Mean-field limits for non-linear Hawkes processes with inhibition on a Erdős-Rényi-graph

We study a multivariate, non-linear Hawkes process $Z^N$ on a $q$-Erdős-Rényi-graph with $N$ nodes. Each vertex is either excitatory (probability $p$) or inhibitory (probability $1-p$). If $p\mathrm{\ne }\frac{1}{2}$, 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, $p=\frac{1}{2}$, we rescale by $N^{1\mathrm{/}2}$ and discuss difficulties, both heuristically and numerically.