Percolation theory of self-exciting temporal processes

TL;DR

This paper explores how the analysis of self-exciting temporal processes, modeled by Hawkes processes, varies with temporal resolution, revealing different universal behaviors and critical points through a percolation framework.

Contribution

It introduces a percolation-based approach to study the impact of temporal resolution on the properties of self-exciting processes, identifying regimes of different universality classes.

Findings

Different avalanche size and duration distributions depend on the resolution parameter.

Pure regimes correspond to critical points of percolation and branching universality classes.

A wide crossover region exhibits hybrid scaling behavior.

Abstract

We investigate how the properties of inhomogeneous patterns of activity, appearing in many natural and social phenomena, depend on the temporal resolution used to define individual bursts of activity. To this end, we consider time series of microscopic events produced by a self-exciting Hawkes process, and leverage a percolation framework to study the formation of macroscopic bursts of activity as a function of the resolution parameter. We find that the very same process may result in different distributions of avalanche size and duration, which are understood in terms of the competition between the 1D percolation and the branching process universality class. Pure regimes for the individual classes are observed at specific values of the resolution parameter corresponding to the critical points of the percolation diagram. A regime of crossover characterized by a mixture of the two universal behaviors is observed in a wide region of the diagram. The hybrid scaling appears to be a likely outcome for an analysis of the time series based on a reasonably chosen, but not precisely adjusted, value of the resolution parameter.