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

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.

27 pages, 8 figures

References in corpus (7)

• Mainshocks are aftershocks of conditional foreshocks: How do foreshock statistical properties emerge from aftershock laws
• Theory of Earthquake Recurrence Times
• Constraints on the Size of the Smallest Triggering Earthquake from the ETAS Model, Baath's Law, and Observed Aftershock Sequences
• Limits of Declustering Methods for Disentangling Exogenous from Endogenous Events in Time Series with Foreshocks, Main shocks and Aftershocks
• Nonlinear theory and tests of earthquake recurrence times
• Renormalization of the ETAS branching model of triggered seismicity from total to observable seismicity
• Hierarchy of Temporal Responses of Multivariate Self-Excited Epidemic Processes

Cited by in corpus (7)

• Linking structure and activity in nonlinear spiking networks
• Does the brain behave like a (complex) network? I. Dynamics
• Cumulants of Hawkes point processes
• How Much is the Whole Really More than the Sum of its Parts? 1 + 1 = 2.5: Superlinear Productivity in Collective Group Actions
• Hierarchy of Temporal Responses of Multivariate Self-Excited Epidemic Processes
• Long Horizon Forecasting With Temporal Point Processes
• Efficient Inference of Flexible Interaction in Spiking-neuron Networks