Exact Results for the Barabási Model of Human Dynamics

Human activity patterns display a bursty dynamics with interevent times following a heavy tailed distribution. This behavior has been recently shown to be rooted in the fact that humans assign their active tasks different priorities, a process that can be modeled as a priority queueing system [A.-L. Barabási, Nature (London) 435, 207 (2005)]. In this Letter we obtain exact results for the Barabási model with two tasks, calculating the priority and waiting time distribution of active tasks. We demonstrate that the model has a singular behavior in the extremal dynamics limit, when the highest priority task is selected first. We find that independently of the selection protocol, the average waiting time is smaller or equal to the number of active tasks, and discuss the asymptotic behavior of the waiting time distribution. These results have important implications for understanding complex systems with extremal dynamics.

Figure 1

Old task priority distribution for the case of a uniform new task priority distribution function, $\rho (x)=1$ and $R(x)=x$ in $0\le x\le 1$, as obtained from (4) (lines) and numerical simulations (points). The case $p=0$ corresponds with the random selection protocol with $R_1(x)=R(x)=x$.

Figure 2

Waiting time probability distribution for the case of a uniform new task priority distribution, $\rho (x)=1$ and $R(x)=x$ in $0\le x\le 1$, as obtained from (8) (pluses) and numerical simulations (open symbols). The inset shows the fraction of tasks with waiting time $\tau =1$ as a function of $p$, as obtained from (8) (line) and numerical simulations (points).

Figure 3

Average waiting time of executed tasks vs the list size for the case of a uniform new task priority distribution function, $\rho (x)=1$ and $R(x)=x$ in $0\le x\le 1$, as obtained from (16) (lines) and numerical simulations (points).