The Hydrodynamic Limit of Hawkes Processes on Adaptive Stochastic Networks

TL;DR

This paper analyzes the large-scale behavior of adaptive networks of Hawkes Processes, deriving a mean-field limit that results in fixed point equations and neural-field type equations under certain conditions, with applications in sociology, neuroscience, and epidemiology.

Contribution

It introduces a new mean-field limit for Hawkes Processes on adaptive networks, including fixed point characterizations and neural-field equations for specific edge dynamics.

Findings

The limiting probability law is a fixed point of a self-consistent Poisson process.

In neuroscience-like cases, the limit yields an autonomous neural-field equation.

The model applies to sociology, neuroscience, and epidemiology contexts.

Abstract

We determine the large size limit of a network of interacting Hawkes Processes on an adaptive network. The flipping of the node variables is taken to have an intensity given by the mean-field of the afferent edges and nodes. The flipping of the edge variables is a function of the afferent node variables. The edge variables can be either symmetric or asymmetric. This model is motivated by applications in sociology, neuroscience and epidemiology. In general, the limiting probability law can be expressed as a fixed point of a self-consistent Poisson Process with intensity function that is (i) delayed and (ii) depends on its own probability law. In the particular case that the edge flipping is only determined by the state of the pre-synaptic neuron (as in neuroscience) it is proved that one obtains an autonomous neural-field type equation for the dual evolution of the synaptic potentiation and neural potentiation.