Novelty and Collective Attention

TL;DR

This paper investigates how collective attention to new information spreads and diminishes among large populations, using data from Digg.com, and models the process with a dynamical approach involving a novelty decay law.

Contribution

It introduces a dynamical model of collective attention based on a single novelty factor and characterizes the decay of attention with a stretched-exponential law.

Findings

Attention decay follows a stretched-exponential law.

A natural time scale governs the fading of attention.

The model effectively describes attention dynamics among millions of users.

Abstract

The subject of collective attention is central to an information age where millions of people are inundated with daily messages. It is thus of interest to understand how attention to novel items propagates and eventually fades among large populations. We have analyzed the dynamics of collective attention among one million users of an interactive website -- \texttt{digg.com} -- devoted to thousands of novel news stories. The observations can be described by a dynamical model characterized by a single novelty factor. Our measurements indicate that novelty within groups decays with a stretched-exponential law, suggesting the existence of a natural time scale over which attention fades.

Figures

Equations

$$N_t=\prod _{s=1}^t(1+r_s{X}_s)N_0\approx \prod _{s=1}^t{e}^{r_s{X}_s}{N}_0=e^{{\sum }_{s=1}^t{r}_s{X}_s}{N}_0,$$

$$\log N_t-\log N_0=\sum _{s=1}^t{r}_s{X}_s.$$

$$\frac{E(\log N_t-\log N_0)}{\mathrm{var}(\log N_t-\log N_0)}=\frac{{\sum }_{s=1}^t{r}_s\mu }{{\sum }_{s=1}^t{r}_s{\sigma }^2}=\frac{\mu }{{\sigma }^2}.$$

$$r_t=\frac{E(\log N_t)-E(\log N_{t-1})}{E(\log N_1)-E(\log N_0)}.$$

$${\int }_0^{\tau }{e}^{-0.4t^{0.4}}dt=\frac{1}{2}{\int }_0^{\infty }{e}^{-0.4t^{0.4}}dt.$$

Full text

The problem of collective attention is at the heart of decision making and the spread of ideas, and as such it has been studied at the individual and small group level by a number of psychologists [1, 2], economists [3], and researchers in the area of marketing and advertising [4, 5, 6]. Attention also affects the propagation of information in social networks, determining the effectiveness of advertising and viral marketing [7]. And while progress on this problem has been made in small laboratory studies and in the theoretical literature of attention economics [8], it is still lacking empirical results from very large groups in a natural, non-laboratory, setting.

To understand the process underlying attention in large groups, consider as an example how a news story spreads among a group of people. When it first comes out, the story catches the attention of a few ones, who may further pass it on to others if they find it interesting enough. If a lot of people start to pay attention to this story, its exposure in the media will continue to increase. In other words, a positive reinforcement effect sets in such that the more popular the story becomes, the faster it spreads.

This growth is counterbalanced by the fact that the novelty of a story tends to fade with time and thus the attention that people pay to it. Therefore, in considering the dynamics of collective attention two competing effects are present: the growth in the number of people that attend to a given story and the habituation that makes the same story less likely to be attractive as time goes on. This process becomes more complex in the realistic case of multiple items or stories appearing at the same time, for now people also have the choice of which stories to attend with their limited attention.

In order to study the dynamics of collective attention and its relation to novel inputs in a natural setting, we analyzed the behavioral patterns of one million people interacting with a news website whose content is solely determined by its own users. Because people using this website assign each news story an explicit measure of popularity, we were able to determine the growth and decay of attention for thousands of news stories and to validate a theoretical model which predicts both the dynamics and the statistical distribution of story lifetimes.

The website under study, digg.com, is a digital media democracy which allows its users to submit news stories they discover from the Internet [9]. A new submission immediately appears on a repository webpage called “Upcoming Stories”, where other members can find the story and, if they like it, add a digg to it. A so-called digg number is shown next to each story’s headline, which simply counts how many users have digged the story in the past.111In fact, digg users are given the option to “bury” a story, which will decrease the story’s digg number. Because this rarely happens, we ignore this possibility and simply assume that a story’s digg number can only grow with time. If a submission fails to receive enough diggs within a certain time limit, it eventually falls out of the “Upcoming” section, but if it does earn a critical mass of diggs quickly enough, it becomes popular and jumps to the digg.com front page.222The actual machine-learning algorithm used to determine whether a story qualifies to appear on the front page is very complex and will not be discussed in this paper [10]. Because the front page can only display a limited number of stories, old stories eventually get replaced by newer stories as the page gets constantly updated. If a story however, becomes very popular it qualifies as a “Top 10” and stays on the right side of the front page for a very long time.

When a story first appears on the front page it attracts much attention, causing its digg number, $N_{t}$, to build up quickly. After a couple of hours its digg rate slows down because of both its lack of novelty and its lack of prominent visibility (reflected in the fact that it moves away from the front page). Thus the digg number of each story eventually saturates to a value, $N_{\infty}$, that depends on both its popularity growth and its novelty decay. In order to determine the statistical distribution of this saturation number, which corresponds to the number of diggs it has accumulated throughout its evolution, we measured the histogram of the final diggs of all 29,864 popular stories in the year 2006. As can be seen from Fig. 1, the distribution appears to be quite skewed, with the normal Q-Q plot of $\log(N_{\infty})$ a straight line. A Kolmogorov-Smirnov normality test of $\log(N_{\infty})$ with mean 6.546 and standard deviation 0.6626 yields a $p$-value of 0.0939, suggesting that $N_{\infty}$ follows a log-normal distribution.

It is then natural to ask whether $N_{t}$, the number of diggs of a popular story after finite time $t$, also follows a log-normal distribution. To answer this question, we tracked the digg numbers of 1,110 stories in January 2006 minute by minute. The distribution of $\log(N_{t})$ again obeys a bell shape curve. As an example, a Kolmogorov-Smirnov normality test of $\log(N_{\mathrm{2\,hours}})$ with mean 5.925 and standard deviation 0.5451 yields a $p$-value as high as 0.5605, supporting the hypothesis that $N_{t}$ also follows a log-normal distribution.

The log-normal distribution can be explained by a simple stochastic dynamical model which we now describe. If $N_{t}$ represents the number of people who know the story at time $t$, in the absence of any habituation, on average a fraction $\mu$ of those people will further spread the story to some of their friends. Mathematically this assumption can be expressed as $N_{t}=(1+X_{t})N_{t-1}$, where $X_{1},X_{2},\dots$ are positive i.i.d. random variables with mean $\mu$ and variance $\sigma^{2}$. The requirement that $X_{i}$ must be positive ensures that $N_{t}$ can only grow with time. As we have discussed above, this growth in time is eventually curtailed by a decay in novelty, which we parameterize by a time dependent factor $r_{t}$ consisting of a series of decreasing positive numbers with the property that $r_{1}=1$ and $r_{t}\downarrow 0$ as $t\uparrow\infty$. With this additional parameter, the full stochastic dynamics of story propagation is governed by $N_{t}=(1+r_{t}X_{t})N_{t-1}$, where the factor $r_{t}X_{t}$ acts as a discounted random multiplicative factor. When $X_{t}$ is small (which is the case for small time steps) we have the following approximate solution:

[TABLE]

where $N_{0}$ is the initial population that is aware of the story. Taking logarithm of both sides, we obtain

[TABLE]

The right hand side is a discounted sum of random variables, which for $r_{t}$ near one (small time steps) can be shown to be described by a normal distribution [11]. It then follows that for large $t$ the probability distribution of $N_{t}$ will be approximately log-normal.

Our dynamic model can be further tested by taking the mean and variance of both sides of Eq. (2):

[TABLE]

Hence if our model is correct, a plot of the sample mean of $\log(N_{t})-\log(N_{0})$ versus the sample variance for each time $t$, should yield a straight line passing through the origin with slope $\mu/\sigma^{2}$. One such plot for 1,110 stories collected in January 2007 is shown in Fig. 2. As can be seen, the points indeed lie on a line with slope 6.947.

The decay factor $r_{t}$ can now be computed explicitly from $N_{t}$ up to a constant scale. Since we have normalized $r_{1}$ to 1, we have

[TABLE]

The curve of $r_{t}$ estimated from the 1,110 stories in January 2007 is shown in Fig. 3(a). As can be seen, $r_{t}$ decays very fast in the first two to three hours, and its value becomes less than 0.03 after three hours. Fig. 3(b,c) show that $r_{t}$ decays slower than exponential and faster than power law. Fig. 3(d) shows that $r_{t}$ can be fit empirically to a stretched exponential relaxation or Kohlrausch-Williams-Watts law [12]: $r_{t}\sim e^{-0.4t^{0.4}}$. The halflife $\tau$ of $r_{t}$ can then be determined by solving the equation

[TABLE]

A numerical calculation gives $\tau=69$ minutes, or about one hour. This characteristic time is consistent with the fact that a story usually lives on the front page for a period between one and two hours.

The stretched exponential relaxation often occurs as the result of multiple characteristic relaxation time scales [12, 13]. This is consistent with the fact that the decay rate of a story on digg.com depends on many factors, such as the story’s topic category, the time of a day when it appears on the front page. The measured decay factor $r_{t}$ is thus an average of these various rates and describes the collective decay of attention.

These measurements, comprising the dynamics of one million users attending to thousands of novel stories, allowed us to determine the effect of novelty on the collective attention of very large groups of individuals, while nicely isolating both the speed of propagation of new stories and their decay. We also showed that the growth and decay of collective attention can be described by a dynamical model characterized by a single novelty factor which determines the natural time scale over which attention fades. The exact value of the decay constant depends on the nature of the medium but its functional form is universal. These experiments, which complement large social network studies of viral marketing [7] are facilitated by the availability of websites that attract millions of users, a fact that turns the internet into an interesting natural laboratory for testing and discovering the behavioral patterns of large populations on a very large scale [14].