Introducing a Probabilistic Structure on Sequential Dynamical Systems, Simulation and Reduction of Probabilistic Sequential Networks

TL;DR

This paper introduces Probabilistic Sequential Networks (PSN), a new model for sequential dynamical systems with algebraic morphisms, establishing their categorical structure and properties related to equilibrium states.

Contribution

It defines PSN with algebraic morphisms, proves their categorical structure, and shows homomorphic PSNs share steady state probabilities under certain conditions.

Findings

PSN form a category with SDS as a subcategory

Homomorphic PSNs have identical equilibrium probabilities

Examples demonstrate morphisms, subsystems, and simulations

Abstract

A probabilistic structure on sequential dynamical systems is introduced here, the new model will be called Probabilistic Sequential Network, PSN. The morphisms of Probabilistic Sequential Networks are defined using two algebraic conditions. It is proved here that two homomorphic Probabilistic Sequential Networks have the same equilibrium or steady state probabilities if the morphism is either an epimorphism or a monomorphism. Additionally, the proof of the set of PSN with its morphisms form the category PSN, having the category of sequential dynamical systems SDS, as a full subcategory is given. Several examples of morphisms, subsystems and simulations are given.