Hidden scaling patterns and universality in written communication

The temporal statistics exhibited by written correspondence appear to be media dependent, with features which have so far proven difficult to characterize. We explain the origin of these difficulties by disentangling the role of spontaneous activity from decision-based prioritizing processes in human dynamics, clocking all waiting times through each agent's “proper time” measured by activity. This unveils the same fundamental patterns in written communication across all media (letters, email, sms), with response times displaying truncated power-law behavior and average exponents near $-\frac{3}{2}$. When standard time is used, the response time probabilities are theoretically predicted to exhibit a bimodal character, which is empirically borne out by our newly collected years-long data on email. These perspectives on the temporal dynamics of human correspondence should aid in the analysis of interaction phenomena in general, including resource management, optimal pricing and routing, information sharing, and emergency handling.

Figure 1

Two clocks for written correspondence. Representation of the communication activity along the axis of time $t$ for an agent $\mathcal{A}$. Arrows pointing into the $t$ axis mark incoming messages from the indicated agents $\mathcal{B}$, $\mathcal{C}$, etc., arrows pointing out of the $t$ axis mark response messages to the same agents. The intervals between such arrows define the interevent times (IETs) of agent $\mathcal{A}$. The response times (RTs) of $\mathcal{A}$ are defined as shown, either clocked through time $t$ (all measured in seconds), or through the activity parameter $s$ which counts the number of outgoing messages from $\mathcal{A}$ (see also the Supplemental Material). The associated RT probability distributions are denoted by $P_R\left(\tau \right)$ and ${\overline{P}}_R\left(\sigma \right)$ when clocked respectively through $s$ or $t$ (with $\tau =\Delta t$ and $\sigma =\Delta s$). The RT distributions in terms of $t$ are nonuniversal, as they depend on the communication medium and the agent; see the lower diagram for $P_R\left(\tau \right)$, showing the $t$-clocked RTs of representative agents communicating through letters (red circles) and email (blue diamonds). In contrast, we find that the same RTs, when clocked through activity $s$, give distributions as in the upper diagram for ${\overline{P}}_R\left(\sigma \right)$, which are almost superposable power laws following Eq. (4.1), with individual exponents $\alpha$ on average near $-\frac{3}{2}$ for all media (letters, email, sms).

Figure 2

Response times re-clocked through activity. Log-log plots of the response-time probability densities ${\overline{P}}_R\left(\sigma \right)$ clocked through activity $s$, for three typical agents for each different written-communication medium (logarithmic binning [42]). Red circles indicate empirical data; blue crosses represent our model predictions. (a) Letters: data from database DL1, on the correspondence of C. Darwin, A. Einstein, and S. Freud; (b) email: data from typical agents in the long term databases DE1 and DE2 (the agent in DE2, with data spanning seven years, is marked by an asterisk); (c) sms: data from typical agents in the database DS1 of Ref. [24]. The probability densities for all media are very well fitted by the truncated power laws in Eq. (4.1) with individual exponents $\alpha$ as follows (going from top to bottom in each column): 1.493, 1.565, 1.886 (letters); 1.519, 1.604, 1.539 (email); 1.491, 1.215, 1.097 (sms). See Table 1 for information on the exponents in the various databases, and the SM for more statistics. The straight dashed lines in the top diagrams are drawn to guide the eye, with the indicated exponents.

Figure 3

Response times clocked through standard time. Log-log plots of empirical response-time probability densities $P_R\left(\tau \right)$ clocked through standard time $t$ (in days for letters, and seconds for email and sms), relative to the same typical agents as in Fig. 2, for all media (logarithmic binning [42]). Red circles indicate empirical data; blue crosses represent computational predictions. See the SM for more statistics. As predicted (see the SM), $P_R\left(\tau \right)$ is affected, for small $\tau$, by the specific features of $P_I\left(\tau \right)$, while the tails of $P_R\left(\tau \right)$ for large $\tau$ follow power laws with the same exponents $\alpha$ as the associated $s$-clocked distributions ${\overline{P}}_R\left(\sigma \right)$, shown in Fig. 2. The bimodality in these $t$-clocked RT probabilities $P_R\left(\tau \right)$ is particularly evident in the RTs for email, in column (b). Also following predictions (see the SM), the crossover in $P_R\left(\tau \right)$ occurs for $\tau \sim T_I$ (green dashed vertical lines), where $T_I$ is the characteristic time of the empirical IET distributions $P_I\left(\tau \right)$ of each agent, shown in the insets. Typical empirical values are $T_I\sim {10}^4$–${10}^5$ sec for email and sms, and $T_I\sim 5$–10 days for letters.