Multivariate Hawkes processes on inhomogeneous random graphs

TL;DR

This paper studies a multivariate Hawkes process modeling neuron interactions on inhomogeneous random graphs, analyzing how spatial inhomogeneity affects system behavior and establishing large population limits.

Contribution

It introduces a framework for Hawkes processes on inhomogeneous random graphs, proving well-posedness and deriving Law of Large Numbers results for large populations.

Findings

Well-posedness of the process established

Law of Large Numbers derived as population size grows

Spatial inhomogeneity impacts long-term dynamics

Abstract

We consider a population of $N$ interacting neurons, represented by a multivariate Hawkes process: the firing rate of each neuron depends on the history of the connected neurons. Contrary to the mean-field framework where the interaction occurs on the complete graph, the connectivity between particles is given by a random possibly diluted and inhomogeneous graph where the probability of presence of each edge depends on the spatial position of its vertices. We address the well-posedness of this system and Law of Large Numbers results as $N\to\infty$. A crucial issue will be to understand how spatial inhomogeneity influences the large time behavior of the system.