Unraveling the Origin of Social Bursts in Collective Attention

TL;DR

This paper investigates the causes of sudden bursts of collective attention on social media during major events, revealing that social heterogeneity and preferential attention drive these phenomena.

Contribution

It introduces a model combining social heterogeneity and preferential attention to explain social bursts, advancing understanding of collective attention dynamics.

Findings

Extreme fluctuations in collective attention are characterized and explained.

Heterogeneity of social interactions influences attention bursts.

Preferential attention towards influential users amplifies collective activity.

Abstract

In the era of social media, every day billions of individuals produce content in socio-technical systems resulting in a deluge of information. However, human attention is a limited resource and it is increasingly challenging to consume the most suitable content for one's interests. In fact, the complex interplay between individual and social activities in social systems overwhelmed by information results in bursty activity of collective attention which are still poorly understood. Here, we tackle this challenge by analyzing the online activity of millions of users in a popular microblogging platform during exceptional events, from NBA Finals to the elections of Pope Francis and the discovery of gravitational waves. We observe extreme fluctuations in collective attention that we are able to characterize and explain by considering the co-occurrence of two fundamental factors: the heterogeneity of social interactions and the preferential attention towards influential users. Our findings demonstrate how combining simple mechanisms provides a route towards complex social phenomena.

Figures

Tables

Table 1. Table 1: Information about data sets used in this study.

Event	Messages	Users	From	To	Days	Keywords
Papal Conclave (Pope Francis)	5,538,257	2,530,554	2013-02-25	2013-03-19	21.43	“pope”, “benedict”, “pontifex”, “resign”, “conclave”, “vatican”
NBA Finals	2,993,898	1,115,981	2015-06-05	2015-06-21	15.55	“#nbafinals”
Cannes Film Festival	1,521,977	521,151	2013-05-06	2013-06-03	27.93	“cannes film festival”, “cannes”, “#cannes2013”, “#festivalcannes”, “#palmdor”, “canneslive”
Gravitational Waves Discovery	859,585	451,739	2016-02-10	2016-02-16	5.71	“ligo”, “#gravitationalwaves”, “#ligo”, “gravitational waves”, “#gravitational waves”, “gravitational #waves”, “onde gravitazionali”, “#OndesGravitationnelles”, “Ondas gravitacionales”, “Ondes Gravitationnelles”, “#ondas #gravitacionales”, “#ondas gravitacionales”
50th Anniv. of M.L. King’s “I have a dream” speech	496,094	391,467	2013-08-25	2013-09-02	7.77	“Martin Luther King”, “#ihaveadream”

Equations

$$P(R_i(t)|i)=B(k_i{p}_A(t),q_i(t)),$$

$$R(t)=\sum _{i=1}^{N(t)}{R}_i(t)\approx p_A(t)\sum _{i=1}^{N(t)}{q}_i(t)k_i,$$

$$S_w=\frac{\underset{t\in w}{\max }R(t)}{⟨R(t){⟩}_{t\in w}}.$$

$$R_N=p_A\sum _{i=1}^N{q}_i{k}_i\propto \sum _{i=1}^N{k}_i^{1+\alpha }.$$

$$\rho (x)\sim k^{-(1+\mu +\alpha )}\sim x^{-(1+\mu +\alpha )/(1+\alpha ))}.$$

$$R_N=\sum _{i=1}^N{x}_i.$$

$$\frac{\sqrt{\mathbb{V}[R_N]}}{\mathbb{E}[R_N]}\sim \frac{1}{\sqrt{N}}.$$

$$(R_N-\mathbb{E}[R_N]{)}^2\sim N^{2/\mu },$$

$$\frac{\sqrt{(R_N-\mathbb{E}[R_N]{)}^2}}{\mathbb{E}(R_N)}\sim N^{(1-\mu )/\mu }.$$

$$\mathbb{E}[R_N]\sim N^{1/\mu }.$$

$$R^{\text{max}}\sim N^{1/\mu },$$

Full text

The ability to filter the most relevant data out of a deluge of information characterizes human intelligence. When this ability is coupled with individual’s behavioral responses, like deciding to take an action based on the processed information, intriguing phenomena Bagrow et al. (2011) such as collective attention might emerge. Like popularity, attention depends on a variety of both endogenous and exogenous factors that have effects on several aspects of human behavior, from timing patterns of activity Barabasi (2005) to peculiar responses to shocks Roehner et al. (2004). The advent of social media and the possibility to record the simultaneous activity of millions of individuals allows the study of this type of phenomena on unprecedented large scales. In fact, such responses are often characterized by information cascades Lerman and Ghosh (2010); Bakshy et al. (2011); Wu et al. (2011); Baños et al. (2013); Goel et al. (2015) and exhibit a rich dynamics with a long memory which is responsible, for instance, for the emergence of power-law distributed physical observables such as waiting times Crane and Sornette (2008); Crane et al. (2010) and responses to social-media items Mathews et al. (2017). This dynamics has been successfully modeled by a special class of self-exciting point processes known as Hawkes processes Hawkes (1971), described by a self-reinforced dynamics where the likelihood of future events increases with the occurrence of a specific event.

Like online popularity Szabo and Huberman (2010); Ratkiewicz et al. (2010); Borghol et al. (2011); Figueiredo et al. (2011); De Vries et al. (2012); Bandari et al. (2012); Pinto et al. (2013), collective attention is characterized by a quickly growing accumulated focus on a specific topic, e.g. presidential elections discussion on socio-technical systems, until a well identified peak of attention is reached, followed by a phase of decreasing interest with a slow decay Naaman et al. (2011); Lehmann et al. (2012); Omodei et al. (2015). The dynamical features of both the rise and decline of attention are still debated, although there is some evidence in support of power-law distributed activity Crane and Sornette (2008); Crane et al. (2010); Ratkiewicz et al. (2010) which is a signature of criticality in complex networked systems Dorogovtsev et al. (2008). On the one hand, some studies succeeded in providing a description of collective attention dynamics while neglecting the effects of the underlying social structure Wu and Huberman (2007). In this case, popularity and the rise of attention are understood as the result of an extrinsic factor – the amount of promotions from the outside world that a content, such as a video or a news, receives – acting upon two intrinsic factors, namely sensitivity to promotion and inherent virality Rizoiu et al. (2017). On the other hand, recent studies highlighted the effects of the topological features, i.e. the underlying network of interactions, as well as of competing dynamics and memory time on the spreading phenomena observed in socio-technical systems Gleeson et al. (2016). Along this direction, many studies proposed different models based on the interplay between social structure and complex spreading dynamics to characterize the collective behavior observed in social media Myers and Leskovec (2014), specially during special events such as the discovery of the “God particle” De Domenico et al. (2013) or in response to real-world exogenous shocks such as disasters He and Lin (2017). The interplay between system’s topology and statistics of exogenous factors – such as news media – determine time-dependent network correlations that have been captured by more complex dynamical models of human activity, such as non-stationary Tannenbaum and Burak (2017) and non-linear Fujita et al. (2018) Hawkes processes and stochastic differential equations with Lévy noise Miotto et al. (2017).

Here, we show that by combining two very simple mechanisms characterizing human activity it is possible to reproduce the most salient statistical features of extreme fluctuations Bouchaud and Georges (1990) during collective attention in online social systems, without focusing on the evolution of the underlying dynamics. More specifically, we consider a preferential attachment process, related to individual’s neighborhood and social connectivity that characterizes the network topology, and a preferential attention process, a cognitive dynamics related to individual’s attention bias towards specific users of the network.

Results

Overview of the data sets. In this work, we analyze the online activity of millions of users posting millions of messages in Twitter, a popular microblogging platform, during some special events. More details about the data are provided in Tab. 1. We focus on events of wide public interest spanning different topics, such as the elections of Pope Francis (religion), NBA finals (sport), the discovery of gravitational waves (science), the Cannes Film Festival (culture). The data sets consist of the second-by-second online activity that for the subsequent analysis has been aggregated at the time scale of $T=1$ min.

Analysis of bursty activity due to collective attention. Let us focus our attention on the evolution of collective attention dynamics over time, during different special events, shown in Fig. 1. A common feature, observed in all events regardless of their type (e.g., political, religious, cultural, scientific), is the bursty behavior of the social system: spikes of activity appear to be randomly placed on top of a more smooth temporal variation. Fig. 1B shows that the spikes are extremely sharp in time, characterized either by an abrupt increase followed by either a decrease of activity within one time unit (1 minute, in the figure) or by a slightly slower decrease of activity resembling the relaxation of a system’s response to some stimulus.

To better understand the nature of such a bursty behavior, we decompose the overall activity into its components due to individual’s lone activity (“Tweet”) – posting messages related to the event which do not involve other users – and to social interactions, such as endorsing (“Retweet”) or replying to (“Reply”) other individual’s posts. Fig. 2 shows that bursts dominated by both individual and social activities exist. Counting the contribution of each activity to many different bursts, compared against random expectations, reveals that the social activities are more frequently responsible for the spikes (see Suppl. Fig. 1–2). The goal of this manuscript is to provide a statistical characterization of this bursty activity and to discuss possible mechanisms that account for them.

Characterizing bursty activity in collective attention. Our goal is to clarify what type of mechanisms can be responsible for the spiky online activity summarized above. Recent studies attempted to relate the overall collective activity to peculiar characteristics of the underlying social structure or the influence of endogenous and exogenous factors Fujita et al. (2018). The extremely fast and socially-dominated nature of spikes point towards a mechanism of reinforcement of collective behaviour taking place endogenously in a social network. Our hypothesis is that the variety of fluctuations observed in empirical data are due to the interplay between topological effects, related to the individual’s neighborhood and social connectivity, and cognitive effects, related to the individual’s bias towards activity from specific users participating into the process. Both effects are known to concentrate the attention in the few most connected users. This motivates us to search for mathematical models that account for the spiky collective attention observed in online platforms such as Twitter and that just depend on individual’s relationships and interactions.

We concentrate on the typical case of spikes generated by social activities in response to previous messages. Once a message $i$ is posted, the $k_{i}$ followers of the source user who posted it can act socially (i.e., in Twitter this might correspond to a Reply or a Retweet, corresponding to a direct comment or an endorsement, respectively). In our simple model, the multiple factors affecting this response are reduced to two: $p_{A}(t)$ the probability of a follower being active and $q_{i}(t)$ the probability of an active follower to react. The extremely short time scales of the spikes suggests that the reactions to a message are dominated by the immediate followers of the source user, instead of long/deep cascades of interactions in the network. With this simplifying assumptions, the probability that the message $i$ triggers $R_{i}$ social activities (responses) at time $t$ is given by

[TABLE]

where $B$ is the Binomial distribution with $k_{i}p_{A}(t)$ samples and probability $q_{i}(t)$. The overall social activities $R(t)$ at time $t$ is obtained summing the number of triggered responses $R_{i}(t)$ over all $N(t)$ messages contributing to social activities at time $t$ as

[TABLE]

where the approximation is based on the expected number of interactions $k_{i}p_{A}(t)q_{i}(t)$, the average of distribution in Eq. (1). In Eq. (2), we consider messages to be randomly placed in the network so that for each message the user associated to $i$ (with $k_{i}$ and $q_{i}$) is randomly chosen. In particular, we consider $k_{i}$ to be a random sample of the degree distribution of the network $\rho(k)$. Our analysis of empirical data reveals that the duration of bursts due to social actions is, on average, shorter than 5 minutes, with 15 minutes as an upper bound (see Suppl. Fig. 2). Due to this extremely short time scales of the duration of the bursts, and similar short time scales for the social reactions to posted messages (see Suppl. Fig. 3), in our model we estimate $N(t)$ (the number of messages contributing to social activities at time $t$) simply as the average number of messages published in a window of time around $t$ (see Ref. Mathews et al. (2017) for a more detailed account of the slow temporal decay of the number of social interactions to a message). It is worth remarking that messages should not be necessarily produced at time $t$, but they can be posted before without triggering social interactions before time $t$.

Equation (2) defines our simple model for collective attention, and different scenarios are obtained by specifying the network (its degree distribution $\rho(k)$) $p_{A}(t),$ and $q_{i}(t)$. The probability of a users to be active $p_{A}(t)$ simply re-scales the number $R(t)$ of social activities and will thus not be relevant in our explanation of the spikes. The two critical parameter in the different scenarios are $k_{i}$ – the number of users that receive the message $i$ – and $q_{i}(t)$ – the probability of user $i$ to act socially (retweet/reply). We consider three different scenarios of increasing complexity:

Homogeneous: $q_{i}(t)=q(t)$ is independent of $i$ and $\rho(k)$ is sharply peaked around an average degree $\langle k\rangle$ (e.g., $\rho(k)\sim Pois(\langle k\rangle)$). In this case, the role of $q(t)$ is to simply re-scale $p_{A}(t)$, which are both assumed to have a smooth temporal dependence not related to the spikes. Fluctuations in this scenario are expected to be small because of the well-behaved degree distribution $\rho(k)$, so that this scenario acts as a null model.

• 2.

Heterogenous: we incorporate to the previous scenario the well-known fact that $\rho(k)$ is a fat-tailed distribution, decaying as $\rho(k)\sim k^{-(1+\mu)}$ for $k\gg 1$. Typically $1<\mu<2$ and in the specific case of Twitter, $\mu\simeq 1.2$ was measured Kwak et al. (2010). Much larger fluctuations are expected in this scenario because of the strong variations in $k_{i}$ for different $i$, i.e., the messages coming from hubs ($k_{i}\gg\langle k\rangle$) are expected to receive much more interactions than messages from typical nodes ($k_{i}\approx\langle k\rangle$).

• 3.

Preferential attention: we incorporate to the previous scenario the fact that reaction to a message is more likely if it comes from a user that is perceived as important or central. The simplest proxy for such an importance is the degree of the message creator and thus we use $q_{i}(t)\propto k_{i}$.

For each of the scenarios, the sum in Eq. (2) effectively considers samples of distribution with short (case 1) or fat (case 2 and 3) tails. The restriction $2<\mu<3$, valid for all degree distributions, ensures that $\langle k\rangle$ is well defined in scenario 2. In contrast, scenario 3 effectively corresponds to drawing samples from a distribution with exponent $\mu-1$ and therefore with a diverging mean (see Materials and Methods, Model with preferential attention).

Revealing the mechanisms behind collective attention. The mechanisms behind collective attention can be revealed by testing to which extent the scenarios above describe the observations. We are interested in the spikes observed in the data, an extreme case of variability of the activity. Here, the data is represented by a time series of length $L$ encoding the collective activity of the social network over time. We divide this time series into non-overlapping windows of size $\ell$ and, for each window $w=1,2,...,L/\ell$, we quantify the spikiness $S_{w}$ in the window as the ratio between the maximum and the mean volume $R(t)$ of social responses, for $t$ in the window:

[TABLE]

The overall number of posted messages in each window is indicated by $N_{T,w}=\sum_{t\in w}N(t)$ and we consider $N(t)$ in Eq. (2). First we discuss the expectations for the dependence of $S_{w}$ on $N_{T,w}$ for the three scenarios of our model. As argued above, the scenarios correspond to random sampling of three fundamentally different types of distribution (short, heavy, and extremely heavy-tails). Accordingly, the $\mathbb{E}[R]$ and $R^{\text{Max}}$ – the expected value of $R(t)$ and its largest value in $\ell$ independent realizations, respectively – scale differently with the number of messages $N_{T,w}$, leading to the following estimations of the spikiness $S_{w}$ (see Materials and Methods, Sum of fat-tailed variables and Maxima):

Homogeneous: $S_{w}\sim 1/\sqrt{N_{T,w}}$, i.e., the usual central-limit-theorem decay (i.e., spikiness is not expected for large values of $N_{T,w}$)

• 2.

Heterogeneous: $S_{w}\sim 1/N_{T,w}^{1-1/\mu}$, i.e., a slower decay of $S_{w}$ (i.e., spikiness persist for larger values of $N_{T,w}$)

• 3.

Preferential attention: $S_{w}$ does not depend on $N_{T,w}$ or, at most, decays slower than algebraic.

The scaling (“$\sim$”) relationships above hold for $N_{T,w}(t)\gg 1$, the usual setting of the generalized central limit theorem (see Materials and Methods). When $N_{T,w}(t)\approx 1$, $R(t)$ will follow the distribution of $q_{i}(t)k_{i}$. As anticipated, the activation probability $p_{A}(t)$ just rescales the triggered social activities $R(t)$ in Eq. (2) and therefore it cancels out in the ratio defining $S_{w}$ in Eq. (3).

In the analysis of the empirical data, typical choices for $\ell$ range from 20 minutes to a few hours: it can not be too small or too large, to allow for a significant number of samples to be analyzed. Each time window consists of $\ell$ measurements, because we have built the data sets at 1 minute resolution.

In order to allow for a meaningful comparison between the data and the results obtained from the model, we generate several independent Monte Carlo realizations of the overall collective activity – including posting messages and social responses – in a window of size $\ell$ and, for each realization, we calculate the corresponding spikiness. This is done for increasing values of $N_{T,w}$ and using $N(t)=N_{T,w}/\ell$. This choice of $N(t)$ is justified by the short time scales of the reactions to tweets – as argued after Eq. (2)– and can be viewed as a lower bound on the number of messages actively generating reactions at time $t$. The value of the spikiness expected from the models and its corresponding variability, respectively $\langle S_{w}\rangle$ and $\sigma_{S_{w}}$, are calculated over the ensemble of Monte Carlo realizations. For each scenario, we build the 90% confidence interval around the expected spikiness and we evaluate whether the pairs $(N_{T,w},S_{w})$ measured from the data lie within this region. The results of our analysis are shown in Fig. 3. The fluctuation analysis reveals some remarkable features of collective attention. The three models introduced in this work account, all together, for observations. The statistics of social activities can widely vary within the same event, as in the case of Pope Francis election, where replies fluctuations are well explained either by the preferential attachment or the preferential attention models. In general, the spikes of Retweets are compatible with the heterogeneous model, while the spikes of Replies are larger than expected by the heterogeneous model and can only be accounted in the preferential attention scenario.

Discussion

During events of special relevance, collective activity is usually more frenetic – i.e. the probability of posting is sufficiently high to guarantee a larger number of messages posted to the social system, typically well above 50 messages per minute – and the overall interest in the subject is driven by external factors. On top of this (smooth) overall tendency, extremely large fluctuations can be observed in form of spikes of activities. This spikes have very short duration (often less than a 1 minute) and reflect a burst of activity and a dramatic concentration of the total social attention. Our main empirical finding is to identify and characterize these spikes. In particular, while spikes can have different origins, most spikes originate from social activities – such as Replies or Retweets in Twitter – in response to messages coming from well connected nodes.

We proposed a simple stochastic model to understand the extreme fluctuations observed in social bursts of collective attention. It incorporates two fundamental mechanisms: the preferential attachment process, related to individual’s neighborhood and social connectivity that characterize the observed network topology, and a preferential attention process, a cognitive dynamics related to individual’s attention bias towards specific users of the network. In this work we considered an heterogeneous connectivity distribution scaling as $k^{-2.2}$, according to independent measurements Kwak et al. (2010), and attention bias linearly proportional to the connectivity $k$. Comparing the model predictions with Twitter data, we find that the more extreme bursts of collective behavior – typically in form of Replies – can be understood only through the combination of those two processes.

Our results show that two simple mechanisms are able to reproduce the statistical features of the appearance of spikes during exceptional events and our approach provides a procedure to measure the existence and the influence of preferential attention during events triggering collective attention.

Materials and Methods

Overview of the data sets. Messages posted by users in Twitter, a popular microblogging platform, have been collected using the streaming real-time provided by Twitter API platform, filtered by the specific keywords reported in Tab. 1. By default, Twitter limits to 1% of the overall number of messages per second that can be retrieved from the streaming API. However, when the fraction of tweets concerning specific keywords is smaller than 1% of the global volume, Twitter does not apply limitations and the complete flow of information is collected. When this is not the case, Twitter provides messages of warning, reporting the cumulative number of missed tweets. For all events considered in this work, the estimated completeness of the sample is above 95%. Because of Twitter policies, the data sets (original tweet IDs) are available upon request. Time series, the main data analyzed in this work, will be made publicly available at this URL: www.

Model with preferential attention. We consider that the probability of a social action is not a constant $q$ but depends on the message $i$ as $q_{i}=qk_{i}^{\alpha}$. From Eq. (2) we obtain

[TABLE]

This can be viewed as a sum of $N$ i.i.d. random variables $x_{i}\equiv k_{i}^{1+\alpha}$ which then, a part from pre-factors, is in the form of Eq. (4). The distribution of $x_{i}$, $\rho(x)$, is related to $\rho(k)$ by $\rho(x)=\rho(k)/dx/dk$. For the power law case $\rho(k)\sim k^{-(1+\mu)}$ we obtain

[TABLE]

This means that the model with preferential attention is equivalent to the model with heterogeneous networks with a modified exponent, obtained by the mapping of the exponents $1+\mu\mapsto 1+\mu^{\prime}=(1+\mu+\alpha)/(1+\alpha)$. In the text we consider the case $\alpha=1$ which leads to a modified exponent $\mu^{\prime}=\mu/2$. In particular, $1<\mu<2$ is mapped to $0.5<\mu^{\prime}<1$. The degree distribution of networks is $\mu>1$ and therefore an effective exponent with $\mu<1$ (the third case discussed below, $0<\mu<1$) can only be achieved through the incorporation of preferential attention.

Sum of fat-tailed variables. Let $x\leq 0$ be a random variable with distribution $\rho(x)$ such that $\rho(x)\sim x^{-(\mu+1)}$ for large $x$, with $\mu>0$ (fat tails). We are interested in the sum of $N$ independent samples of $x$

[TABLE]

Following Ref. Bouchaud and Georges (1990), the following cases can be described:

$\mu\geq 2$. In this case, which includes also distributions $\rho(x)$ with short tails (such as the Poisson distribution in scenario 1.), both moments $\langle x\rangle$ and $\langle x^{2}\rangle$ exist and for large $N$ the usual central limit theorem applies such that $\mathbb{E}[R_{N}]=\langle x\rangle N$ and variance $\mathbb{V}[R_{N}]=2\sigma_{x}^{2}N$. Therefore, fluctuations are small and decay with $N$ as

[TABLE]

• 2.

$1<\mu<2$. In this case, $\langle x\rangle$ exists but $\langle x^{2}\rangle$ does not. The expected value $\mathbb{E}[R_{N}]=\langle x\rangle N$ holds, but the fluctuations increase dramatically. In particular, $\mathbb{V}[R_{N}]$ diverges with $N$ as

[TABLE]

and therefore

[TABLE]

This still decays to zero because $(1-\mu)/\mu<0$.

• 3.

$0<\mu<1$. In this case $\langle x\rangle$ is not defined and

[TABLE]

Maxima. Our measure of spikiness $S_{w}$ defined in Eq. (3) consider the block-$\ell$ maximum of $R$, denoted by $R^{\text{Max}}$ (i.e., the largest value of $R(t)$ in $\ell$ independent realizations). For $\rho(x)\sim x^{-(1+\mu)}$, the tails of the distribution of $R(t)$ behave as the tails of $\rho(x)$ and therefore, from extreme value theory, we expect the scaling

[TABLE]

for $0<\mu<2$ and $R^{\text{max}}\sim\sqrt{N}$ for $\mu>2$ (including the Poisson distribution).

Acknowledgments

M.D.D. acknowledges partial financial support from the Max Planck Institute for the Physics of Complex Systems. EGA was funded by the University of Sydney bridging Grant G199768.

Competing financial interests

The authors declare no competing financial interests.