  Stochastic Thermodynamics uses Markovian jump processes to model random
transitions between observable mesoscopic states. Physical currents are
obtained from anti-symmetric jump observables defined on the edges of the graph
representing the network of states. The asymptotic statistics of such currents
are characterized by scaled cumulants. In the present work, we use the
algebraic and topological structure of Markovian models to prove a gauge
invariance of the scaled cumulant-generating function. Exploiting this
invariance yields an efficient algorithm for practical calculations of
asymptotic averages and correlation integrals. We discuss how our approach
generalizes the Schnakenberg decomposition of the average entropy-production
rate, and how it unifies previous work. The application of our results to
concrete models is presented in an accompanying publication.
