  We consider the so-called \emph{discrete $p$-Laplacian}, a nonlinear
difference operator that acts on functions defined on the nodes of a possibly
infinite graph. We study the associated nonlinear Cauchy problem and identify
the generator of the associated nonlinear semigroups. We prove higher order
time regularity of the solutions. We investigate the long-time behavior of the
solutions and discuss in particular finite extinction time and conservation of
mass. Namely, on one hand, for small $p$ if an infinite graph satisfies some
isoperimetric inequality, then the solution to the parabolic $p$-Laplace
equation vanishes in finite time; on the other hand, for large $p,$ these
parabolic $p$-Laplace equations always enjoy conservation of mass.
