Simple dynamics on graphs

TL;DR

This paper investigates how the structure of interaction graphs influences the complexity of finite dynamical systems, proving that for systems with at least three states, the graph does not enforce complex dynamics, and providing partial results for the boolean case.

Contribution

It establishes that for systems with at least three states, the interaction graph cannot force complex dynamics, and offers partial characterizations for boolean systems.

Findings

For |A| ≥ 3, all graphs admit systems with simple convergence.

Convergence to fixed points occurs within logarithmic steps for these systems.

Identifies classes of unsigned digraphs with polynomial or linear convergence times in the boolean case.

Abstract

Does the interaction graph of a finite dynamical system can force this system to have a "complex" dynamics ? In other words, given a finite interval of integers $A$, which are the signed digraphs $G$ such that every finite dynamical system $f:A^n\to A^n$ with $G$ as interaction graph has a "complex" dynamics ? If $|A|\geq 3$ we prove that no such signed digraph exists. More precisely, we prove that for every signed digraph $G$ there exists a system $f:A^n\to A^n$ with $G$ as interaction graph that converges toward a unique fixed point in at most $\lfloor\log_2 n\rfloor+2$ steps. The boolean case $|A|=2$ is more difficult, and we provide partial answers instead. We exhibit large classes of unsigned digraphs which admit boolean dynamical systems which converge toward a unique fixed point in polynomial, linear or constant time.