Impact of interactions on human dynamics

Queueing theory has been recently proposed as a framework to model the heavy tailed statistics of human activity patterns. The main predictions are the existence of a powerlaw distribution for the interevent time of human actions and two decay exponents α = 1 and α = 3/2. Current models lack, however, a key aspect of human dynamics, i.e. several tasks require, or are determined by, interactions between individuals. Here we introduce a minimal queueing model of human dynamics that already takes into account humanhuman interactions. To achieve large scale simulations, we obtain a coarse-grained version of the model, allowing us to reach large interevent times and reliable scaling exponents estimations. Using thiswe show that the interevent distribution of interacting tasks exhibit the scaling exponents α = 2, 3/2, and a series of numerable values between 3/2 and 1. This work demonstrates that, within the context of queueing models of human dynamics, interactions change the exponent of the power-law distributed interevent times. Beyond the study of human dynamics, these results are relevant to systems where the event of interest consists of the simultaneous occurrence of two (or more) events. © 2008 Elsevier B.V. All rights reserved. Understanding the timing of human activities is extremely important to model human related activities, such as communication systems [1] and the spreading of computer viruses [2]. In the recent yearswe have experienced an increased research activity in this areamotivated by the increased availability of empirical data. We now count withmeasurements of human activities covering several individuals and several events per individual [3–7]. Thanks to this datawe are in a position to investigate the laws and patterns of human dynamics, using a scientific approach. Barabási has taken an important step in this direction reconsidering queueing theory [8,9] as a framework to model human dynamics [5]. Within this framework, the to do list of an individual is modeled as a finite length queue with a task selection protocol, such as highest priority first. The main predictions are the existence of a power law distribution of interevent times Pτ ∼ τ and two universality classes characterized by exponents α = 1 [5,10,11] and α = 3/2 [6,11]. These universality classes have been corroborated by empirical data for email [5,11] and regular mail communications [6, 11], respectively, motivating further theoretical research [12–14]. The models proposed so far have been limited, however, to single individual dynamics. In practice people are connected in social networks and several of their activities are not performed independently. This reality forces us to model human dynamics in the presence of interactions between individuals. Our past experiencewith phase transitions has shown us that interactions and their nature are a key factor determining the universality classes and their corresponding scaling exponents [15]. Furthermore, beyond the study of human dynamics, there are several systems where the event of interest consists of the simultaneous occurrence of two (or more) events. For example, collective phenomena in disordered media, such as the interaction of two (or more) particles in cluster formation. ∗ Corresponding author. E-mail address: [email protected] (J.G. Oliveira). 0378-4371/$ – see front matter© 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2008.08.022 188 J.G. Oliveira, A. Vazquez / Physica A 388 (2009) 187–192 Fig. 1. System of two agents with a common interacting task I and an aggregate task O representing a set of individual tasks. Fig. 2. Probability distribution of the interevent time τ of the interacting task I, as obtained from the direct numerical simulations of the model. Each dataset was obtained after 1011 model time steps, corresponding with total number of I plus O task executions. Note that as LA and/or LB increases it becomes computationally harder to have a good estimate of Pτ because the execution of the I task becomes less frequent. The inset shows the distribution for L = 3 as obtained from the original model with 1012 steps (green diamonds), and the coarse-grained model with N = 109 (red plus), derived to obtain more reliable estimation of the exponents. To investigate the impact of human-human interactions on the timing of their activities, we consider a minimal model consisting of two agents, A and B (Fig. 1). Each agent ismodeled by a priority list containing two tasks, interacting task (I) and aggregate non-interacting task (O). The interacting task models a common activity such as meeting each other, requiring the simultaneous execution of that task by both agents. On the other hand, the non-interacting task represents an aggregate meta-activity accounting for all other tasks the agents execute, which do not require an interaction between them. To each task, we assign random priorities xij (i = I,O; j = A, B) extracted from a probability density function (pdf) fij(x) (see Fig. 1). The rules governing the dynamics are as follows. Initial condition:We start with a random initial condition, assigning a priority to the I and O tasks from their corresponding pdf. Updating step: At each time step, both agents select the task with higher priority in their list. If (i) both agents select the interacting task then it is executed, (ii) otherwise each agent executes the O task, representing the execution of any of their non-interacting tasks. Our aim is to determine the impact of the interaction between the agents and the shape of fij(x) on the scaling exponent α of the interevent time distribution of the interacting task I. For simplicity, we focus on the following priority distribution. Consider the case where each agent has Lj (j = A, B) tasks, one I task and Lj − 1 non-interacting tasks, their priorities following a uniform distribution in the interval [0, 1]. The pdf of the highest priority among Lj− 1 tasks is in this case given by (Lj − 1)xLj−2, resulting in fij(x) = { 1, i = I (Lj − 1)xLj−2, i = O. (1) This example shows that the priorities pdf of task I and O are in general different. All the results shown belowwere obtained using the pdf in Eq. (1). To investigate the interevent time distribution, we perform extensive numerical simulations. Fig. 2 shows the interevent time distribution as obtained fromdirect simulations of themodel introduced above. It becomes clear that for large LA and/or LB we do not obtain a good statistics, even after waiting for 1011 updating steps. This observation is a consequence of the behavior of fOj(x) when LA and/or LB are large (Fig. 3). Focusing on agent A, as LA increases fOA(x) gets more concentrated J.G. Oliveira, A. Vazquez / Physica A 388 (2009) 187–192 189 Fig. 3. Probability density function of the non-interacting aggregate task priority of user A, as obtained from Eq. (1). With increasing the queue length LA , fOA(x) concentrates more and more in the vicinity of x = 1 . around priority one, while the priority of the I task remains uniformly spread between zero and one. This fact results in increasingly large interevent times between the execution of the I task. To speed-off the numerical simulations, we derive a coarse-grained version of the model, allowing us to analyze the scaling behavior of the interevent time distribution over several orders of magnitude (inset of Fig. 2). We start by noticing that, given (xIA, xIB), the joint pdf of (xOA, xOB) factorizes and the probability q(xIA, xIB) that both agents execute I right after O is given by