  We consider the Cauchy problem for the evolutive discrete p-Laplacian in
infinite graphs, with initial data decaying at infinity.
  We prove optimal sup and gradient bounds for nonnegative solutions, when the
initial data has finite mass, and also sharp evaluation for the confinement of
mass, i.e., the effective speed of propagation. We provide estimates for some
moments of the solution, defined using the distance from a given vertex.
  Our technique relies on suitable inequalities of Faber-Krahn type, and looks
at the local theory of continuous nonlinear partial differential equations. As
it is known, however, not all of this approach can have a direct counterpart in
graphs. A basic tool here is a result connecting the supremum of the solution
at a given positive time with the measure of its level sets at previous times.
  We also consider the case of slowly decaying initial data, where the total
mass is infinite.
