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Exact Results for the Baraba´si Model of Human Dynamics

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Exact Results for the Baraba´si Model of Human DynamicsAlexei Va´zquezDepartment of Physics and Center for Complex Networks Research, University of Notre Dame, Notre Dame, Indiana 46556, USA(Received 14 June 2005; published 6 December 2005)Human activity patterns display a bursty dynamics with interevent times following a heavy taileddistribution. This behavior has been recently shown to be rooted in the fact that humans assign their activetasks different priorities, a process that can be modeled as a priority queueing system [A.-L. Baraba´si,Nature (London) 435, 207 (2005)]. In this Letter we obtain exact results for the Baraba´si model with twotasks, calculating the priority and waiting time distribution of active tasks. We demonstrate that the modelhas a singular behavior in the extremal dynamics limit, when the highest priority task is selected first. Wefind that independently of the selection protocol, the average waiting time is smaller or equal to thenumber of active tasks, and discuss the asymptotic behavior of the waiting time distribution. These resultshave important implications for understanding complex systems with extremal dynamics.DOI: 10.1103/PhysRevLett.95.248701 PACS numbers: 89.75.Da, 02.50.Le, 89.65.EfSeveral problems of practical interest require us tounderstand human activity patterns [1–3]. Typical ex-amples are the design of telephone systems or web servers,where it is critical to know how many users would use theservice simultaneously. The traditional approach to char-acterize the timing of human activities is based in twoassumptions: the execution of each task is independentfrom the others, and each task is executed at a constantrate [1–4]. A specific task, such as sending Email ormaking phone calls, is then modeled as a Poisson process[4], characterized by a homogeneous activity pattern. Moreprecisely, the time interval between two consecutive ex-ecutions of a task follows an exponential distribution. Anincreasing amount of empirical evidence is indicating,however, that human activity patterns are rather heteroge-neous, with short periods of high activity separated by longperiods of inactivity [1,5–11]. This heterogeneity is char-acterized by a heavy tail in the distribution of the timeinterval between two consecutive executions of the giventask [5,10,11].In practice the execution of one task is not independentof the others. Humans keep track of a list of active tasksfrom where they decide what to do next, the selection ofone task implying the exclusion of the others. This picturelead Baraba´si to model the task management by a human asa queueing system, where the human plays role of theserver [5]. Queueing systems [12] have already receivedsome attention in the physics literature [13–16]. Thisinterest is motivated by the observation of a nonequilib-rium phase transition from a noncongested phase with astationary number of active tasks to a congested phasewhere the number of active tasks grows in time. In thenoncongested phase the mean waiting time before theexecution of an active task is finite. When approachingthe phase transition point the mean waiting time diverges,while it grows with time in the congested phase.The Baraba´si model belongs, however, to a new class ofqueueing models with a fixed number of active tasks. Inthis case the behavior of interest comes from the taskselection protocol. In the extremal dynamics limit, whenthe highest priority task is selected first, numerical simu-lations and heuristic arguments show that most of the tasksare executed in one step, while the waiting time distribu-tion of tasks waiting more than one step exhibits a heavytail [5]. Yet, further research is required to obtain thescaling behavior in the vicinity of this singular point.In this work we obtain exact results for the Baraba´simodel, allowing us to prove previous conjectures based onheuristic arguments and numerical simulations, and creat-ing a solid background for future research. We calculatethe priority and waiting time distribution of those tasksremaining in the list for the case of two active tasks. Wecorroborate the observation of a singular behavior in thelimit when the task with the highest priority is selectedfirst, and derive the corresponding scaling behavior. Wealso obtain an upper bound for the average waiting time,which is independent of the selection protocol. Based onthis result we discuss the asymptotic behaviors of thewaiting time distribution. All the results presented herewere checked by numerical simulations, providing a per-fect match with the theoretical curves.Baraba´si model.—The Baraba´si model is defined asfollows. A human keeps track of a list with L active tasksthat he/she must do. A priority x  0 is assigned to eachactive task when it is added to the list, with a probabilitydensity function (PDF) x . The list is started at t  0 byadding L new tasks to it. At each discrete time step t>0the task in the list with the highest priority is selected withprobability p, and with probability 1  p a task is selectedat random. The selected task is executed, removed from thelist, and a new task is added. The control parameter pinterpolates between the random selection protocol at p 0 and the highest priority first selection protocol at p  1.The numerical simulations indicate that the case L  2already exhibits the relevant features of the model [5].Furthermore, if we focus on a single task, such as sendingEmail, we can model the active tasks list as a list with twotasks, one corresponding to sending Email and the other toPRL 95, 248701 (2005)PHYSICAL REVIEW LETTERSweek ending9 DECEMBER 20050031-9007=05=95(24)=248701(4)$23.00 248701-1 © 2005 The American Physical Societydoing something else. Within this scenario the waiting timecoincides with the time between two consecutive execu-tions of the corresponding task. Thus, the L  2 caseprovides us with a minimal model to study the statisticalproperties of the time between the consecutive execution ofspecific tasks.Consider the Baraba´si model with L  2. The task thathas been just selected and its priority reassigned will becalled the new task, while the other task will be called theold task. Let x and Rx Rx0dxx be the priorityPDF and distribution function of the new task, which aregiven. In turn, let 1x; t and R1x; t Rx0dx1x; t bethe priority PDF and distribution function of the old task atthe tth step.


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