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We study a multivariate, non-linear Hawkes process $Z^N$ on the completegraph with $N$ nodes. Each vertex is either excitatory (probability $p$) orinhibitory (probability $1-p$). We take the mean-field limit of $Z^N$, leadingto a multivariate point process $\bar Z$. If $p\neq\tfrac12$, we rescale theinteraction intensity by $N$ and find that the limit intensity process solves adeterministic convolution equation and all components of $\bar Z$ areindependent. In the critical case, $p=\tfrac12$, we rescale by $N^{1/2}$ andobtain a limit intensity, which solves a stochastic convolution equation andall components of $\bar Z$ are conditionally independent.

Mean-field limits for non-linear Hawkes processes with excitation and inhibition