Prominent examples of flip processes

Abstract

Flip processes, introduced in [Garbe, Hladk\'y, \v{S}ileikis, Skerman: From flip processes to dynamical systems on graphons], are a class of random graph processes defined using a rule which is just a function $\mathcal{R}:\mathcal{H}_k\rightarrow \mathcal{H}_k$ from all labelled graphs of a fixed order $k$ into itself. The process starts with an arbitrary 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 $G_{i-1}[v_1,\ldots,v_k]$ by $\mathcal{R}(G_{i-1}[v_1,\ldots,v_k])$. Using the formalism of dynamical systems on graphons associated to each such flip process from ibid. we study several specific flip processes, including the triangle removal flip process and its generalizations, 'extremist flip processes' (in which $\mathcal{R}(H)$ is either a clique or an independent set, depending on whether $e(H)$ has less or more than half of all potential edges), and 'ignorant flip processes' in which the output $\mathcal{R}(H)$ does not depend on $H$.