Abstract
Human communication in social networks is dominated by emergent statistical laws such as nontrivial correlations and temporal clustering. Recently, we found longterm correlations in the user's activity in social communities. Here, we extend this work to study the collective behavior of the whole community with the goal of understanding the origin of clustering and longterm persistence. At the individual level, we find that the correlations in activity are a byproduct of the clustering expressed in the powerlaw distribution of interevent times of single users, i.e. short periods of many events are separated by long periods of no events. On the contrary, the activity of the whole community presents longterm correlations that are a true emergent property of the system, i.e. they are not related to the distribution of interevent times. This result suggests the existence of collective behavior, possibly arising from nontrivial communication patterns through the embedding social network.
Introduction
Various constituents of social systems have been found to follow remarkable statistical regularities. Only the recent availability of relevant data made it possible to unravel such features. Tracking bank notes or cell phones it has been shown that humans follow simple and reproducible mobility patterns^{1,2}. The communication via emails occurs in bursts, exhibiting a broad distribution of times between successive messages of individuals (interevent times)^{3,4}. Recently, we have found that the act of sending messages of individual users in two online communities present longterm correlations^{5} characterized by powerlaw correlation functions obtained via standard Detrended Fluctuation Analysis.
In the present work we examine the relation between the two empirical findings of broad interevent time distributions^{3,4} and the longterm persistence identified in the communication activity^{5}. Therefore, we investigate the communication activity of actants in a social online community with special consideration of the timing and study longterm correlations in the communication as well as clustering of successive messages. Here, the term clustering is used when the events tend to occur in burst, i.e. packages of many events are separated by long periods without events. In the case of powerlaw interevent times this takes place on all scales. In other words, in the case of (temporal) clustering, the interevent time distributions are more inhomogeneous than in the case of Poissonian statistics.
Longterm correlations have been found in the dynamics of many physical, technological and natural systems. They are characterized by a divergent correlation time, i.e. a powerlaw decaying autocorrelation function (for a review see^{6}). Such correlations lead to a pronounced mountainvalleystructure on all time scales – comprising indeterministic epochs of small and large values^{7}. This type of persistence represents a surprising regularity since it is present in many different data such as DNAsequences, human heartbeat, climatological temperature, etc.^{8,9,10}. Longterm persistence in human related data has been reported for highway traffic^{11,12}, Wikipedia access^{13}, Ethernet traffic^{14}, finance and economy^{15,16,17}, written language^{18,19}, as well as physiological records^{9,20,21}. Human brain activity^{22,23,24} and human motor activity^{25} also comprise longterm correlations as well as city growth^{26,27,28,29}, biological networks^{30} and the spreading of disease^{31}.
The distributions of interevent times (times between successive messages) have been found to be rather broad, described by powerlaws^{3}. If many short intervals are separated by few long ones, the activity as messages per unit time comprises persistence, i.e. epochs of large and small activity. Since such distributions have been described with powerlaws, we wish to investigate the relation between the longterm correlations in activity^{5} and the broad (powerlaw) distribution of interevent times^{3}. We will test two possible scenarios: (i) In the first scenario, the longterm correlations found in the communication activity^{5} result from Levy type distributions, i.e. correlations are only due to the powerlaw interevent time distribution (with exponents in the specific range)^{32}. In the second scenario, (ii) the activity comprises ‘real’ correlations, i.e. the interevent time distributions do not follow a powerlaw, but the communication activity is temporally not independent, namely longterm correlated.
We study the activity of sending messages based on detailed temporal data from a social online community and obtain the longterm correlation exponent H via DFA. The exponent H depends on the overall activity of the members; the more active the members the larger the fluctuation exponents. This exponents reaches a value H ≈ 0.90 for the most active users from an uncorrelated value H ≈ 0.5 for the less active ones. Then, we compare the value of H with the corresponding exponents of randomized data and a theoretical prediction relating correlations with clustering in the interevent times. From the consistency of the comparison of this three measures, we conclude that the longterm correlations found in the activity of sending messages for single users is a direct consequence of the powerlaw distributed interevent time of the individuals. Thus, the burstiness in the user activity explains the longterm correlations.
More interesting results are found when we consider the activity of the whole community as a sum of the activity of its members. Again we find nontrivial longterm correlations with exponents H in the same range as the individual users. However, the origin of this correlations is not related to the interevent activity. This is probed by shuffling the activity data but preserving the distribution of interevent times. In this case, this shuffling destroys the longterm correlations, implying that the correlations are not a byproduct of the broad distribution of interevent times. We conclude that the whole system acts as a true longterm correlated system where correlations are not directly related to the Levy distributions of events.
We analyze the data of an online community (www.pussokram.com, POK^{33,34,35},) covering the complete lifetime of the community over 492 days from February 2001 until June 2002. We record the activity among almost 30,000 members with more than 500,000 messages sent. This internetsite has been used for general social interactions and dating. The data consist of the time when the messages are sent and anonymous identification numbers of the senders and receivers. The data has been analyzed by us in^{5,36}. In contrast to similar network data sets consisting only of snapshots, i.e. temporally aggregated social networks expressing who sent messages to whom, the advantage of this data set is that it provides the exact time when the messages were sent. For a discussion see^{37}.
Before shutdown, the members could log in and meet virtually. In such communities, there are different ways of interacting. Usually, it is possible to choose favorites, i.e. certain members, that a person somehow feels committed to. Such platforms also offer the possibility to discuss in groups with other members about specific topics. We focus on messages sent among the members – they are similar to emails but have the advantage that they are sent within a closed community where there are no messages coming from or going outside. Figure 1 illustrates patterns of sending messages for typical single users [a–d] and for the whole community [e]. The data is publically available at http://lev.ccny.cuny.edu/∼hmakse/soft_data.html. We would like to note that we do not consider here the QX dataset which we analyzed in^{5,36}, since it covers only 2 months and the scaling of the distribution of interevent times is not reliable and we could not measure the shape of this distribution consistently.
Results
Study of correlations in individual activity
Applying DFA^{21,38,39} we have found in^{5,36} that the individual activity records, x(t), i.e. messages per unit time (records of messages per day or per week), exhibit longterm correlations. The fluctuation function provided by DFA scales as
where the exponent H is also known as the Hurst exponent. In the case of longterm correlations – which are characterized by a powerlaw decaying autocorrelation function:
where 〈·〉 denotes the average, σ_{x} is standard deviation of x(t) and γ is the correlation exponent (0 ≤ γ ≤ 1) – one finds 1/2 ≤ H ≤ 1, whereas larger exponents correspond to more pronounced longterm correlations. For uncorrelated or shortterm correlated records (γ ≥ 1, or in general γ ≥ d, d is the substrate dimension) the asymptotic fluctuation exponent is H = 1/2. In the range 0 ≤ γ ≤ 1 both exponents are related via
For an overview, we refer to^{6,39}. DFAn removes polynomial trends of the order n – 1 from the original record x(t), i.e. DFA2 copes with linear trends.
It is important to note that the DFA fluctuation function Eq. (1) is not applied to the activity x(t), but to the integrated signal y(t) = Σ^{t}x(t′). Thus, x(t) would be the analogous to the steps in a random walk and y(t) the displacement. DFA incorporates an additional detrending of the data. The integration leads to the appearance of longterm correlation when the interval between each step is powerlaw distributed. We will come back to this result when explaining the longterm correlations in terms of the burstiness.
We have measured the fluctuation exponents by applying least squares fits to log F(Δt) vs. log Δt on the scales 10 < Δt < 70 weeks conditional to the member's activity level, e.g. their total number of messages, M^{5}. Figure 2 depicts the DFA results. We find that the less active members, sending very few messages in the period of data acquisition, exhibit uncorrelated behavior. The more messages the members send, the more correlated is their activity. The fluctuation exponent H increases with M and reaches values up to H = 0.91±0.04 (value obtained for sending messages, we disregard the last points, M > 400, which have too large errors bars). The uncorrelated behavior(H ≈ 0.5) for small activity can be understood since when M ≈ 1–10 there is not enough time in the data acquisition window to capture longterm correlations. Thus, the change from H = 0.5 to H = 0.91 might be most probably due to a crossover behavior due to finite acquisition time. In^{36} we propose a model which reproduces the dependence of the fluctuation exponents on the activity level of the members. For receiving messages we find almost identical results^{36}. We use weekly resolution in order to cope with possible weekly oscillations^{4,40,41,42}.
Similar longterm correlations have been found in^{43,44} in traded values of stocks and email communication. The fluctuation exponent increases with the mean trading activity of the corresponding stock or with the average number of emails similarly as in our results.
Study of clustering in individual activity
The timing of human communication activity has been found to comprise bursts where many events occur in relatively short periods which are separated by long periods with few or no events at all. Such patterns can be characterized with the interevent times, i.e. the times, dt, between successive messages. For email communication it has been argued that their probability density follows a powerlaw,
with exponent µ ≈ 1^{3,45,46}. As an origin for such heavy tails in human dynamics a queuing model has been suggested^{3} according to which each individual performs tasks from a priority list. It has been confirmed that such a process can reproduce bursts of activity or clustering, see e.g.^{47,48}. In contrast, analyzing the same email data, a lognormal distribution has been found to be more appropriate to describe the interevent time distribution^{49,50}. We would like to remark that fitting fat tailed distributions is disputed^{51,52,53,54}. There is neither a consensus on a typical functional form nor on a proper fitting technique. Recently, a cascading Poisson process based on daily and weekly cycles has been proposed as origin of slowerthanexponential decays of P(dt)^{4,42}. We studied the cascading Poisson process in^{36}.
In^{55}, memory in the sequences of dt has been studied for different data sets, characterizing the interevent times in terms of a burstiness parameter, which is based on the distribution and in terms of a memory coefficient, which is the autocorrelation function at lag 1. In addition, the authors locate the corresponding data sets in a phase diagram defined by these two quantities. Nevertheless, we would like to note that the quantification of longterm correlations in the dt can be hindered by noise^{56,57}.
Next, we study the POK data, i.e. the interevent times dt between successive messages of individual members and relate their statistics to the longterm correlations. The finding of longterm correlations opens the question of the origin of such a persistence pattern in the social communication. From a statistical physics point of view, we consider two possible scenarios:

1
In the first scenario, the intervals between the messages follow a powerlaw^{3,58}. Accordingly, the activity pattern comprises many short intervals and few long ones, implying persistent epochs of small and large activity. This fractallike clustering in the activity can – depending on the exponent – lead to longterm correlations with H > 1/2 (see the analogous problem of the origin of longterm correlations in DNA sequences as discussed in^{59}). This scenario implies a direct link between the correlations in the activity and the distribution of interevent times which can be obtained analytically^{60}. We call this scenario “Levy correlations” since the actual activity may not be correlated perse, but correlations arise as a byproduct of integrating a signal with a powerlaw distribution of interevents in the DFA formalism.

2
In the second scenario, the intervals between the messages may or may not follow a powerlaw distribution, but the values of the interevent times are not independent of each other and comprise ‘real’ longterm persistence. For example, the distribution of interevent times could be stretched exponential (see recent work on the study of extreme events of climatological records exhibiting longterm correlations^{56,61}) and then the only way to explain longterm correlations in the activity are correlations in the interevent times. We call this scenario “true correlations” since the correlations are not related to the distribution of interevents but they reflect ‘real’ correlations in the dynamics of the communication activity.
A possible way to discern between these two scenarios is to shuffle the temporal activity, keeping the interevent distribution intact. While in the case of Levy type correlations shuffling the interevent times should not influence the longterm correlation properties of x(t), in the case of ‘real’ longterm correlations shuffling the interevent times should destroy the (asymptotic) longterm correlations since the memory is due to the arrangement of the interevent times. In what follows, we investigate the activity of individual members and the activity of the whole POK community.
Study of interevent distribution of individual members
Figure 2 exhibits the fluctuation exponents for individual members when we shuffle the data but preserve the distribution of interevent times. This is done according to the following steps: (i) Extract the set of interevent times of each user. (ii) Shuffle the extracted data. (iii) Rebuild the record of events. Since the sum of the interevent times does not add up to the entire period of data acquisition, the first event is chosen so that the remaining time is split into two, one part in the beginning and the other one at the end. (iv) Repeat the analysis.
The corresponding exponents also reach high values, almost as high as for the original data and do not drop for very active members. This agreement is a first indication of Levy correlations in single user activity.
Further evidence is found by studying the distribution of interevent times in the activity of each individual. Figure 3 shows the probability density, P(dt), of times between messages of the same users sent in the online community. A powerlaw regime of approximately two decades can be seen with an exponent µ ≈ 1.5, which differs from the exponent reported for email communication^{3,46}, i.e. µ ≈ 1. A reason for these different findings might be that in the case of^{3} only one user is considered and that µ depends on the activity level of the users, as we show below. In addition, here we study all messages from a closed community. The exponent we find is closer to the one reported for reply times (waiting times), i.e. the time individuals spend between receiving and sending to the same communication partner. For reply times of emails and land mail µ_{w} ≈ 1.5 has been reported^{3,62}.
Since we found a dependence of the fluctuation exponent H on the activity level M, i.e. the total number of messages each member sends, we suspect that also µ might depend on M. Thus, in Fig. 4 we plot for sending messages in POK (daily resolution) the P(dt) for groups of different activities, i.e. different total number of messages M. We find that for the most active members P(dt) decays rather steeply, while for the least active members P(dt) decays much slower. Due to the finite size of the data it is not quite clear which functional form the curves follow. If one assumes a powerlaw decay then the exponents are roughly in the range 1 ≤ µ ≤ 3.
As discussed above, the powerlaw distribution of interevent times, Eq. (3), can lead to longterm correlations in activity, without requiring temporal dependencies between the intervals themselves. It can be shown that the longterm persistence properties of this point process are characterized by the fluctuation exponent which theoretically depends on µ according to^{23,32,60,63}:
see Fig. 5. Apart from detrending, DFA provides an integration of the original record. So if there are long periods of no activity due to powerlaw interevent times, then, this is reflected in longterm persistence in the signal calculated by DFA. Thus, the existence of longterm correlations is due to the long periods distributed via Levy distributions as expressed by the direct relation between correlations and Levy interevent activity, Eq. (4).
Applying least squares fits (in the straight range) to the P(dt) for sending in POK (Fig. 4) we obtain values for µ as a function of the activity level M and determine the corresponding fluctuation exponents, H_{µ}, as expected from Eq. (4). We would like to note that the curves in Fig. 4 are not always straight lines leading to large uncertainty regarding the estimated values of µ.
Figure 2 depicts the fluctuation exponents H_{µ} from Eq. (4) in comparison with the values obtained from DFA. We find H ≈ H_{µ} for a big part of the M range. The exponents H_{µ} are also close to H of the shuffled records where the interevent times are preserved. The fact that when we shuffle the signal, respecting the corresponding distribution of interevent, gives rise to the same correlation function, indicates that the origin of the longterm correlation obtained in DFA are due to the Levy correlations. This is further corroborated by the agreement between H from DFA and the prediction H_{µ}. From Fig. 2 we see that the three curves are in a reasonable agreement. This supports that the correlations in single user activity can be due to the powerlaw distribution of the interevent times, which is in favor of Levy type correlations.
Study of whole community activity
Next, we investigate the activity of the community as a whole. While we have studied the activity of single users, it is of interest to investigate the activity of the whole community by considering the number of messages sent by all members in a specified period of time. Figure 1(e) shows such activity temporally aggregated to one day. The interest arises since we would like to test the existence of correlations emerging from collective behavior in the communication patterns at the level of the whole community.
For this study, we disregard who sends the messages to whom and only consider the instants when any message was sent. In order to have a sufficiently long record to apply DFA, we aggregate the data to messages per hour (instead of daily or weekly resolution). As can be seen in Fig. 1(e), the record contains oscillations^{4}. Since such periodicities lead to erratic fluctuation functions^{39}, we subtract the hourly averages over all days: x_{tot}(t) → x_{tot}(t) − 〈x_{tot}〉_{t} _{mod 24}.
The DFA fluctuation functions are shown in Fig. 6. The hump on scales around 20 hours in the results of DFA1 and DFA2 are residual oscillations, i.e. they were not completely removed. On larger scales this effect vanishes and we find a fluctuation exponent H_{tot} ≈ 0.9. The straight line in the case of DFA0 is due to the fact that the maximum exponent is 1^{39}. More importantly, when the record of the whole community is shuffled but preserving the interevent distribution, the asymptotic scaling is F ∼ (Δt)^{1/2}. That is, in contrast to the result for individual activity, when we shuffle the signal of the whole community, we obtain the uncorrelated exponent: (dashed lines in Fig. 6). The fact that the correlations vanish (H = 0.9 → H = 0.5) when the data is shuffled indicates that the longterm correlations found in the activity of the community as a whole are not due to Levy correlations. Instead, correlations in the whole community are “true correlations” appearing as a manifestation of collective behavior of the scale of the entire community.
Another surprise appears when we calculate the distribution of interevent times for the whole community. Here we define interevent the time between the sending of two consecutive messages of any member in the community. This contrasts with the same study done at the single user level (Fig. 4) when interevent is defined as the time between two events of the same user. In a sense, P(dt) for the entire community captures the collective behavior emerging from the entire community as information travels through the network.
In Fig. 7 the resulting probability density is displayed. We find a plateau up to 50 seconds followed by a powerlaw decay according to Eq. (3) with µ ≈ 2.25. Thus, the distribution of interevent activity of the community as a whole is also a Levy type like the single user activity, albeit with a larger exponent. Such a larger exponent reflects the fact that P(dt) is narrower for the community than for the individuals, as expected.
When we convert the exponent µ ≈ 2.25 to the H_{µ} through the Levy distribution model, Eq. (4), we find H_{µ} ≈ 0.88. Thus surprisingly, Eq. (4) may also explain the persistence as in the individual activity. However, the main evidence of Fig. 6, that is, the fact that the correlations vanish when we shuffle the data, probe that, even if Eq. (4) provides a good estimation of H, the longterm correlations are due to ‘real’ correlations and are not an artifact of the integration of a Levy type activity with DFA.
The longterm correlations found in the behavior of the entire community is more understandable than in the activity of single members, since the activity of the community is based on the communication patterns of the messages and information flowing through the whole system. The existence of H ≈ 0.9 at the whole level and the indications that the correlations are real ones is an interesting instance of the emergence of critical behavior in the collective dynamics of the system as a whole.
We conclude that while at the individual level we find Levy correlations, the activity of the whole community comprises ‘real’ correlations, which is due to the (possibly correlated) superposition of the individuals activity into a collective selforganized information flow in the system. Such a behavior is reminiscent of critical systems in phase transitions.
Discussion
We have studied the timing of communication in a social online community and find longterm persistence in the activity of sending messages at the single user level and the whole community level. Furthermore, we have addressed the question of the origin of these longterm correlations and whether these are Levy type or ‘real’ correlations. While in the case of Levy type correlations the interevent times need to be powerlaw distributed, ‘real’ longterm correlations are independent of the distributions, since they are due to interdependencies in the activities.
Our work, then, still leaves unanswered the question of the cause of the longterm persistence in the communication patterns at the whole community level. One possibility is that the temporal correlations are related to correlations in the network structure^{64,65}. The persistence could also be due to social effects, i.e. the dynamics in the social network^{66} induces persistent fluctuations, such as cascades. An example could be that a group of friends tries to make an appointment and therefore sends many subsequent messages in a relatively short time^{67}. After agreeing, the communication activity among the group drops. The activity patterns of individuals could be understood as a superposition of many such cascades. On the other hand, it could be purely due to a state of mind^{23}, solipsistic, emerging from moods. More research is needed to thoroughly understand the interesting properties of human activity and its motives.
In conclusion, we have determined 3 exponents to characterize communication activity: (i) H, the fluctuation exponent of the original data, (ii) H_{shuf}, the fluctuation exponent when the data is shuffled preserving the interevent times, (iii) H_{µ}, the fluctuation exponent which is expected from powerlaw distributed interevent times. We find that H ≈ H_{shuf} ≈ H_{µ} ≈ 0.9 which supports the hypothesis of Levy correlations in the single user activity, while we find H ≈ 0.9 ≠ H_{shuf} ≈ 0.5 for the collective behavior of the whole community revealing nontrivial longterm correlations and selforganization at the level of the whole system.
We should mention a third scenario which we leave for future work. It is possible that the correlations comprise more complex features. It has been shown that nonlinear correlations in multifractal data sets lead to powerlaw distributed interevent times (of peaks over threshold)^{68}. In fact, the authors of^{68} find in their Fig. 1(c) a similar dependence of µ on the total number of events as we do for H_{µ} in our Fig. 2. Additional analysis is needed to fully characterize the multifractal properties^{69,70,71} of communication activity via emails or messages in online communities.
Change history
06 November 2015
A correction has been published and is appended to both the HTML and PDF versions of this paper. The error has not been fixed in the paper.
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Acknowledgements
We thank C. Briscoe, J.F. Eichner, L.K. Gallos and H.D. Rozenfeld for useful discussions. This work was supported by National Science Foundation Grants NSFSES0624116 and NSFEF0827508 and ARL. F.L. acknowledges financial support from The Swedish Bank Tercentenary Foundation. S.H. thanks the European EPIWORK project, the Israel Science Foundation, ONR and DTRA for financial support.
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Rybski, D., Buldyrev, S., Havlin, S. et al. Communication activity in a social network: relation between longterm correlations and interevent clustering. Sci Rep 2, 560 (2012). https://doi.org/10.1038/srep00560
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DOI: https://doi.org/10.1038/srep00560
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