trajectories are then studied from the perspective of dynamical systems. Among others topics, we study continuity properties of these trajectories with respect to time and initial graphon, existence and stability of fixed points and speed of convergence (whenever the infinite time limit exists). We give an example of a flip process with a periodic trajectory.
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