Affine relationships between steady currents

Perturbing transition rates in a steady nonequilibrium system, e.g. modelled by a Markov jump process, causes a change in the local currents. Their susceptibility is usually expressed via Green-Kubo relations or their nonequilibrium extensions. However, we may also wish to directly express the mutual relation between currents. Such a nonperturbative interrelation was discovered by P.E. Harunari et al. in [1] by applying algebraic graph theory showing the mutual linearity of currents over different edges in a graph. We give a novel and shorter derivation of that current relationship where we express the current-current susceptibility as a difference in mean first-passage times. It allows an extension to multiple currents, which remains affine but the relation is not additive.