Eur. Phys. J. B (2013) 86: 124
https://doi.org/10.1140/epjb/e2013-30493-9

Regular Article

Hierarchy of temporal responses of multivariate self-excited epidemic processes

¹ Department of Management, Technology and Economics, ETH Zurich, Scheuchzerstrasse 7, 8092 Zurich, Switzerland
² Mathematical Department, Nizhny Novgorod State University, Gagarin prosp. 23, 603950 Nizhny Novgorod, Russia
³ Center for Law & Economics, ETH Zurich, Haldeneggsteig 4, 8092 Zurich, Switzerland

^a e-mail: dsornette@ethz.ch

Received: 21 June 2012
Received in final form: 8 November 2012
Published online: 1 April 2013

Abstract

Many natural and social systems are characterized by bursty dynamics, for which past events trigger future activity. These systems can be modelled by so-called self-excited Hawkes conditional Poisson processes. It is generally assumed that all events have similar triggering abilities. However, some systems exhibit heterogeneity and clusters with possibly different intra- and inter-triggering, which can be accounted for by generalization into the “multivariate” self-excited Hawkes conditional Poisson processes. We develop the general formalism of the multivariate moment generating function for the cumulative number of first-generation and of all generation events triggered by a given mother event (the “shock”) as a function of the current time t. This corresponds to studying the response function of the process. A variety of different systems have been analyzed. In particular, for systems in which triggering between events of different types proceeds through a one-dimension directed or symmetric chain of influence in type space, we report a novel hierarchy of intermediate asymptotic power law decays  ~1/t^1−(m+1)θ of the rate of triggered events as a function of the distance m of the events to the initial shock in the type space, where 0 < θ < 1 for the relevant long-memory processes characterizing many natural and social systems. The richness of the generated time dynamics comes from the cascades of intermediate events of possibly different kinds, unfolding via random changes of types genealogy.