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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 $p\neq\tfrac12$, wetake 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 limitintensity process solves a deterministic convolution equation and allcomponents of $\bar Z$ are independent. The fluctuations around the mean fieldlimit converge to the solution of a stochastic convolution equation. In thecritical case, $p=\tfrac12$, we rescale by $N^{1/2}$ and discuss difficulties,both heuristically and numerically.

Mean-field limits for non-linear Hawkes processes with inhibition on a Erd\H{o}s-R\'{e}nyi-graph