On Metzler positive systems on hypergraphs

TL;DR

This paper extends Metzler matrix theory to Metzler tensors for hypergraphs, analyzing positive dynamical systems and designing control laws for stability, with applications to ecological and epidemic models.

Contribution

It introduces Metzler tensors for hypergraphs, extending matrix analysis tools to tensor-based systems, and develops control strategies for stability in these systems.

Findings

Metzler tensors characterized and their properties described.

Control laws designed to stabilize Metzler positive systems on hypergraphs.

Applications demonstrated on Lotka-Volterra and SIS epidemic models.

Abstract

In graph-theoretical terms, an edge in a graph connects two vertices while a hyperedge of a hypergraph connects any more than one vertices. If the hypergraph's hyperedges further connect the same number of vertices, it is said to be uniform. In algebraic graph theory, a graph can be characterized by an adjacency matrix, and similarly, a uniform hypergraph can be characterized by an adjacency tensor. This similarity enables us to extend existing tools of matrix analysis for studying dynamical systems evolving on graphs to the study of a class of polynomial dynamical systems evolving on hypergraphs utilizing the properties of tensors. To be more precise, in this paper, we first extend the concept of a Metzler matrix to a Metzler tensor and then describe some useful properties of such tensors. Next, we focus on positive systems on hypergraphs associated with Metzler tensors. More importantly, we design control laws to stabilize the origin of this class of Metzler positive systems on hypergraphs. In the end, we apply our findings to two classic dynamical systems: a higher-order Lotka-Volterra population dynamics system and a higher-order SIS epidemic dynamic process. The corresponding novel stability results are accompanied by ample numerical examples.