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Percolation theory of self-exciting temporal processes.

Daniele Notarmuzi¹, Claudio Castellano², Alessandro Flammini¹

¹Center for Complex Networks and Systems Research, Luddy School of Informatics, Computing, and Engineering, Indiana University, Bloomington, Indiana 47408, USA.

Physical Review. E

|March 19, 2021

Summary

Temporal resolution impacts activity patterns. Different avalanche distributions arise from Hawkes processes depending on how bursts are defined, revealing competition between percolation and branching process models.

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Area of Science:

Complex systems
Statistical physics
Network science

Background:

Inhomogeneous activity patterns are prevalent in natural and social phenomena.
Understanding these patterns requires analyzing bursts of activity over time.
The definition of individual activity bursts can influence observed macroscopic behaviors.

Purpose of the Study:

To investigate how temporal resolution affects the properties of inhomogeneous activity patterns.
To analyze the formation of macroscopic activity bursts using a percolation framework.
To understand the dependence of activity pattern characteristics on the resolution parameter.

Main Methods:

Utilized time series data from a self-exciting Hawkes process.
Employed a percolation framework to study burst formation.
Analyzed avalanche size and duration distributions as a function of temporal resolution.

Main Results:

The same Hawkes process yields different avalanche size and duration distributions based on temporal resolution.
Observed competition between 1D percolation and branching process universality classes.
Identified pure regimes at critical resolution points and a crossover regime with mixed behaviors.

Conclusions:

Temporal resolution is a critical factor in characterizing activity patterns.
The interplay between percolation and branching processes governs observed behaviors.
Hybrid scaling is a common outcome when temporal resolution is not precisely adjusted.