Lower bound for entropy production rate in stochastic systems far from equilibrium

We show that the Schnakenberg's entropy production rate in a master equation is lower bounded by a function of the weight of the Markov graph, here defined as the sum of the absolute values of probability currents over the edges. The result is valid for time-dependent nonequilibrium entropy production rates. Moreover, in a general framework, we prove a theorem showing that the Kullback-Leibler divergence between distributions $P(s)$ and $P'(s):=P(m(s))$, where $m$ is an involution, $m(m(s))=s$, is lower bounded by a function of the total variation of $P$ and $P'$, for any $m$. The bound is tight and it improves on Pinsker's inequality for this setup. This result illustrates a connection between nonequilibrium thermodynamics and graph theory with interesting applications.