Nilpotent dynamics on signed interaction graphs and weak converses of Thomas' rules

TL;DR

This paper investigates how the structure of signed interaction graphs constrains the dynamics of finite systems, showing that complex behaviors cannot always be inferred from the graph alone, especially under degree-bounded conditions.

Contribution

It proves that degree-bounded systems with certain graph structures exhibit simple dynamics, and establishes weak converses of Thomas' rules relating cycles in the graph to fixed points.

Findings

If the interaction graph is not a cycle, the system can converge to a fixed point in n+1 steps.

Degree-bounded systems with complex dynamics cannot be inferred solely from the interaction graph.

Presence of positive or negative cycles in the graph influences the number of fixed points possible.

Abstract

A finite dynamical system with $n$ components is a function $f:X\to X$ where $X=X_1\times\dots\times X_n$ is a product of $n$ finite intervals of integers. The structure of such a system $f$ is represented by a signed digraph $G$, called interaction graph: there are $n$ vertices, one per component, and the signed arcs describe the positive and negative influences between them. Finite dynamical systems are usual models for gene networks. In this context, it is often assumed that $f$ is {\em degree-bounded}, that is, the size of each $X_i$ is at most the out-degree of $i$ in $G$ plus one. Assuming that $G$ is connected and that $f$ is degree-bounded, we prove the following: if $G$ is not a cycle, then $f^{n+1}$ may be a constant. In that case, $f$ describes a very simple dynamics: a global convergence toward a unique fixed point in $n+1$ iterations. This shows that, in the degree-bounded case, the fact that $f$ describes a complex dynamics {\em cannot} be deduced from its interaction graph. We then widely generalize the above result, obtaining, as immediate consequences, other limits on what can be deduced from the interaction graph only, as the following weak converses of Thomas' rules: if $G$ is connected and has a positive (negative) cycle, then $f$ may have two (no) fixed points.