This talk is about a class of classical random processes on graphs that include the discrete Bak-Sneppen process, introduced in 1993, and the several versions of the contact process. These processes are parametrized by a probability 0≤p≤1 that controls a local update rule. Numerical simulations reveal a phase transition when p goes from 0 to 1, which I will discuss in the talk. Analytically little is known about the phase transition threshold, even for one-dimensional chains. In this talk we consider a power-series approach based on representing certain quantities, such as the survival probability or expected hitting times, as a power-series in p. We prove that the coefficients of those power series stabilize as the length n of the chain grows, and I will give a sketch of this proof in the talk. This stabilization of coefficients is a phenomenon that has been used in the physics community but was not yet proven. We show that for local events A, B of which the support is a distance d apart we have cor(A,B)=O(pd). The stabilization is useful because it allows for the (exact) computation of coefficients for arbitrary large systems which can then be analyzed using the wide range of existing methods of power series analysis.