  We consider a finite-state, continuous-time Markov process, represented in
the "linear framework" by a directed graph with labelled edges which specifies
the infinitesimal generator of the process. If the graph is strongly connected,
the process has a unique steady-state probability distribution, $p$, which may
not be one of thermodynamic equilibrium. If the label (rate) of any edge
(transition) is perturbed, to reach the new steady-state probability
distribution $p'$, we find that the Kullback-Leibler (KL) divergence between
these distributions is bounded by the change in the thermodynamic affinity,
$\Delta A(C)$, of any cycle, $C$, that includes the altered transition,
D$_{KL}$$(p'||p) \leq |\Delta A(C)|$, irrespective of the structure of the
graph. It follows that, if an equilibrium distribution is shifted away from
equilibrium by perturbing a single rate, then the free energy difference
between these distributions is similarly bounded $F^{neq}-F^{eq}\leq |\Delta
A(C)|$. Our analysis reveals universal, energy-induced bounds on the
information-processing capabilities of Markov systems operating arbitrarily far
from thermodynamic equilibrium.
