  We consider a class of random processes on graphs that include the discrete
Bak-Sneppen (DBS) process and the several versions of the contact process (CP),
with a focus on the former. These processes are parametrized by a probability
$0\leq p \leq 1$ that controls a local update rule. Numerical simulations
reveal a phase transition when $p$ goes from 0 to 1. Analytically little is
known about the phase transition threshold, even for one-dimensional chains. In
this article we consider a power-series approach based on representing certain
quantities, such as the survival probability or the expected number of steps
per site to reach the steady state, as a power-series in $p$. We prove that the
coefficients of those power series stabilize as the length $n$ of the chain
grows. This 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 $\mathrm{cor}(A,B) = \mathcal{O}(p^d)$. The
stabilization 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.
