  Generalized mass-action systems are power-law dynamical systems arising from
chemical reaction networks. Essentially, every nonnegative ODE model used in
chemistry and biology (for example, in ecology and epidemiology) and even in
economics and engineering can be written in this form. Previous results have
focused on existence and uniqueness of special steady states (complex-balanced
equilibria) for all rate constants, thereby ruling out multiple (special)
steady states. Recently, necessary conditions for linear stability have been
obtained.
  In this work, we provide sufficient conditions for the linear stability of
complex-balanced equilibria for all rate constants (and also for the
non-existence of other steady states). In particular, via sign-vector
conditions (on the stoichiometric coefficients and kinetic orders), we
guarantee that the Jacobian matrix is a $P$-matrix. Technically, we use a new
decomposition of the graph Laplacian which allows to consider orders of
(generalized) monomials. Alternatively, we use cycle decomposition which allows
a linear parametrization of all Jacobian matrices. In any case, we guarantee
stability without explicit computation of steady states. We illustrate our
results in examples from chemistry and biology: generalized Lotka-Volterra
systems and SIR models, a two-component signaling system, and an enzymatic
futile cycle.
