Modeling bursts and heavy tails in human dynamics

The dynamics of many social, technological and economic phenomena are driven by individual human actions, turning the quantitative understanding of human behavior into a central question of modern science. Current models of human dynamics, used from risk assessment to communications, assume that human actions are randomly distributed in time and thus well approximated by Poisson processes. Here we provide direct evidence that for five human activity patterns, such as email and letter based communications, web browsing, library visits and stock trading, the timing of individual human actions follow non-Poisson statistics, characterized by bursts of rapidly occurring events separated by long periods of inactivity. We show that the bursty nature of human behavior is a consequence of a decision based queuing process: when individuals execute tasks based on some perceived priority, the timing of the tasks will be heavy tailed, most tasks being rapidly executed, while a few experiencing very long waiting times. In contrast, priority blind execution is well approximated by uniform interevent statistics. We discuss two queuing models that capture human activity. The first model assumes that there are no limitations on the number of tasks an individual can hadle at any time, predicting that the waiting time of the individual tasks follow a heavy tailed distribution $P\left({\tau }_w\right)\sim {\tau }_w^{-\alpha }$ with $\alpha =3∕2$. The second model imposes limitations on the queue length, resulting in a heavy tailed waiting time distribution characterized by $\alpha =1$. We provide empirical evidence supporting the relevance of these two models to human activity patterns, showing that while emails, web browsing and library visitation display $\alpha =1$, the surface mail based communication belongs to the $\alpha =3∕2$ universality class. Finally, we discuss possible extension of the proposed queuing models and outline some future challenges in exploring the statistical mechanics of human dynamics.

Figure 1

The difference between the activity patterns predicted by a Poisson process (top) and the heavy tailed distributions observed in human dynamics (bottom). (a) Succession of events predicted by a Poisson process, which assumes that in any moment events take place with probability $q$. The horizontal axis denotes time, each vertical line corresponding to an individual event. Note that the interevent times are comparable to each other, long delays being virtually absent. (b) The absence of long delays is visible on the plot showing the delay times $\tau$ for 1000 consecutive events, the size of each vertical line corresponding to the gaps seen in (a). (c) The probability of finding exactly $n$ events within a fixed time interval is $\mathcal{P}(n;q)=e^{-qt}{\left(qt\right)}^n∕n\text{!}$, which predicts that for a Poisson process the interevent time distribution follows $P\left(\tau \right)=q{e}^{-q\tau }$, shown on a log-linear plot in (c) for the events displayed in (a) and (b). (d) The succession of events for a heavy tailed distribution. (e) The waiting time $\tau$ of 1000 consecutive events, where the mean event time was chosen to coincide with the mean event time of the Poisson process shown in (a)–(c). Note the large spikes in the plot, corresponding to very long delay times. (b) and (e) have the same vertical scale, allowing to compare the regularity of a Poisson process with the bursty nature of the heavy tailed process. (f) Delay time distribution $P\left(\tau \right)\simeq {\tau }^{-2}$ for the heavy tailed process shown in (d) and (e), appearing as a straight line with slope $-2$ on a log-log plot. The signal shown in (d)–(f) was generated using $\gamma =1$ in the stochastic priority list model discussed in Appendix .

Figure 2

(Color online) (a) The interevent time distribution between (a) two consecutive visits of a web portal by a single user; (b) two consecutive emails sent out by a user and (c) two consecutive library loans made by a single individual. For (a)–(c) we show as a straight line the $\alpha =1$ scaling. (d) The interevent time distribution between two consecutive transactions made by a stock broker. The distribution follows a power law with the exponential cutoff $P\left(\tau \right)\sim {\tau }^{-1.3}\exp (-\tau ∕{\tau }_0)$. (e)–(g) The distribution of the exponents $\left(\alpha \right)$ characterizing the interevent time distribution of users browsing the web portal (e), individual loans from the library (f) and the emails sent by different individuals (g). The exponent $\alpha$ was determined only for users whose total activity levels exceeded certain thresholds, the values used being 15 web visits (e), 15 emails (f) and 10 books (g). (h) and (l) We numerically generate for 10 000 individuals interevent time distributions following a power law with exponent $\alpha =1$. The distribution of the measured exponents follows a normal distribution similar to the distribution observed in (e)–(g). If we double the time window of the simulation (h) the deviation around the average becomes much smaller (l). (i)–(k) The distribution of the number of events in the studied systems: number of HTML hits for each user (i), the number of books checked out by each user (j) and the number of emails sent by different individuals (k), indicating that the overall activity patterns of individuals is also heavy tailed.

Figure 3

(Color online) Distribution of the response and arrival time intervals of the email user shown in Fig. 2b. (a) Given two email users A and B, the response times of user A to B are the time intervals between A receiving an email from B and A sending an email to B. The response time distribution of user A is then computed taking into account the response times to all users he/she communicates with. The continuous line is a power law fit with exponent $\alpha =1.0$. (b) Given a user A, the interarrival times are the time intervals between the two consecutive arrivals of an email to user A, independently of the sender. The arrival time distribution of user A is obtained taking into account all the interarrival times for that user. The continuous line is a power law fit with exponent 0.98. (c)–(f) The real waiting time distribution of two email users (left and right), where ${\tau }_{\text{real}}$ represents the time between the time the user first sees an email and the time she sends a reply to it. The black symbol shown in the upper left-hand corner represents the messages that were replied to right after the user has noticed it. (c) and (d) The line is a fit to (8) resulting in $p=0.999999\pm 0.000005$ for both users. (e) and (f) The line is a fit to a log-normal distribution.

Figure 4

(Color online) Distribution of the response times for the letters replied to by Einstein, Darwin, and Freud, as indicated on each plot. Note that the distributions are well approximated with a power law tail with exponent $\alpha =3∕2$. While for Darwin and Einstein the datasets provide very good statistics (the power law regime spanning 4 orders of magnitude), the plot corresponding to Freud’s responses is not so impressive, yet still being well approximated by the power law distribution. Note that while in most cases the identified reply is indeed a response to a received letter, there are exceptions as well: many of the very delayed replies represent the renewal of a long lost relationship.

Figure 5

(Color online) Waiting time distribution for tasks in the queuing model discussed in Sec. 5 with continuous priorities. The numerical simulations were performed as follows: At each step we generate an arrival ${\tau }_a$ and service time ${\tau }_s$ from an exponential distribution with rate $\lambda$ and $\mu$, respectively. If ${\tau }_a<{\tau }_s$ or there are no tasks in the queue then we add a new task to the queue, with a priority $x∊[0,1]$ from uniform distribution, and update the time $t\to t+{\tau }_a$. Otherwise, we remove from the queue the task with the largest priority and update the time $t\to t+{\tau }_s$. The waiting time distribution is plotted for three $\rho =\lambda ∕\mu$ values: $\rho =0.9$ (circles), $\rho =0.99$ (squares), and $\rho =0.999$ (diamonds). The data has been rescaled to emphasize the scaling behavior $P\left({\tau }_w\right)={\tau }_w^{-3∕2}f({\tau }_w∕{\tau }_0)$, where ${\tau }_0\sim {(1-\sqrt{\rho })}^{-2}$. In the inset we plot the distribution of waiting times for $\rho =1.1$, after collecting up to ${10}^4$ (plus) and ${10}^5$ (diamonds) executed tasks, showing that the distribution of waiting times has a power law tail even for $\rho >1$ (supercritical regime). Note, however, that in this regime a high fraction of tasks are never executed, staying forever on the priority list whose length increases linearly with time, a fact that is manifested by a shift to the right of the cutoff of the waiting time distribution.

Figure 6

(Color online) Waiting time probability distribution function for the model discussed in Sec. 6 for $L=2$ and a uniform new task priority distribution function, $\eta \left(x\right)=1$, in $0⩽x⩽1$, as obtained from (6) (lines) and numerical simulations (symbols), for $p=0.9$ (squares), $p=0.99$ (diamonds), and $p=0.999$ (triangles). The inset shows the fraction of tasks with waiting time $\tau =1$, as obtained from (6) (lines) and numerical simulations (symbols).

Figure 7

(Color online) Rescaled plot of the average priority of the lowest task priority in the list for $L=2$ (a) and $L=3$ (b) and different values of $p$ (see legend). The inset in (b) shows the exponent ${\theta }_L$ for different $L$ (points), indicating that ${\theta }_L={\theta }_3∕2^{L-3}$ for $L>2$ (continuous line). (c) Rescaled plot of the waiting time distribution for $L=3$. Similar plots are obtained for larger vales of $L$ (data not shown).

Figure 8

(Color online) Waiting time distribution for tasks in the queuing model discussed in Appendix , with a maximum queue length $L$. The waiting time distribution is plotted for three $L$ values: $L=10$ (circles), $L=100$ (squares), and $L=1000$ (diamonds). The data has been rescaled to emphasize the scaling behavior $P\left({\tau }_w\right)={\tau }_w^{-3∕2}f({\tau }_w∕{\tau }_0)$, where ${\tau }_0\sim L^2$. In the inset we plot the waiting time for $\rho ={10}^6$, showing the crossover to the model discussed in Sec. 6 in the limit $\rho \to \infty$ and $L$ fixed.