[Paper Review] Anomalous Scaling of Offspring and Generation Numbers in Branching Processes

А. И. Саичев, Agnès Helmstetter+1 more|arXiv (Cornell University)|May 1, 2003

Complex Systems and Time Series Analysis12 citations

TL;DR

This paper studies the statistical scaling of total offspring and generations in a stochastic branching process with power-law distributed fertility, relevant to earthquake aftershock sequences. For power-law fertility with exponent γ < 2, it derives heavy-tailed asymptotic distributions: p_r(r) ~ 1/r^(1+1/γ) for total offspring and p_g(g) ~ 1/g^(1+1/(γ−1)) for total generations, validated by numerical simulations.

ABSTRACT

We consider a general stochastic branching process, which is relevant to earthquakes as well as to many other systems, and we study the distributions of the total number of offsprings (direct and indirect aftershocks in seismicity) and of the total number of generations before extinction. We apply our results to a branching model of triggered seismicity, the ETAS (epidemic-type aftershock sequence) model. The ETAS model assumes that each earthquake can trigger other earthquakes (``aftershocks''). An aftershock sequence results in this model from the cascade of aftershocks of each past earthquake. Due to the large fluctuations of the number of aftershocks triggered directly by any earthquake (``fertility''), there is a large variability of the total number of aftershocks from one sequence to another, for the same mainshock magnitude. We study the regime where the distribution of fertilities mu is characterized by a power law ~1/\mu^(1+gamma). For earthquakes, we expect such a power-law distribution of fertilities with gamma = b/alpha based on the Gutenberg-Richter magnitude distribution ~10^(-bm) and on the increase ~10^(alpha m) of the number of aftershocks with the mainshock magnitude m. We derive the asymptotic distributions p_r(r) and p_g(g) of the total number r of offsprings and of the total number g of generations until extinction following a mainshock. In the regime \gamma<2 relevant for earhquakes, for which the distribution of fertilities has an infinite variance, we find p_r(r)~1/r^(1+1/gamma) and p_g(g)~1/g^(1+1/(gamma -1)). These predictions are checked by numerical simulations.

Motivation & Objective

• To understand the statistical behavior of total offspring and generations in a stochastic branching process with highly variable triggering (fertility).
• To model earthquake aftershock sequences using the ETAS model, where each event can trigger further events through a cascade.
• To investigate the impact of power-law distributed fertilities (with exponent γ) on the scaling of total offspring and generations until extinction.
• To derive asymptotic distributions for total offspring r and total generations g in the regime γ < 2, where fertility variance is infinite.
• To validate theoretical predictions through numerical simulations of the branching process.

Proposed method

• Formalize a general stochastic branching process where each individual (earthquake) produces a random number of offspring (aftershocks) with a power-law distributed fertility μ ~ 1/μ^(1+γ).
• Apply the model to the ETAS framework, where mainshocks trigger aftershocks, and each aftershock can in turn trigger further events.
• Use extreme value and branching process theory to derive the asymptotic distributions of total offspring r and total generations g.
• Derive the scaling behavior p_r(r) ~ 1/r^(1+1/γ) for total offspring and p_g(g) ~ 1/g^(1+1/(γ−1)) for total generations under the condition γ < 2.
• Perform numerical simulations to verify the predicted power-law scaling of the distributions.
• Relate the model parameters to seismological observables: γ = b/α, where b is the Gutenberg-Richter b-value and α is the productivity scaling exponent.

Experimental results

Research questions

• RQ1 How does the distribution of total offspring scale in a branching process with power-law distributed fertility when the variance is infinite?
• RQ2 What is the asymptotic distribution of the total number of generations until extinction in such a branching process?
• RQ3 How do the scaling exponents of the offspring and generation distributions depend on the power-law exponent γ of the fertility distribution?
• RQ4 To what extent do the theoretical predictions match numerical simulations in the regime γ < 2?
• RQ5 Can the model explain the large variability in total aftershock counts across different seismic sequences despite identical mainshock magnitudes?

Key findings

• For γ < 2, the distribution of total offspring r follows a heavy tail: p_r(r) ~ 1/r^(1+1/γ), indicating extreme variability in total aftershock counts.
• The distribution of total generations g scales as p_g(g) ~ 1/g^(1+1/(γ−1)), showing anomalous scaling due to the heavy-tailed fertility distribution.
• The derived scaling laws are confirmed by numerical simulations, supporting the theoretical predictions in the regime of infinite fertility variance.
• The scaling exponent for total offspring depends inversely on γ, with heavier tails emerging as γ approaches 0.
• The generation distribution exhibits a steeper power law than the offspring distribution, reflecting the slower decay of rare, long-lasting sequences.
• The model links seismological parameters: γ = b/α, where b is the Gutenberg-Richter b-value and α is the magnitude dependence of aftershock productivity, enabling empirical validation.