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

Continuum limit of $p$-Laplacian evolution problems on graphs:$L^q$ graphons and sparse graphs