Log-normal statistics in e-mail communication patterns

TL;DR

This study analyzes email communication patterns, finding that interevent times follow a log-normal distribution and waiting times are best described by a mixture of two log-normals, challenging previous power-law models.

Contribution

It provides the first systematic Bayesian analysis showing log-normal distributions fit email timing data better than power-laws or queuing models.

Findings

Interevent times follow a log-normal distribution.

Waiting times are best modeled by a mixture of two log-normals.

Power-law models are rejected based on Bayesian analysis.

Abstract

Following up on Barabasi's recent letter to Nature [435, 207--211 (2005)], we systematically investigate the time series of e-mail usage for 3,188 users at a university. We focus on two quantities for each user: the time interval between consecutively sent e-mails (interevent time), and the time interval between when a user sends an e-mail and when a recipient sends an e-mail back to the original sender (waiting time). We perform a standard Bayesian model selection analysis that demonstrates that the interevent times are well-described by a single log-normal while the waiting times are better described by the superposition of two log-normals. Our analysis rejects the possibility that either measure could be described by truncated power-law distributions with exponent $\alpha \simeq 1$. We also critically evaluate the priority queuing model proposed by Barab\'asi to describe the distribution of the waiting times. We show that neither the assumptions nor the predictions of the model are plausible, and conclude that a theoretical description of human e-mail communication patterns remains an open problem.