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Characterization of flip process rules with the same trajectories

Eng Keat Hng

TL;DR

The paper resolves when two flip-process rules induce the same graphon trajectory, by proving that rules of the same order have identical trajectories if and only if their orbit-aggregated coefficients $q_{J,\mathcal{R}}$ agree for all $J\in\mathcal{J}_k$, equivalently if their velocity operators agree on the kernels and on graphons. The authors develop a robust framework based on rooted densities $T_{F^{a,b}}$, the velocity operator $\mathfrak{V}_{\mathcal{R}}$, and a linear-independence argument for induced densities via parametrized graphon representations and twinfree reductions. This yields a complete classification of equivalence classes of flip-process rules and shows that unique rules are exactly the symmetric deterministic rules or the order-$2$ rules. The work thus clarifies how local replacement rules shape macroscopic graphon dynamics and provides a foundation for comparing flip processes via their deterministic graphon trajectories and, potentially, via their one-step distributions.

Abstract

Garbe, Hladký, Šileikis and Skerman [Ann. Inst. Henri Poincaré Probab. Stat., 60 (2024), pp. 2878-2922] recently introduced a general class of random graph processes called flip processes and proved that the typical evolution of these discrete-time random graph processes corresponds to certain continuous-time deterministic graphon trajectories. We obtain a complete characterization of the equivalence classes of flip process rules with the same graphon trajectories. As an application, we characterize the flip process rules which are unique in their equivalence classes. These include several natural families of rules such as the complementing rules, the component completion rules, the extremist rules, and the clique removal rules.

Characterization of flip process rules with the same trajectories

TL;DR

The paper resolves when two flip-process rules induce the same graphon trajectory, by proving that rules of the same order have identical trajectories if and only if their orbit-aggregated coefficients agree for all , equivalently if their velocity operators agree on the kernels and on graphons. The authors develop a robust framework based on rooted densities , the velocity operator , and a linear-independence argument for induced densities via parametrized graphon representations and twinfree reductions. This yields a complete classification of equivalence classes of flip-process rules and shows that unique rules are exactly the symmetric deterministic rules or the order- rules. The work thus clarifies how local replacement rules shape macroscopic graphon dynamics and provides a foundation for comparing flip processes via their deterministic graphon trajectories and, potentially, via their one-step distributions.

Abstract

Garbe, Hladký, Šileikis and Skerman [Ann. Inst. Henri Poincaré Probab. Stat., 60 (2024), pp. 2878-2922] recently introduced a general class of random graph processes called flip processes and proved that the typical evolution of these discrete-time random graph processes corresponds to certain continuous-time deterministic graphon trajectories. We obtain a complete characterization of the equivalence classes of flip process rules with the same graphon trajectories. As an application, we characterize the flip process rules which are unique in their equivalence classes. These include several natural families of rules such as the complementing rules, the component completion rules, the extremist rules, and the clique removal rules.
Paper Structure (11 sections, 13 theorems, 38 equations)

This paper contains 11 sections, 13 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Main Theorem
  3. Proving Theorem \ref{['thm:rule-vel-char']}
  4. Key concepts and main lemmas
  5. Proof of Theorem \ref{['thm:rule-vel-char']}
  6. Proofs of lemmas
  7. Proving Theorem \ref{['thm:rule-unique-symm-det']}
  8. Proof of Theorem \ref{['thm:rule-unique-symm-det']}
  9. Concluding remarks
  10. Characterization by flip process distributions
  11. Trajectories with varying speeds

Key Result

Theorem 1.1

For every rule $\mathcal{R}$ there is a function $\Phi_{\mathcal{R}}^{} \colon \mathcal{W}_{0} \times [0,\infty) \to \mathcal{W}_{0}$ such that for every $\varepsilon,T>0$ and flip process $(G_{\ell})_{\ell\in\mathbb{N}_0}$ with rule $\mathcal{R}$ where $G_0$ has $n$ vertices, with probability at le

Theorems & Definitions (31)

  • Theorem 1.1: Theorem 5.1 in GarbeHladkySileikisSkerman
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1: Definition 4.4 in GarbeHladkySileikisSkerman
  • Theorem 2.2: Theorem 4.5 in GarbeHladkySileikisSkerman
  • Theorem 2.3
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3: Lemma 4.8 in GarbeHladkySileikisSkerman
  • Lemma 3.4: Lemma 4.15 in GarbeHladkySileikisSkerman
  • ...and 21 more