A Multiscale Survival Process for Modeling Human Activity Patterns

Human activity plays a central role in understanding large-scale social dynamics. It is well documented that individual activity pattern follows bursty dynamics characterized by heavy-tailed interevent time distributions. Here we study a large-scale online chatting dataset consisting of 5,549,570 users, finding that individual activity pattern varies with timescales whereas existing models only approximate empirical observations within a limited timescale. We propose a novel approach that models the intensity rate of an individual triggering an activity. We demonstrate that the model precisely captures corresponding human dynamics across multiple timescales over five orders of magnitudes. Our model also allows extracting the population heterogeneity of activity patterns, characterized by a set of individual-specific ingredients. Integrating our approach with social interactions leads to a wide range of implications.


S1. event time
In real dataset such as E-mail and online group chatting, the distribution of real interevent time often appears a peak ranging from ∆t = 16 hours to ∆t = 24 hours. This phenomenon can be explained by sociological behavior involving the time (usually 16 hours) between when people leave work and when they return to their offices. In such situation, we often use event time instead of physical time to avoid the influence. Assume that we have all the records of users' behavior form time 0 to T , the time point of the behaviors of a certain user is denoted as t 1 ,t 2 , .... Define a function N(t) that equals the total number of events of all the users(i.e., an message sent) before time t.
If the physical time is t i , we define the corresponding event time t i as follows, The time interval between t i and t j in event time is denote as follows, Event time uses a "tick" of system (i.e., an message sent) as the basic time unit to uniform the time period.

S2. baselines
In the experiment, we compare our model with several baselines, they are defined as follows: Poisson Process: The waiting time distribution of Poisson process is actually Exponential Distribution. The probability density function(p.d.f ) is intensity function is in which λ 0 is a parameter. We can estimate this parameter by maximum likelihood estimator (MLE) and the result is Pareto Distribution: Pareto Distribution represents the power law feature. The probability density function(p.d.f ) is intensity function is in which τ m and α are two parameters. Denote that Pareto Distribution requires that all interevent time τ > τ m , so when used in real data, we often crop data and discard the data less than τ m . The logarithmic likelihood function is so the parameter can be learned use maximum likelihood estimator (MLE).
Weibull Distribution: Weibull Distribution is a widely used and studied distribution, is has lots of modification form , here we use the basic model with two parameters. The probability density function(p.d.f ) is intensity function is Log-normal Distribution: Log normal distribution is similar to normal distribution, the probability density function(p.d.f ) is in which µ and σ are two parameters. The estimation of parameter is similar to normal distribution, the result is as follows, 2/3

S3. statistical tests
In experiment, we use three statistical tests as metrics of goodness of fit. We will give a brief introduction of each statistical test in the following: Kolmogorov-Smirnov test: Kolmogorov-Smirnov test (KS test) is a nonparametric test of the equality of continuous, one-dimensional probability distributions that can be used to compare a sample with a reference probability distribution. The statistic is, where sup x is the supremum of the set of distances. F n is the empirical distribution function for n iid observations X i , defined as, where I [−∞,x] (X i ) is the indicator function, equal to 1 if X i ≤ x and equal to 0 otherwise.
Chi-squared test: Chi-squared test, also referred to as χ 2 test, is a statistical hypothesis test in which the sampling distribution of the test statistic is a chi-square distribution when the null hypothesis is true. The statistic is, where O i is the number of observations of type i, E i is the expected (theoretical) number of type i.

Cramér-von Mises test:
Cramér-von Mises test (Cramér test) is a criterion used for judging the goodness of fit of a cumulative distribution function F * compared to a given empirical distribution function F n . The statistic is, where x 1 , x 2 , · · · , x n are the observed values in increasing order.