---
abstract: 'Recent work has shown that the distribution of inter-event times for e-mail communication exhibits a heavy tail which is statistically consistent with a cascading Poisson process. In this work we extend the analysis to higher-order statistics, using the Fano and Allan factors to quantify the extent to which the empirical data depart from the known correlations of Poissonian statistics. The analysis shows that the higher-order statistics from the empirical data is indistinguishable from that of randomly reordered time series, thus demonstrating that e-mail correspondence is no more bursty or correlated than a Poisson process. Furthermore synthetic data sets generated by a cascading Poisson process replicate the burstiness and correlations observed in the empirical data. Finally, a simple rescaling analysis using the best-estimate rate of activity, confirms that the empirically observed correlations arise from a non-homogeneus Poisson process.'
address: |
     (a) Departamento de Física, PUC-Rio and National Institute of Science and Technology for Complex Systems, Rua Marquês de São Vicente 225, CEP 22453-900 RJ, Rio de Janeiro, Brazil\
    (b) Department of Chemical and Biological Engineering, Northwestern University, Evanston, IL 60208, USA\
    (c) Department of Physiology, Feinberg Medical School, Northwestern University, 303 East Chicago Ave. Chicago, IL 60611, USA 
author:
- 'C. Anteneodo$^{a}$, R. Dean Malmgren$^{b}$, and D. R. Chialvo$^{c}$'
title: 'Poissonian bursts in e-mail correspondence '
---

The assessment of human activity patterns is crucial for many applications, such as optimization of information traffic, service scheduling and human resource planning. In particular, the temporal dynamics of e-mail correspondence sparked recent interest [@eckmann; @barabasi0; @barabasi1; @malmgren0; @barabasi2; @malmgren1; @other] because of its importance as a communication medium and the availability of very large databases. Recent research [@barabasi0; @barabasi1; @zhou; @malmgren0; @malmgren1] has shown that the probability distribution of the time elapsed between consecutively sent e-mails by a single user exhibits heavy tails.

The origin of such heavy-tailed statistics is controversial and the focus of much debate. One explanation for the existence of the inter e-mail times distribution is a priority queuing model [@barabasi0]. Another contrasting view is a cascading Poisson process [@malmgren1; @malmgren2]. In this latter model, there is a primary non-homogeneous Poisson process, which explicitly incorporates daily and weekly modulations, each of whose events triggers a secondary process which is also Poissonian but with a much larger characteristic rate. According to this model, “bursts” of e-mail activity occur in non-overlapping homogeneous Poisson cascades (as opposed to the overlapping cascades of Ref. [@gruneis], for instance) separated by long periods of inactivity defined by the primary process. The resulting inter-event time distribution predicted by the model is therefore heavy-tailed due to the mixture of several different scales of rates of activity.

While the cascading Poisson process has been shown to be statistically consistent with empirical inter-event time distributions of several individuals [@malmgren1], it is unclear whether higher-order statistical patterns are present in the data and whether the cascading Poisson process adequately captures these patterns. Here, we quantify the burstiness and correlations in the empirical and synthetically generated point process data sets using standard statistical measures. We show that the burstiness and correlations in e-mail communication patterns are Poissonian, which are in fact reproduced by the cascading Poisson process.

![ Time series of inter e-mail intervals for a representative user, and its autocorrelation function (inset). []{data-label="fig:timeseries"}](figure1.eps){width="75.00000%"}

We study here a database considered previously [@eckmann; @barabasi0; @barabasi1; @malmgren0; @malmgren1] comprised of 3,188 e-mail accounts over an 83-day period at a European University. From this database, we restrict our analysis to the 394 “typical” users that send at least 40 messages over 83 days [@malmgren0; @malmgren1], referred here as “empirical” time series. An example is presented in Figure 1 (for User 467) which depicts some of the typical features, namely that there are long pauses between e-mails of the order of 16 and 48 hours and that the short intervals are uncorrelated as the inset shows. The second set of time series to be analyzed, was generated by the cascading Poisson model in Ref [@malmgren1], one per user, referred to here as “synthetic” time series.

For a given user, the sequence of e-mails in time can be seen as a point process [@cox]. Typically, point processes are characterized with inter-event times or counting statistics, so one might naturally characterize higher-order statistical patterns of a point process with multivariate distributions of these quantities. Multivariate distributions, however, are difficult to assess for time series with few events, as in the present case. We therefore use the Fano and Allan factors, two standard metrics for point processes that provide reliable results for time series with few events [@cox; @gruneis; @thurner; @ACh], to gain insight into the higher-order statistical structure of e-mail correspondence.

![ Fano and Allan factors as a function of the time window $T$ for User 467, a typical e-mail user (top panels). For comparison, we plot the mean and standard error of the Fano and Allan factors for 30 realizations of shuffled time series as well as the mean and standard error of the Fano and Allan factors for 30 realizations of the homogeneous Poisson process with the same rate. The bottom panels show the respective logarithmic local slopes of the empirical Fano and Allan factor curves. []{data-label="fig:af_467"}](figure2.eps){width="75.00000%"}

The Fano and Allan factors are calculated by dividing the whole observation time interval into $W$ non-overlapping time windows of equal length $T$ and counting the number of events $N_k$ in each time window, indexed by $k$. The Fano factor ($FF$) is the ratio of the variance to the mean of the number of events in each time window, $FF=( \langle N_k^2\rangle-\langle
N_k\rangle^2)/\langle N_k\rangle$, and it represents a measure of the dispersion—burstiness—of the resulting time series relative to a homogeneous Poisson process with the same rate. The Allan factor ($AF$) quantifies the difference in variance of counts of adjacent time windows, $AF=(\langle (N_{k+1}-N_k)^2 \rangle)/(2\langle N_k\rangle), $ and it is a measure of the correlation of counts between successive time windows relative to the expectation from a homogeneous Poisson process with the same rate.

If the time series were generated by a homogeneous Poisson process, then the number of counts $N_k$ in each time window would be independent and identically distributed random variables drawn from a Poisson distribution. In such a case, $FF(T)=AF(T)=1$, regardless of the time window length $T$ (Fig. \[fig:af\_467\], top panels, grey circles). Deviations from unity therefore quantify departures from Poissonian statistics. For example, $FF(T)>1$ would indicate that the time series is more bursty than expected from a homogeneous Poisson process at a particular time-scale $T$. Indeed, oftentimes researchers identify scale-free features in point processes with a power-law increase in the Fano and Allan factors [@thurner].

![ Fano and Allan factors as a function of the length of the counting time window $T$, for the empirical and synthetic time series for User 467. The synthetic time series is constructed with the same number of events as the empirical time series from a cascading Poisson process with best-estimate parameters obtained in Ref. [@malmgren0]. []{data-label="fig:af_467syn"}](figure3.eps){width="75.00000%"}

We begin by analyzing the Fano and Allan factors as a function of the length of the time window for a representative e-mail user, to be complemented later in the manuscript with the averages for the entire database. The results for this user’s activity are plotted in Fig. [\[fig:af\_467\]]{}, (top panels, black circles, User 467 ). Notice that for time windows shorter than a few minutes the point process of e-mails is essentially Poisson, denoted by the fact that both indices remain close to unity. For longer times the $FF$ and $AF$ curves noticeably depart from unity, which might suggest that there are some non-Poissonian effects in e-mail communication. Nevertheless, notice that the non-unitary region exhibits no power-law scaling. This can be also seen in the calculation of the local slope of the log-log plot, which changes continuously denoting absence of scaling (Fig. [\[fig:af\_467\]]{}, lower panels). This finding is relevant at the light of alternative models [@barabasi0] of this type of processes.

To further evaluate the empirical time series’ departure from Poissonian statistics, we analyzed the surrogate time series obtained by randomly reordering the sequence of inter-event times. If the empirical time series exhibits the same behavior in $FF$ and $AF$ as the shuffled time series, then the observed departure from Poissonian statistics is only due to the distribution of inter-event times and not due to their particular ordering; that is, the inter-event times are independent, as is illustrated in Fig. \[fig:af\_467\] (upper panel, white circles). Thus, the observed “departure” from Poissonian statistics seen here is merely an artifact of the heavy-tailed inter-event time distributions, and not of some higher-order statistical structure.

Now, we proceed to compare the Fano and Allan factors of the empirical time series of this typical user with a synthetic time series generated from the cascading Poisson process from Ref. [@malmgren1]. The model uses the best-estimate parameters for this specific user and the same number of events as the empirical time series. As Fig. \[fig:af\_467syn\] shows, the agreement between the empirical and synthetic $FF$ and $AF$ curves is remarkable, indicating that the cascading Poisson process is capturing not only the density distribution (already discussed at length in Ref. [@malmgren1]) but also the higher-order statistical features of e-mail communication. This agreement is also consistent with (and anticipated by) the similarities between the empirical and shuffled time series (Fig.  \[fig:af\_467\]).

![ Fano and Allan factors as a function of the time window $T$ for the rescaled time series of User 467. For comparison, we compute the Fano and Allan factors for 30 realizations of a homogeneous Poisson process with unit rate and the same number of events. The large fluctuations observed at long time-scales are due to poor statistics (e.g. $W=8$ in the longest time window). []{data-label="fig:af_467rsc"}](figure4.eps){width=".75\textwidth"}

The analysis up to now indicates that the origin of the observed higher order correlations is related with the non-homogeneity in the rate of e-mail activity. If this is the case, we should be able to rescale time such that the resulting point process appears to have originated from a homogeneous Poisson process. Specifically, any non-homogeneous Poisson process with occurrence rate $\rho(t)$, can be mapped onto a homogeneous Poisson process through a simple transformation of the timescale, namely $\tilde{t}=\int_0^t
\rho(t)$ [@cox]. In this new time scale, the Poisson process has unit rate, $\tilde{\rho}(\tilde{t})=1$. In the particular case of a cascading Poisson process with known best-estimate parameters $\rho(t)$ and $\rho_a$ and where we know which events are associated with which process[@malmgren1], we rescale the times between consecutive events accordingly. The results presented in Fig. \[fig:af\_467rsc\] confirm our hypothesis: the rescaled inter-event time sequence exhibit $FF$ and $AF$ values close to unity for time window lengths up to about ten times the characteristic time. By comparison, the results corresponding to thirty realizations of a homogeneous Poisson process with unit rate are presented as well.

![ Summary of agreement between the empirical time series, shuffled time series (left panels) and cascading Poisson model (right panels). The top panels show the mean and standard deviation (whiskers) Fano factor as a function of the time window length for the time series averaged over all 394 users. The bottom panels show the mean logarithmic distance between the $FF$ results for the empirical and shuffled time series (left) or cascading Poissonian model (right) for all users. []{data-label="fig:allshuf"}](figure5.eps){width="75.00000%"}

We know move beyond the analysis of User 467 to present the results of our analysis of all 394 users. The results presented in Fig. \[fig:allshuf\] show the same analysis presented in Figs. 2-4 now computed for all 394 e-mails users in the database. In Fig. \[fig:allshuf\] (left top panel) it can be seen that the mean $FF$ curve for all users replicates well the mean $FF$ curve for the shuffled time series for each of the 394 users. Thus, at the population level, higher order correlations of the process exhibit the same basic features: for short time windows (up to a few minutes), the $FF$ is unitary for all users, a fingerprint of a Poissonian process; and, for longer time windows, both the empirical and shuffled data sets exhibit a slight increase. As discussed before, this is indicative of correlations in the empirical data which are trivially related to the distribution of inter-event times. In that figure the mean logarithmic difference is also plotted, this is quantified, for each user and time window size with log-distances, $R_i(T)=\log_{10}[FF_{e,i}(T)/FF_{s,i}(T) ]$,where subscript $e$ indicates the empirical time series and $s$ the series compared, either shuffled, synthetic or homogeneous Poisson. We do not find any significant deviations between the empirical and shuffled $FF$ curves. The analysis of the entire database also confirms the similarities between the empirical and synthetic time series. This is presented in right top panel in Fig. \[fig:allshuf\], demonstrating that the mean behavior of the synthetic $FF$ curves is consistent with the empirical mean $FF$ curve with no significant systematic deviations between them.

![ Top panel: Mean and standard deviation (whiskers) of Fano factor as a function of the time window length for the rescaled time series averaged over all 394 users. Bottom panel: shows the mean distance between the $FF$ of the rescaled and that of a homogeneous Poisson process.[]{data-label="fig:allrsc"}](figure6.eps){width="75.00000%"}

Similar conclusions can be reached from the results of the $FF$ curves of rescaled data compared to a homogeneous Poisson process for all 394 users considered here. This is summarized in Fig. \[fig:allrsc\]. The statistical distance indicates that after properly rescaling time, a homogeneous Poisson process is in fact recovered.

Summarizing, the analysis described here shows that the sequences of inter e-mail times are uncorrelated for time scales shorter than a few minutes. At longer time scales, although the process exhibits correlations, they are trivial as they are found to be originated by daily and weekly cycles of activity. The trivial origin of these correlations is further confirmed by a rescaling transformation which leads to a homogeneous Poisson process. There was no evidence of scale free process at any of the levels of e-mail activity analyzed. To conclude, the present findings are consistent with the cascading non-homogeneous Poisson model as the best description for the human activity patterns behind the e-mail sequences.

[**Acknowledgements:**]{} We are grateful to J-P Eckmann for providing the data. CA acknowledges Northwestern University for the kind hospitality and Brazilian agencies CNPq and Faperj for partial financial support, DRC acknowledges support by NIH NINDS of USA (Grants NS58661).

[9000]{}

J.P. Eckmann, E. Moses, D. Sergi, [*Proc. Natl. Acad. Sci. USA*]{} [**101**]{}, 14333-14337 (2004).

A.-L. Barabási, [*Nature*]{} [**435**]{}, 207 (2005).

A. Vazquez, J.G. Oliveira, Z. Dezso, K.I Goh, I. Kondor, A.-L. Barabási, [*Phys. Rev. E*]{} [**73**]{}, 036127 (2006).

T. Zhou, X.P. Han, B.H. Wang, preprint *arXiv:physics/0801.1389v1*.

D.B. Stouffer, R.D. Malmgreen, L.A.N. Amaral, preprint *arXiv:physics/0510216*; D.B. Stouffer, R.D. Malmgreen, L.A.N. Amaral, preprint *arXiv:physics/0605027*.

A.-L. Barabási, K.-I. Goh, A. Vazquez, preprint *arXiv:physics/0511186*.

R.D. Malmgren, D.B. Stouffer, A.E. Motter, L.A.N. Amaral, [*Proc. Natl. Acad. Sci*]{} [**105**]{}, 18153-18158 (2008).

H. Ebel, L.-I. Mielsch, and S. Bornholdt, [*Phys. Rev. E*]{} [**66**]{}, 035103 (2002); R. Guimerà, L. Danon, A. Díaz-Guilera, F. Giralt, and A. Arenas, [*Phys. Rev. E*]{} [**68**]{}, 065103 (2003); G. Kossinets and D.J. Watts, [*Science*]{} [**311**]{}, 88-90 (2006); A. Vazquez, [*Physica A*]{} [**373**]{}, 747-752 (2007); A. Chatterjee, preprint *arXiv:cond-mat/0307533*. R.D. Malmgren, J.M. Hofman, L.A.N. Amaral, D.J. Watts, preprint *arXiv:0905.0106*.

F. Grüneis, M. Nakao, M. Yamamoto, [*Biol. Cybernetics*]{} [**62**]{}, 407-413 (1990); F. Grüneis, [*Physica A*]{} [**123**]{}, 149-160 (1984); F. Grüneis, H.J. Baiter, [*Physica A*]{} [**136**]{}, 432-452 (1986).

D.R. Cox, V. Isham, [*Point Processes*]{} (Chapman and Hall, London, 1980).

S. Thurner, S.B. Lowen, M.C. Feurstein, C. Heneghan, H.G. Feichtinger, M.C. Teich, [*Fractals*]{} [**5**]{}, 565-595 (1997). C. Anteneodo, D.R. Chialvo, preprint *arXiv:0904.2177*.
