Modelling bursty time series

TL;DR

This paper introduces a simple task-queuing model that explains bursty human activity patterns with power-law interevent times, providing exact asymptotic results and a scaling law linking interevent times and autocorrelation.

Contribution

The study presents an analytically solvable model that captures bursty dynamics and derives a universal scaling law between interevent time exponents and autocorrelation.

Findings

Interevent time distribution follows a power-law with exponents between 1 and 2.

A universal scaling law alpha + beta = 2 relates interevent and autocorrelation exponents.

Slowly decaying autocorrelation indicates long-range dependence only if the scaling law is violated.

Abstract

Many human-related activities show power-law decaying interevent time distribution with exponents usually varying between 1 and 2. We study a simple task-queuing model, which produces bursty time series due to the nontrivial dynamics of the task list. The model is characterised by a priority distribution as an input parameter, which describes the choice procedure from the list. We give exact results on the asymptotic behaviour of the model and we show that the interevent time distribution is power-law decaying for any kind of input distributions that remain normalizable in the infinite list limit, with exponents tunable between 1 and 2. The model satisfies a scaling law between the exponents of interevent time distribution (alpha) and autocorrelation function (beta): alpha + beta = 2. This law is general for renewal processes with power-law decaying interevent time distribution. We conclude that slowly decaying autocorrelation function indicates long-range dependency only if the scaling law is violated.