Inferring transition rates on networks with incomplete knowledge

Across many fields, a problem of interest is to predict the transition rates between nodes of a network, given limited stationary state and dynamical information. We give a solution using the principle of Maximum Caliber. We find the transition rate matrix by maximizing the path entropy of a random walker on the network constrained to reproducing a stationary distribution and a few dynamical averages. A main finding here is that when constrained only by the mean jump rate, the rate matrix is given by a square-root dependence of the rate, $\omega_{ab} \propto \sqrt{p_b/p_a}$, on $p_a$ and $p_b$, the stationary state populations at nodes a and b. We give two examples of our approach. First, we show that this method correctly predicts the correlated rates in a biochemical network of two genes, where we know the exact results from prior simulation. Second, we show that it correctly predicts rates of peptide conformational transitions, when compared to molecular dynamics simulations. This method can be used to infer large numbers of rates on known networks where smaller numbers of steady-state node populations are known.