Eur. Phys. J. B 83, 271-282 (2011)
https://doi.org/10.1140/epjb/e2011-20298-3

Generating functions and stability study of multivariate self-excited epidemic processes

¹ Department of Management, Technology and Economics, ETH Zurich, Kreuzplatz 5, 8032 Zurich, Switzerland
² Mathematical Department, Nizhny Novgorod State University, Gagarin prosp. 23, Nizhny Novgorod, 603950, Russia

Corresponding author: ^a dsornette@ethz.ch

Received: 18 April 2011
Revised: 18 July 2011
Published online: 23 September 2011

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.