also determine the stabilization rate for data exhibiting slow decay ‘at infinity’, in a suitable sense. To the best of our knowledge such results are new in the framework of discrete nonlinear diffusion equations on graphs.\
We apply an alternative approach, more local than the one in [@Mugnolo:2013], [@Hua:Mugnolo:2015] where the global arguments of semigroup theory are extended to graphs, actually in a more general setting which is out of the scope of this paper. We comment below in the Introduction on the inherent difficulty and even partial unfeasibility of a local approach in graphs. It is therefore an interesting and not trivial problem to understand how much of this body of techniques can be used in this environment. This paper can be seen as a cross section of this effort; specifically we look at the interplay between spread of mass and sup estimates, following ideas coming from the theory of continuous partial differential equations, with the differences required by the discrete character of graphs.

We recall the following notation: for any $R\in {{\boldsymbol}{N}}$, we let $$B_{R}(x_{0})
  =
  \{
  x\in V
  \mid
  d(x,x_{0})
  \le R
  \}
  \,.$$ Here $d$ is the standard combinatorial distance in $G$ so that $d$ only 