Hierarchy of Temporal Responses of Multivariate Self-Excited Epidemic   Processes
  We present the first exact analysis of some of the temporal properties of multivariate self-excited Hawkes conditional Poisson processes, which constitute powerful representations of a large variety of systems with bursty events, for which past activity triggers future activity. The term "multivariate" refers to the property that events come in different types, with possibly different intra- and inter-triggering abilities. We develop the general formalism of the multivariate generating moment 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 $\sim 1/t^{1-(m+1)\theta}$ 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 $0 < \theta <1$ 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 a kind of inter-breeding genealogy. 
A. Saichev and D. Sornette
cond-mat.stat-mech physics.geo-ph