Generating Functions and Stability Study of Multivariate Self-Excited Epidemic Processes

TL;DR

This paper develops a theoretical framework using generating functions to analyze the stability of multivariate self-excited Hawkes processes, revealing how mutual triggering extends system stability across various complex systems.

Contribution

It introduces a general theory of multivariate generating functions for Hawkes processes and characterizes stability domains based on the excitation network topology.

Findings

Mutual triggering significantly enlarges the stability domain.

The stability depends on the topological structure of mutual excitations.

The theory applies to diverse systems like earthquakes, social networks, and financial contagion.

Abstract

We present a stability study of the class of multivariate self-excited Hawkes point processes, that can model natural and social systems, including earthquakes, epileptic seizures and the dynamics of neuron assemblies, bursts of exchanges in social communities, interactions between Internet bloggers, bank network fragility and cascading of failures, national sovereign default contagion, and so on. We present the general theory of multivariate generating functions to derive the number of events over all generations of various types that are triggered by a mother event of a given type. We obtain the stability domains of various systems, as a function of the topological structure of the mutual excitations across different event types. We find that mutual triggering tends to provide a significant extension of the stability (or subcritical) domain compared with the case where event types are decoupled, that is, when an event of a given type can only trigger events of the same type.