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Hierarchy of temporal responses of multivariate self-excited epidemic processes

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  • Alexander Saichev
  • Thomas Maillart
  • Didier Sornette

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. Copyright EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2013

Suggested Citation

  • Alexander Saichev & Thomas Maillart & Didier Sornette, 2013. "Hierarchy of temporal responses of multivariate self-excited epidemic processes," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 86(4), pages 1-19, April.
  • Handle: RePEc:spr:eurphb:v:86:y:2013:i:4:p:1-19:10.1140/epjb/e2013-30493-9
    DOI: 10.1140/epjb/e2013-30493-9
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    References listed on IDEAS

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    1. BAUWENS, Luc & HAUTSCH, Nikolaus, 2006. "Modelling financial high frequency data using point processes," LIDAM Discussion Papers CORE 2006080, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. Yannick Malevergne & Didier Sornette, 2006. "Extreme Financial Risks : From Dependence to Risk Management," Post-Print hal-02298069, HAL.
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    1. Didier Sornette & Thomas Maillart & Giacomo Ghezzi, 2014. "How Much Is the Whole Really More than the Sum of Its Parts? 1 ⊞ 1 = 2.5: Superlinear Productivity in Collective Group Actions," PLOS ONE, Public Library of Science, vol. 9(8), pages 1-15, August.
    2. Maillart, Thomas & Sornette, Didier, 2019. "Aristotle vs. Ringelmann: On superlinear production in open source software," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 523(C), pages 964-972.

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