  In this paper we study numerical approximations of the evolution problem for
the nonlocal $p$-Laplacian with homogeneous Neumann boundary conditions. First,
we derive a bound on the distance between two continuous-in-time trajectories
defined by two different evolution systems (i.e. with different kernels and
initial data). We then provide a similar bound for the case when one of the
trajectories is discrete-in-time and the other is continuous. In turn, these
results allow us to establish error estimates of the discretized $p$-Laplacian
problem on graphs. More precisely, for networks on convergent graph sequences
(simple and weighted graphs), we prove convergence and provide rate of
convergence of solutions for the discrete models to the solution of the
continuous problem as the number of vertices grows. We finally touch on the
limit as $p \to \infty$ in these approximations and get uniform convergence
results.
