  We study a multivariate, non-linear Hawkes process $Z^N$ on a
$q$-Erd\H{o}s-R\'{e}nyi-graph with $N$ nodes. Each vertex is either excitatory
(probability $p$) or inhibitory (probability $1-p$). If $p\neq\tfrac12$, we
take the mean-field limit of $Z^N$, leading to a multivariate point process
$\bar Z$. We rescale the interaction intensity by $N$ and find that the limit
intensity process solves a deterministic convolution equation and all
components of $\bar Z$ are independent. The fluctuations around the mean field
limit converge to the solution of a stochastic convolution equation. In the
critical case, $p=\tfrac12$, we rescale by $N^{1/2}$ and discuss difficulties,
both heuristically and numerically.
