Dominant Vertices in Regulatory Networks Dynamics

TL;DR

This paper introduces the concept of dominant vertices in discrete-time regulatory networks, demonstrating how these vertices determine the system's long-term behavior and providing algorithms to identify them.

Contribution

It defines dominant vertices in regulatory networks, proves their role in system dynamics, and offers algorithms to identify these vertices based on network structure.

Findings

Dominant vertices determine the asymptotic state of the system.

The proposed algorithm accurately identifies dominant sets in theoretical examples.

Structural properties influence the inheritance of dominance in strongly connected networks.

Abstract

Discrete-time regulatory networks are dynamical systems on directed graphs, with a structure inspired on natural systems of interacting units. There is a natural notion of determination amongst vertices, which we use to classify the nodes of the network, and to determine what we call "sets of dominant vertices". In this paper we prove that in the asymptotic regime, the projection of the dynamics on a dominant set allows us to determine the state of the whole system at all times. We provide an algorithm to find sets of dominant vertices, and we test its accuracy on three families of theoretical examples. Then, by using the same algorithm, we study the relation between the structure of the underlying network and the corresponding dominant set of vertices. We also present a result concerning the inheritability of the dominance between strongly connected networks.