# Graph-Theoretic Stability Conditions for Metzler Matrices and Monotone Systems

## TL;DR

This paper develops graph-theoretic stability conditions for Metzler matrices and monotone systems, linking cycle structures in the interconnection graph to system stability through input-to-state stability and small-gain theory.

## Contribution

It introduces novel graph-based stability criteria for Metzler systems using input-to-state gains and extends these results to nonlinear monotone systems.

## Key findings

- Cyclic small-gain theorem is necessary and sufficient for Metzler system stability.
- New graph-theoretic conditions based on sum-interconnection gains.
- Structural properties of the interconnection graph influence stability.

## Abstract

This paper studies the graph-theoretic conditions for stability of positive monotone systems. Using concepts from input-to-state stability and network small-gain theory, we first establish necessary and sufficient conditions for the stability of linear positive systems described by Metzler matrices. Specifically, we derive two sets of stability conditions based on two forms of input-to-state stability gains for Metzler systems, namely max-interconnection gains and sum-interconnection gains. Based on the max-interconnection gains, we show that the cyclic small-gain theorem becomes necessary and sufficient for the stability of Metzler systems; based on the sum-interconnection gains, we obtain novel graph-theoretic conditions for the stability of Metzler systems. All these conditions highlight the role of cycles in the interconnection graph and unveil how the structural properties of the graph affect stability. Finally, we extend our results to the nonlinear monotone system and obtain similar sufficient conditions for global asymptotic stability.