  We study Markov population processes on large graphs, with the local state
transition rates of a single vertex being linear function of its neighborhood.
A simple way to approximate such processes is by a system of ODEs called the
homogeneous mean-field approximation (HMFA). Our main result is showing that
HMFA is guaranteed to be the large graph limit of the stochastic dynamics on a
finite time horizon if and only if the graph-sequence is quasi-random. Explicit
error bound is given and being of order $\frac{1}{\sqrt{N}}$ plus the largest
discrepancy of the graph. For Erd\H{o}s R\'{e}nyi and random regular graphs we
show an error bound of order the inverse square root of the average degree. In
general, diverging average degrees is shown to be a necessary condition for the
HMFA to be accurate. Under special conditions, some of these results also apply
to more detailed type of approximations like the inhomogenous mean field
approximation (IHMFA). We pay special attention to epidemic applications such
as the SIS process.
