Prominent examples of flip processes
Abstract
Flip processes, introduced in [Garbe, Hladký, Šileikis, 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 $G_{i1}$ and replacing the induced graph $G_{i1}[v_1,\ldots,v_k]$ by $\mathcal{R}(G_{i1}[v_1,\ldots,v_k])$. 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 $H$.
 Publication:

arXiv eprints
 Pub Date:
 June 2022
 DOI:
 10.48550/arXiv.2206.03884
 arXiv:
 arXiv:2206.03884
 Bibcode:
 2022arXiv220603884A
 Keywords:

 Mathematics  Combinatorics;
 Mathematics  Probability;
 05C80;
 G.2.2