  In this paper we study continuum limits of the discretized $p$-Laplacian
evolution problem on sparse graphs with homogeneous Neumann boundary
conditions. This extends the results of [24] to a far more general class of
kernels, possibly singular, and graph sequences whose limit are the so-called
$L^q$-graphons. More precisely, we derive a bound on the distance between two
continuous-in-time trajectories defined by two different evolution systems
(i.e. with different kernels, second member and initial data). Similarly, we
provide a bound in the case that one of the trajectories is discrete-in-time
and the other is continuous. In turn, these results lead us to establish error
estimates of the full discretization of the $p$-Laplacian problem on sparse
random graphs. In particular, we provide rate of convergence of solutions for
the discrete models to the solution of the continuous problem as the number of
vertices grows.
