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Heterogenous scaling in interevent time of on-line bookmarking

In this paper, we study the statistical properties of bookmarking behaviors in Delicious.com. We find that the interevent time distributions of bookmarking decays powerlike as interevent time increases at both individual and population level. Remarkably, we observe a significant change in the exponent when interevent time increases from intra-day to inter-day range. In addition, dependence of exponent on individual Activity is found to be different in the two ranges. These results suggests that mechanisms driving human actions are different in intra- and inter-day range. Instead of monotonically increasing with Activity, we find that inter-day exponent peaks at value around 3. We further show that less active users are more likely to resemble poisson process in bookmarking. Based on the temporal-preference model, preliminary explanations for this dependence have been given . Finally, a universal behavior in inter-day scale is observed by considering the rescaled variable.


Introduction
With increasing availability of data from internet applications, recent years have witnessed expanding interest to characterize and model human behaviors. Many on-line human activities such as email communications [1,2,3,4], web surfing [5,6,7], movie rating [8], online game [9], blog posting [10] and off-line activities such as letter communications [11,4,12] and text message [13] are under active investigation which provide understanding of our society. One of the main results from these empirical studies is the heavytailed nature of the interevent time distribution: the time interval between two consecutive human actions, which we denote as τ , follows a power-law distribution, i.e. P (τ ) ∼ τ −β . Moreover, some studies claimed that there exist a few universality classes in human dynamics characterized by universal exponents [2], which lead to scientific debates [3,4,8,7]. Other studies show that the exponents of inter-event time distribution depend on Activity (the frequency an individual takes actions), which implies that exponent of individual is not a well representative for human behaviors [8,7], but an universal behavior can be anyway found by considering the rescaled variable τ / τ [4,7]. It is noted that this strong dependence can only be observed in the inter-day range and it becomes much weaker in the intra-day range [15]. These results also suggest that we may classify human activities by different time range.
In this paper, we study in details interevent time statistics in Delicious.com, which is a typical web 2.0 application. Through Delicious, users save and manage bookmarks, while sharing interesting bookmarks with friends. It should be noted that there is a close relation between web surfing and bookmarking: in most case, we surf on web, bookmark interesting webpages, and continue surfing. Heavy-tails were already observed in the distribution of time between consecutive visits to URLs [6,7]. On the other hand, the data set of Delicious is widely adopted as training sets for recommender systems [17,18]. Understanding the temporal pattern in Delicious may give insight to devise time-aware recommender algorithms, which utilize time stamps of data to increase recommendation accuracy [19,20].
The paper is organized as follows. In Section 2, we provide detailed descriptions of the data set studied. In Section 3, we give examples of individual interevent time distributions which show heavy-tailed nature and heterogenous scaling in intra-day and inter-day range. In Section 4, we give the global interevent time distribution in these two ranges and distinguish them by estimating the decay exponents respectively. In Section 5, through comprehensive analysis of exponent dependence on Activity, we show different trend observed in the intra-and inter-day scale. Then a data collapse among the inter-day distributions is observed by considering the rescaled variable τ / τ . Finally, we conclude and summerize the results in Section 6.

number distribution
Our data set consists of 54204641 bookmarking activities by 220867 users over a period of 31 months (between 2004/04/01 and 2007/11/01). Here we use only the identifier (ID) of the users and the time when the bookmarks were saved. The resolution of time stamps is in seconds. Figure 1 shows the distribution of k, the number of bookmarks saved by a single user. As we can see, P (k) is broad and the tail of the distribution decays as power-law as k increases, giving P (k) ∼ k −2.41 . This result resembles the distribution of messages number in Ebay [7], and is significantly different from that of logging action in wikipedia (which follows power-law over the whole range [7]) and post number in blog (which is so called "double power law" [10,21]). Interestingly, in spite of the difference in these distributions, statistics on interevent times of them are very similar as we will see below.

Interevent time distribution for individuals
In our context, the interevent time τ is defined as the time interval between consecutive bookmarks by the same user. Figure 2 shows the cumulative distribution of interevent time obtained from six users. As we can see, all curves show a crossover around τ ≈ 1 day, which correspond to a change in exponent between intra-and inter-day range. Although power-law decays are observed in both ranges, the change in exponent (which is also noticed in other systems [15]) suggests that the mechanism driving intra-and interday activities are different. Moreover, changes in exponent are observed even within the intra-day range for some users. As shown in figs 2e and 2f, a slight increase in the decay exponent is observed at τ ≈ 1 hour.

The global distribution of interevent time
The global distribution of interevent time are plotted in fig. 3. In order to have a clear picture in the intra-day range, we express τ in fig. 3a with a resolution of minutes. In fig. 3b, we express it with a resolution of days where the circadian oscillation are masked which makes the decay in the inter-day range clearer. Both distributions in the intra-and inter-day range present a powerlike decay with exponents β inter ≃ 1.07 for intra-day range and β intra ≃ 2.41 for inter-day range. This significant difference between the exponents of the two ranges is consistent with the results obtained from the distribution of individuals in Sec 3. The exponent in the intra-day distribution in this case is the same as the one obtained from consecutive visits of URL [6]. It is reasonable, since bookmarking often follows web surfing as we mentioned above. The exponent of the inter-day distribution is very large compared to other systems [8,15,7] as we know that would make some difference in the following analyses. It should be noted that we fit the inter-day distributions with the so-called "shifted power-law" [22]: When h is large this distribution tends to resemble exponential distribution, and it approaches a power law when h → 0 [22]. As we know, human activities based on poisson process (where event occurrence are independent and have identical probability) can lead to an exponential distribution of interevent time. Therefore, h somehow measures the extent that human activity resembles poisson process, i.e. homogeneous probability to bookmark webpages over time. For instance, the distribution in fig. 3b is fitted by eq. (1) with h ≈ 3.32. We will further discuss the fitting values of h for distributions with different individual Activity in the next section.

Activity and exponents
Based on the heterogeneity, we perform investigation on both the intraand inter-day distributions. Firstly, we define the average Activity A i of user i as where n i is the total number of bookmarks saved by user i and T i is the time interval between the first and the last bookmark of user i. We consider only users with at least 20 bookmarks and T i which is more than 10. There are 173108 users who satisfy these conditions. Figure 4 shows the distribution of Activity A i which approximately follow a log-normal distribution, as shown by the fitting line. As we can see, the A i of most user is between 0.01 and 1 with most probable value around 0.2 per day. To examine interevent time distribution in relation with user Activity, we then sort users in an ascending order of A i and divide the entire population into 10 groups, each of which have M users (M ≈ N/10 where N is the total number of users). Accordingly, the mean activity of each group obeys the inequality A 1 < A 2 ... < A 10 . We then investigate the dependence of exponent on Activity in both the intra-and inter-day range. In fig. 5, we plot the interevent time distribution of group 1, 5 and 9 (which respectively correspond to A = 0.09, 0.37, 1.12 per day). In the intra-day range, we find that the exponents only weakly dependent on A (as shown by fig. 5a  a slight decrease is observed with A ). On the contrary, in the inter-day range, the exponents increase from 2.15 to 2.91 with A from 0.37 to 1.12. This result shows that behavioral heterogeneity in intra-and inter-day range is also evident from exponent dependence on A . In fig. 6, we show the inter-day exponent of the interevent time distribution as a function of A i in fig 6. It is noted that the exponents here increase much faster than the ones in previous studies in spite of the same tendency [8,7,15]. The exponent of group 1 is 2.15 and the one of group 3 already increase to 2.74. The reason for this steep increase has to do with the large exponent of the global distribution which is 2.41 as we mentioned above. The advantage is that it leads us to observe this dependence on a broader range. As we can see, the exponent reaches the maximum at group 6 and then it decreased slightly. It suggests there is a limiting value (β ≃ 3) of decay exponents. Actually, in Radicchi's study [7], the exponents of last group of America On-Line and Wikipedia also decrease. We further plot in the inset of fig. 6 the fitted values of h from eq. (1) for the distributions of the 10 groups, which shows a monotonic decrease of h with A . For instance, h ≈ 7 for group 1 and h ≈ 0.7 for group 10, indicating a substantial decrease of h. As mentioned before, it shows that poisson process plays a more significant role in the behavior pattern of less active users. We can understand these results based on the temporal-preference model which has two "choose rule" [10]: (1) the more the user performs an activity recently, the more likely he will do it next; (2) there exists occasions that user choose what to do randomly with independent probability. In relevance to the model, the behaviors of the inactive users are relatively more likely to follow the random rule. Similar dependence of exponents on A is actually observed in this model [10]. In fig. 7, these distributions are rescaled with the average interevent time τ of the respective group of users. As we can see, the scaling produces a data collapse between the different curves. This observations, which are already noticed in other systems [16,7,23], implies that the mechanism underlying individual behaviors within the inter-day range is still the same in spite of different exponents. Furthermore, the Activity of individual users play an important role in this mechanism.

conclusion
In this paper, we show the heavy-tailed nature of the distribution of interevent times at both individual and population level. Our results further verified heterogenous human dynamics in intra-and inter-day ranges. On one hand, there is a significant difference between the exponents in the intraand inter-day distribution, which are 1.07 and 2.41. On the other hand, the inter-day exponents are strongly dependent on individual Activity, while this dependence is absence in intra-day range. Moreover, our study suggests that there is a maximum value of β ≈ 3 for the increase of exponent with Activity. The distributions in intra-day range seem to be in universality classes of human dynamics characterized by exponents of β = 1 [10,6,7,2,12], if we ignore the weak dependence of exponents on Activity. There are already models which produce the interevent time distribution with similar exponents [14,24]. The strong dependence of exponent on Activity in the interday range implies that it is inappropriate to classify the distributions in this range by only their exponents. However the universality still exists in interday human dynamic. One evidence is the similar exponent dependence on Activity which are observed in many systems [8,7,15]. The data collapse between these different distributions provide further confirmation. In general, the exponents of inter-day distribution are often greater than that of intra-day distribution and smaller than 3 [8,7,15]. The temporal-preference model, which is built to simulate the behavior pattern of blog-posting, can already give a preliminary explanation for the inter-day dynamics. However the current model is oversimplified. To improve and give better agreements with most systems, it is the key to understand the relation between action repetition and memory [25,26].