Markov modeling of online inter-arrival times ∗ Corentin Vande Kerckhove † Bal´ azs Gerencs´ er ‡ Julien M. Hendrickx §† Vincent Blondel † Abstract In this paper, we investigate the arising communication patterns on social media, and in particular the series of events happening for a single user. While the distribution of inter-event times is often assimilated to power-law density functions, a debate persists on the nature of an underlying model that explains the observed distribution. In the present, we propose an intuitive explanation to understand the observed dependence of subsequent waiting times. Our contribution is twofold. The first idea consists of separating the short waiting times – out of scope for power-law distributions – from the long ones. The model is further enhanced by introducing a two-state Markovian process to incorporate memory. 1 Introduction One of the popular research topics on networked humanity is to understand how people interact and communicate [1]. Scholars investigated the arising communication patterns, and in particular the series of events happening for a single user. The distribution of waiting times separating two consecutive events – also denoted by the inter-event distribution – is often found to have a density fitting a power-law function [2, 3, 4]. There are studies concerning other distributions, for instance about fitting the Weibull distribution for call patterns [5]. Currently we stay with power-law densities as reference for the online activities being analyzed. Also a debate persists on the nature of an underlying model that explains the observed distribution, and whether the model should incorporate an inter-event dependence. In this paper, we aim to target these questions by focusing on social media activities. We * This work was supported by the DYSCO Network (Dynamical Systems, Control, and Optimization), funded by the Interuniversity Attraction Poles Programme, initiated by the Belgian Federal Science Policy Office, and by the Concerted Research Actions (ARC) ”Large graphs and networks” and ”Revealflight” of the French Community of Belgium. † C. V. Kerckhove, J. M. Hendrickx and V. Blondel are with ICTEAM Institute, Universit´ e catholique de Louvain, Belgium corentinvdk@gmail.com, julien.hendrickx@uclouvain.be and vincent.blondel@uclouvain.be ‡ B. Gerencs´ er is with MTA Alfr´ ed R´ enyi Institute of Mathematics, Hungary and E¨ otv¨osLor´ and Uni- versity, Department of Probability and Statistics, Hungary gerencser.balazs@renyi.mta.hu He was supported by NKFIH (National Research, Development and Innovation Office) grants PD 121107 and KH 126505. § The work of J. Hendrickx is supported by a WBI.World excellence scholarship. 1 arXiv:1509.04857v3 [stat.AP] 7 Dec 2018