# Sample space reducing cascading processes produce the full spectrum of scaling exponents

## TL;DR

This paper introduces a generalized cascading process based on Sample Space Reducing processes that can produce a wide range of power-law exponents, explaining phenomena like cosmic ray spectra and network cascades.

## Contribution

The authors define a new cascading process that extends SSR processes, capable of generating arbitrary power-law exponents and applying energy conservation to recover known physical spectra.

## Key findings

- Power-law distributions with exponents matching the cascade's multiplication parameter.
- Energy conservation in SSR cascades reproduces Fermi's cosmic ray spectrum with exponent -2.
- Applicable to fragmentation, network diffusion, and spreading phenomena.

## Abstract

Sample Space Reducing (SSR) processes are simple stochastic processes that offer a new route to understand scaling in path-dependent processes. Here we define a cascading process that generalises the recently defined SSR processes and is able to produce power laws with arbitrary exponents. We demonstrate analytically that the frequency distributions of states are power laws with exponents that coincide with the multiplication parameter of the cascading process. In addition, we show that imposing energy conservation in SSR cascades allows us to recover Fermi's classic result on the energy spectrum of cosmic rays, with the universal exponent -2, which is independent of the multiplication parameter of the cascade. Applications of the proposed process include fragmentation processes or directed cascading diffusion on networks, such as rumour or epidemic spreading.