# Topological properties of epidemic aftershock processes

## TL;DR

This paper explores the topological properties of epidemic aftershock processes modeled by the ETAS framework, linking branching process theory with seismic event clustering to better understand earthquake triggering mechanisms.

## Contribution

It extends the ETAS model from classical branching processes, analyzing how its topological properties depend on key parameters and providing a null-model for empirical seismic clustering analysis.

## Key findings

- Topological properties depend on average branching ratio and exponent ratio.
- ETAS model bridges Poisson and scale-free tree behaviors.
- Provides a null-model for empirical aftershock sequence analysis.

## 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 $\alpha/b$ 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.