  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.
