The Dynamics of Semilattice Networks

TL;DR

This paper develops analytical tools to understand the long-term behavior of semilattice networks, a class of finite dynamical systems, by classifying their limit cycles and providing bounds based on their dependency graphs.

Contribution

It offers a complete classification of limit cycles in semilattice networks with strongly connected graphs and establishes polynomial bounds for their dynamics.

Findings

Complete classification of limit cycles in certain networks

Polynomial bounds for general semilattice networks

Analytical tools linking graph properties to dynamics

Abstract

Time-discrete dynamical systems on a finite state space have been used with great success to model natural and engineered systems such as biological networks, social networks, and engineered control systems. They have the advantage of being intuitive and models can be easily simulated on a computer in most cases; however, few analytical tools beyond simulation are available. The motivation for this paper is to develop such tools for the analysis of models in biology. In this paper we have identified a broad class of discrete dynamical systems with a finite phase space for which one can derive strong results about their long-term dynamics in terms of properties of their dependency graphs. We classify completely the limit cycles of semilattice networks with strongly connected dependency graph and provide polynomial upper and lower bounds in the general case.