  We consider a system of $N$ particles whose interactions are characterized by
a (weighted) graph $G^N$. Each particle is a node of the graph with an internal
state. The state changes according to Markovian dynamics that depend on the
states and connection to other particles. We study the limiting properties,
focusing on the dense graph regime, where the number of neighbors of a given
node grows with $N$. We show that when $G^N$ converges to a graphon $G$, the
behavior of the system converges to a deterministic limit, the graphon mean
field approximation. We obtain convergence rates depending on the system size
$N$ and cut-norm distance between $G^N$ and $G$. We apply the results for two
subcases: When $G^N$ is a discretization of the graph $G$ with individually
weighted edges; when $G^N$ is a random graph obtained through edge sampling
from the graphon $G$. In the case of weighted interactions, we obtain a bound
of order $O(1/N)$. In the random graph case, the error is of order
$O(\sqrt{\log(N)/N})$ with high probability. We illustrate the applicability of
our results and the numerical efficiency of the approximation through two
examples: a graph-based load-balancing model and a heterogeneous bike-sharing
system.
