Limits of memory coefficient in measuring correlated bursts

TL;DR

This paper investigates the limitations of the memory coefficient in measuring correlated bursts in event sequences, deriving an analytical form and explaining empirical inconsistencies in human activity data.

Contribution

The authors derive an analytical expression for the memory coefficient under certain assumptions, revealing its limitations in capturing correlated bursts.

Findings

Memory coefficient is limited in measuring correlated bursts.

Analytical form explains the association between memory coefficient and burst size distribution.

Empirical inconsistencies in human activity data are reconcilable with the model.

Abstract

Temporal inhomogeneities in event sequences of natural and social phenomena have been characterized in terms of interevent times and correlations between interevent times. The inhomogeneities of interevent times have been extensively studied, while the correlations between interevent times, often called correlated bursts, are far from being fully understood. For measuring the correlated bursts, two relevant approaches were suggested, i.e., memory coefficient and burst size distribution. Here a burst size denotes the number of events in a bursty train detected for a given time window. Empirical analyses have revealed that the larger memory coefficient tends to be associated with the heavier tail of burst size distribution. In particular, empirical findings in human activities appear inconsistent, such that the memory coefficient is close to $0$, while burst size distributions follow a power law. In order to comprehend these observations, by assuming the conditional independence between consecutive interevent times, we derive the analytical form of the memory coefficient as a function of parameters describing interevent time and burst size distributions. Our analytical result can explain the general tendency of the larger memory coefficient being associated with the heavier tail of burst size distribution. We also find that the apparently inconsistent observations in human activities are compatible with each other, indicating that the memory coefficient has limits to measure the correlated bursts.