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

TL;DR

This paper investigates the mean-field limit of a multivariate non-linear Hawkes process on an Erdős-Rényi graph with mixed excitatory and inhibitory nodes, deriving deterministic and stochastic limit equations and discussing critical case challenges.

Contribution

It introduces a new mean-field limit framework for non-linear Hawkes processes with inhibition on random graphs, including the critical case where excitatory and inhibitory nodes are balanced.

Findings

Limit intensity solves a deterministic convolution equation.

Components of the limit process are independent.

Fluctuations converge to a stochastic convolution equation.

Abstract

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.