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\begin{document}

\title{Topological properties of epidemic aftershock processes}



\author[1]{Jordi Baró Urbea}%
\affil[1]{Centre de Recerca Matemàtica}%


\vspace{-1em}



  
  \date{June 10, 2024}


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\selectlanguage{english}
\begin{abstract}
Earthquakes in seismological catalogs and acoustic emission events in
lab experiments can be statistically described as a linear Hawkes point
process, where the spatio-temporal rate of events is a linear
superposition of background intensity and the aftershock clusters
triggered by preceding activity. Traditionally, statistical seismology
has interpreted this model as the outcome of an epidemic branching
process, where one-to-one causal links can be established between
mainshocks and aftershocks. Declustering techniques have been used to
infer the underlying triggering trees and relate their topological
properties with epidemic branching models. Here, we review how the
standard Epidemic Type Aftershock Sequence (ETAS) model extends from the
Galton-Watson (GW) branching processes and bridges two extreme cases:
Poisson sampling and scale-free power-law trees. We report the most
essential topological properties expected in GW epidemic trees: the
branching probability, the distribution of tree size, the expected
family size, and the relation between average leaf-depth and tree size.
We find that such topological properties depend exclusively on two
sampling parameters of the standard ETAS model: the average branching
ratio N\_b and the exponent ratio \selectlanguage{greek}α/\selectlanguage{english}b determining the branching
probability distribution. From these results, one can use the
memory-less GW as a null-model for empirical triggering processes and
assess the validity of the ETAS model to reproduce the statistics of
natural and artificial catalogs.%
\end{abstract}%



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