  Previous studies have inferred robust stability of reaction networks by
utilizing linear programs or iterative algorithms. Such algorithms become
tedious or computationally infeasible for large networks. In addition, they
operate like black-boxes without offering intuition for the structures that are
necessary to maintain stability. In this work, we provide several graphical
criteria for constructing robust stability certificates, checking robust
non-degeneracy, {verifying persistence}, and establishing global stability. By
characterizing a set of stability-preserving graph modifications that includes
the enzymatic modification motif, we show that the stability of arbitrarily
large nonlinear networks can be examined by simple visual inspection. We show
applications of this technique to ubiquitous motifs in systems biology such as
Post-Translational Modification (PTM) cycles, the Ribosome Flow Model (RFM),
T-cell kinetic proofreading and others.
