A Dynamical Model of Twitter Activity Profiles

DOI: 10.1145/2700171.2791029 Published: ACM Hypertext 2015

A Dynamical Model of Twitter Activity Proﬁles

∗

Hoai Nguyen Huynh

Erika Fille Legara

1. Institute of High Performance Computing

Institute of High Performance

Computing Agency for Science Technology and Research,

Agency for Science Technology and Research,

Singapore 2. Complexity Institute Nanyang Technological

Singapore legaraeft@ihpc.a-

star.edu.sg

University, Singapore huynhhn@ihpc.a-

star.edu.sg

ABSTRACT

The advent of the era of Big Data has allowed many re- searchers to dig into various socio-technical systems, includ- ing social media platforms. In particular, these systems have provided them with certain veriﬁable means to look into cer- tain aspects of human behavior. In this work, we are speciﬁ- cally interested in the behavior of individuals on social media platforms—how they handle the information they get, and how they share it. We look into Twitter to understand the dynamics behind the users’ posting activities—tweets and retweets—zooming in on topics that peaked in popularity. Three mechanisms are considered: endogenous stimuli, ex- ogenous stimuli, and a mechanism that dictates the decay of interest of the population in a topic. We propose a model involving two parameters η⋆and λ describing the tweeting behaviour of users, which allow us to reconstruct the ﬁnd- ings of Lehmann et al. (2012) on the temporal proﬁles of popular Twitter hashtags. With this model, we are able to accurately reproduce the temporal proﬁle of user engage- ments on Twitter. Furthermore, we introduce an alternative in classifying the collective activities on the socio-technical system based on the model.

Categories and Subject Descriptors

J.2 [Computer Applications]: Physical Sciences and En- gineering; J.4 [Computer Applications]: Social and Be- havioral Sciences; I.6 [Computing Methodologies]: Sim- ulation and Modeling

∗https://sites.google.com/site/nelive/ †http://www.erikalegara.net/ ‡http://www.chrismonterola.net/

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for proﬁt or commercial advantage and that copies bear this notice and the full cita- tion on the ﬁrst page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or re- publish, to post on servers or to redistribute to lists, requires prior speciﬁc permission and/or a fee. Request permissions from Permissions@acm.org. HT’15, September 1–4, 2015, Guzelyurt, TRNC, Cyprus. © 2015 ACM. ISBN 978-1-4503-3395-5/15/09 ...$15.00. DOI: http://dx.doi.org/10.1145/2700171.2791029.

†

‡

Christopher Monterola

Institute of High Performance

Computing Agency for Science Technology and Research,

Singapore monterolac@ihpc.a-

star.edu.sg

Keywords

Social networks; information diﬀusion; modelling

INTRODUCTION The study of information diﬀusion from gossip spread- ing [15, 16], to the propagation of viral memes [20, 23, 10], fads, and trends [24, 2, 22], and even word-of-mouth mar- keting [8, 13] has become increasingly interesting especially in this era of “Big Data.” Current technologies and methods have allowed researchers to look more closely into the social network fabric—the medium at which the proliferation of various entities takes place. Questions relating to how fast information travels or what kind of information captures the most audience have piqued the interest of many researchers [3, 25, 17, 11, 5]. Various approaches have been implemented to shed light into these. Researchers have looked into the role of a network’s degree of connectivity, modularity, and various centrality measures, among other things [25, 17, 11, 5]. Eﬀorts have also been put in understanding the degree of social “inﬂuence” of entities on each other [19, 9, 6]. Many have also investigated the nature of topics that are being diﬀused in a social system.

In this work, we propose a model that aims to capture the various aspects of these approaches—we do not only look at the network structure in isolation, but also augment it with particulars on the nature of the information being spread, and the individuals’ tendencies to spread such information or “inject” new ones. Particularly, we investigate the observa-

tions described in [14] on the dynamical classes of collective attention in Twitter where they deﬁned four groups depend- ing on the temporal features of their popularity dynamics. We initially introduce two free parameters intrinsic to the users’ behaviours, λ and η⋆, where λ quantiﬁes the rate of decay at which a user would spread a given information and η⋆is the threshold an agent has that determines whether or not he/she propagates information from the users he/she follows. The rules deﬁned are then implemented in an em- pirical Twitter network obtained from the Stanford Large Network Dataset Collection [18].

This paper is structured as follows: we ﬁrst describe the data and model in Sec. 2, then present the results and dis- cussions in Sec. 3, and ﬁnally summarise and establish our conclusions in Sec. 4.

2. DATA AND THE MODEL

2.1 Data The dataset utilised here is a set of 115 hashtags used by Lehmann et al. in [14]. It contains the time series of number of tweets and distinct users for each of the hashtags. Each time series centers around a day on which the number of relevant tweets attain their maximum “popularity,” and spans from seven days before to seven days after the day of the peak. The full data collected in [14] contain 130 million Twitter messages appearing in the period of approximately 6 months from November 20, 2008 to May 27, 2009. We point the readers to reference [14] for further details on the dataset utilised here for model ﬁtting and veriﬁcation.1

2.2 The model

2.2.1 Deﬁnitions and rules The model is deﬁned on a general network N with N nodes, each node representing a user. Each user i has Fi “fol- lowers” and Li “leaders” whom he/she follows. This leader- follower relationship results to a directed network. It is also worth noting that although the Twitter network structure is dynamically changing in the real-world, here we only con- sider a static structure given the relatively short time frame we are considering, which is two weeks. Note that when a user i follows another user j, the follower sees all the tweets that j posts; if, on the other hand, user i visits the proﬁle page of j, i will not only see the tweets, but also the retweets and replies that user j posts.

Three mechanisms are incorporated in our model. Two of which, exogenous and endogenous, deﬁne the manner at which information is propagated in the system [14, 7]. The endogenous process involves a re-posting of someone else’s tweet (“retweet”), thereby propagating/diﬀusing the same tweet across the social network. On the other hand, when new information is “injected” in the social network system, an exogenous process is said to have taken place. In addition to these two mechanisms, a third one is regarded as well that accounts for the decay of the level of the activities involv- ing a speciﬁc topic on Twitter. To encapsulate, our model incorporates these three processes: (1) injection of new in- formation into the network, (2) spreading of information in the network, and (3) decay of information after a peak.

The key features of the model proposed are quantiﬁed in two parameters η⋆and λ—characterising the spreading of information and the decay of activities in the network, re- spectively. The parameter η⋆quantiﬁes the threshold of in- ﬂuence of leaders on their followers, determining whether or not a follower would take action such as retweeting and/or replying to a tweet, consequently exposing his/her own fol- lowers to the information. In other words, η⋆encapsulate the level of contagion of a piece of information in the net- work. On the other hand, the parameter λ quantiﬁes the rate of decay of interest of a user in the information after a certain point in time. It could be seen that, in our model, the build-up in activities before a topic’s peak in popular- ity is solely reﬂected by the parameter η⋆, while the decay in activities after the peak is the interplay between the two parameters η⋆and λ.

1This is an aggregated dataset containing the daily number of tweets for each hastag and was generously provided by Bruno Gon¸calves, see Acknowledgement.

To make the model results comparable with the data we have at hand, we use the scale of one day as one time unit. The rules and ﬂowchart of implementation of the model are described in Fig. 1. The model is updated sequentially, i.e. the state of a user i at time t only depends on the state of the network before time t but not at time t.

2.2.2 Assumptions The model constructed makes the following assumptions on the tendency of a user to tweet and retweet. A user posts an original2 tweet if he/she is exposed to some new informa- tion outside of his/her Twitter network, i.e. from external sources (or has some original ideas to share). A user who follows a lot of other users tends to rely solely on his/her social network for information and, hence, retweets more of- ten than “injects” new information from external sources. On the contrary, a user who has a huge following tends to be more active in posting original ideas or new tweets rather than just reposting others’. These assumptions on tenden- cies are illustrated in Fig. 2.

Let us consider a user i (i = 1, 2, . . . , N) who follows Li leaders l(i, j) (j = 1, 2 . . . , Li) and who has Fi followers. The probability that user i is exposed to external sources is

ρi(t) = Aiχ(t −t0), (1)

in which Ai represents the activeness of i in following news and propagating to other people, and χ(t −t0) the coverage by the media. In general, the temporal proﬁle of external media coverage satisﬁes the limiting conditions

   

1 ≥χ(x) ≥0 ∀x

χ(0) = 1

. (2)

  

lim |x|→∞χ(x) = 0

We, however, assume that within a narrow window of time around the event, the media coverage is consistent and stays approximately constant so that χ(x ∼0) ≈1. By the as- sumption described above, the activeness Ai takes the form

)

(

Ai = Fi Fmax ×

1 − Li Lmax + Fi

(3)

to reﬂect the assumption that a user having more followers tends to be active in following news and can introduce inter- esting stuﬀ, but that is oﬀset by having many leaders—as in such case, the user tends to rely on the leaders for informa- tion rather than tweeting so himself/herself as illustrated in Fig. 2(a) (see, for example, [12]).

Upon external exposure, the probability of a user i to tweet Ti depends on: (1) the interest of user σi in the nature of the information or the particular topic under considera- tion, (2) the level of interest τi(t −t0) as a function of time, and (3) his hesitancy to tweet Hi.

Ti = σiτi(t −t0) −Hi. (4)

The level of interest τi(t −t0) is high during and before the event, and decays with rate λ after the event

{

1 if x ≤0

τ(x) =

exp (−λx) if x > 0 . (5)

2“Original” in this sense is used in a loose fashion. It only means that the post is not a retweet.

for every time step: for each user of the network:

“Injects” new tweet with

probability ρ True AND user has not tweeted

t < t0 t ≥ t0

Tweet with probability decaying exponentially; rate λ

Tweet with probability T

False OR user has tweeted

η ≥ η⋆ η < η⋆

t < t0 t ≥ t0

Retweet with probability decaying exponentially; rate λ

Retweet with probability R

No tweet

Figure 1: Rules of the model proposed in this work. t0 is the day of the peak and η is the amount of activities by the user’s leaders accumulated after his last tweet.

The hesitancy to tweet (also retweet) depends on the num- ber of leaders and followers a user has as illustrated in Fig. 2(b). The less leaders or followers a user has, the more hesitant he is to retweet because of the lack of engagement and/or motivation to do so. Hence,

Hi = 1 Li + Fi + 1. (6)

Here, we also assume that σ = 1 indicating that we only focus on the topics that are of interest to the users.

Next, we deﬁne the average inﬂuence of all leaders of a user i as

Li ∑

Ii = 1

Fl(i,j), (7)

j=1

in which Fl(i,j) is the number of followers that the leader l(i, j) has.

In addition, we quantify the amount of exposure user i has to the inﬂuence of his/her leaders in the following equation:

∑

Yi(t) =

Fl(i,j). (8)

all leaders l(i,j) having

tweeted recently before t

And the necessary condition for retweeting is

Yi ≥η⋆Ii. (9)

Upon this condition is met, the user i retweets with proba- bility

Ri(t −t0) = σiτi(t −t0) −Hi, (10)

which takes the same form as Eq. (4) in which Hi represents the hesitancy as described in Eq. (6).

The number of leaders who tweeted recently, i.e. after the user’s last tweet and before current time t, is denoted as

ηi(t). The total number of possible retweets by user i at time t is given by

√

η⋆ × Yi η⋆Ii , (11)

ηi(t)

νi(t) =

in which we only take the integer part and take 0 as 1 be- cause the number of retweets is at least 1 if the user retweets.

If the user retweets, it does not necessarily mean that he would retweet all n tweets. The probability to retweet R means that he tweets at least one tweet. Therefore, it could be calculated that each of his n possible retweets carries probability r = 1 −

n√

1 −R. By identifying the two key parameters λ and η⋆, we can expect to observe four diﬀerent types of users’ behaviour in response to an event, as illustrated in Fig. 3. The four types correspond to four quadrants in the (λ, η⋆) parameter space, namely lowly contagious-slow decaying, lowly contagious- fast decaying, highly contagious-slow decaying and highly contagious-fast decaying.

RESULTS AND DISCUSSIONS The empirical network we use for simulation was obtained from Stanford Large Network Dataset Collection [18]. The entire network is a combination of 1, 000 ego networks with 81, 306 nodes and 1, 768, 149 links, a diameter of 7, and a clustering coeﬃcient of 0.5653. We run the simulation start- ing from δt days before a topic peaks in popularity t0 (we also refer to this one as “event”) until 7 days after t0. δt can vary from 0 to 7, mimicking the fact that the amount of activities related to an event becomes signiﬁcant up to δt days before the event. δt = 0 corresponds to sudden events while a large value of δt indicates an anticipated one. It is noteworthy that by varying δt, we eﬀectively include a third parameter in our model, which characterises the injection of information into the network.

unlikely to be

exposed to external media

regular

Number of Leaders

propagator

Very likely to be

exposed to external media initiator/seeder

not so active

Number of Followers

(a) Tweeting behaviour of diﬀerent types of Twitter users based on their number of leaders and followers. Each type corresponds to the likelihood of being exposed to external media.

hesitant to retweet very willing to

Number of Leaders

retweet

hesitancy

hesitant to retweet willing to retweet

Number of Followers

(b) Retweeting hesitancy of diﬀerent types of Twitter users based on their number of leaders and followers. The arrows indicate the directions of increasing hesitancy, i.e. when the number of leaders or followers decreases.

Figure 2: Behaviour patterns of diﬀerent types of users according to their number followers and lead- ers.

We then scan the (λ, η⋆) parameter space in the steps of ∆λ = 0.1 (λ ∈[0; 4]) and ∆η⋆= 1 (η⋆∈[1; 60]) to produce diﬀerent time series for the number of tweets as well as the number of (distinct) users everyday and identify the ones that reproduce the empirical observations by using the dis- tance metric introduced below. Since this is a Monte-Carlo simulation that involve generation of random numbers, we perform 50 runs with distinct seeds for the random number

low high

slowly spread

Level of Contagion

last long

fast decay

quickly go viral

last long

fast decay

Interest Decay Rate, λ fast slow

Figure 3: Distribution of diﬀerent types of event in the (λ, η⋆) parameter space.

generator for each set-up, i.e. the triplet (δt, η⋆, λ), and take the average results.

3.1 Validation of the model We compare the data generated by our model to the em- pirical data by calculating the matching score of the two pro- ﬁles which are quantiﬁed by the fraction of users or tweets on a single day. In details, let P = (P1, P2, . . . , PN) be the proﬁle of the tweets produced by our model, i.e. Pi is the fraction of tweets on day ti within the entire period from t1 to tN. By deﬁnition, we have

i=N ∑

Pi = 1. (12)

i=1

Similarly, Q = (Q1, Q2, . . . , QN) is the corresponding proﬁle of the tweets in the data collected by [14].

We compare P and Q by introducing the metric

v u u t

( Pi −Qi max (Pi, Qi)

i=N ∑

δ(P , Q) = 1

, (13)

i=1

which quantiﬁes the (normalised)“distance”between the two proﬁles. It is obvious that when the two proﬁles are iden- tical P ≡Q, i.e. Pi = Qi ∀i = 1, 2, . . . , N, the distance is δ(P , Q) = 0. This is a normalised measure so that the maximum possible value of δ is 1.

( Pi −Qi max (Pi, Qi)

In Eq. (13), when Pi = Qi = 0, the term

does not have any contribution to δ. Finally, we set a tolerance threshold θ = 0.04 such that all the terms with Pi + Qi ≤θ do not have any contribution to δ.

Using the metric introduced above and after visually veri- fying the plots (Fig. 4), we consider measures with δ(P , Q) ≤ 0.08 good and discard the rest. Of the 115 hashtags, about 80% (88/115) result to good ﬁts—both for the number of users and number of retweets. The remaining 20% fall into

1.00

⌘⇤= 60 λ = 0.3 λ = 2.9 ⌘⇤= 01 ⌘⇤= 28 λ = 4.0

⌘⇤= 60 λ = 1.0

0.75

0.50

Tweets/Retweets Users

0.25

#superbowlads #nsotu #watchmen #dbi 0.00

1.00

λ = 3.4 ⌘⇤= 01 ⌘⇤= 31 λ = 4.0 λ = 0.4 ⌘⇤= 25 ⌘⇤= 21 λ = 1.2

0.75

0.50

0.25

0.00

1.00

0.75

0.50

0.25

#superbowlads #nsotu #watchmen #dbi 0.00

1.00

0.75

0.50

0.25

0.00

#hoppusday #macheist #plurk #poynterday

A A B B

P P S S

−8 −4 0 4 8−8 −4 0 4 8−8 −4 0 4 8−8 −4 0 4 8 D

Day

#hoppusday #macheist #plurk #poynterday

A A B B

P P S S

−8 −4 0 4 8−8 −4 0 4 8−8 −4 0 4 8−8 −4 0 4 8 D

Day

Figure 4: Time series of activities (top) and users (bottom). Results from the model (blue) shown together with the data (red) presented in [14] for classes A, B, P, and S, respectively.

the groups of activities distributed before and symmetric around the peak day [14], which have signiﬁcant amounts of activities distributed prior to the events. This demonstrates that the proposed model, in spite of it being capable of cap- turing the main features in the collective attention build-up and decay of users before and after the event day, requires additional framework that would quantify the“sense of time” of the users—whether or not an event is approaching [1]. This aspect will be investigated and reported elsewhere.

It is worth noting that while it is not straightforward to know how many times a user would tweet or retweet in a day, we have shown that our assumptions in Sec. 2.2.2 for the users’ activities work well in estimating both the number of users and retweets in most cases. Moreover, the fact that we could reproduce the temporal proﬁles of activities (see

Fig. 4) using our model with only two user-intrinsic param- eters and an eﬀective third parameter for external factors, justiﬁes and validates our assumptions and hypotheses in identifying the key mechanisms of information spreading in social networks.

3.2 Classiﬁcation of hashtag types With the estimated parameter values, we generate the plot for the distribution of the hashtags on the two-dimensional parameter space of η⋆and λ, as shown in Fig. 5. From the plot, we can observe the clustering pattern correspond- ing to diﬀerent types of event shown in Fig. 3, with only a few outliers. It is quite evident that there is a clustering of large points at the bottom left corner of the plot, which correspond to the events that quickly go viral and last long.

Those events appear many days before the peak and gen- erate signiﬁcant amount of activities afterward. The other three clusters contain small points signifying the events start not so long before their peak of activities.

As illustrated by the colors of the data points in Fig. 5, we can also observe that the distribution of the points cor- respond very well to the classiﬁcation of dynamical classes reported in [14], i.e. the points for each of the four classes can be segregated into distinct clusters (with exception of a few points in class of activities concentrating before the peak, see below). The four classes are called A, B, P and S, respectively, in this work for convenience of the discussion. Class A describes events where the associated activities are concentrated after a topic peaks in popularity. Class B, on the other hand, refers to the events where the activities oc- cur before the peaks. Class P consists of events where the activities are concentrated on a single day. Finally, Class S contains events that have signiﬁcant activites before, on and after the peak day. Our results show that the clusters de- scribed above also reveal the existence of subclasses within each of the classes. In Fig. 5, we can generally identify 7 clusters of data points (or hashtags) which show very good correspondence to the classiﬁcation in [14].

From the ﬁttings, we can observe two subgroups in the class with activities concentrating after the peak, i.e. class A (after). One group shows long range behaviours in which the activities span over a long period of time reﬂected by slow decay of interest (small λ) but high spreading threshold (large η⋆). The other group shows short range behaviours in which the activities span over a very short period of time reﬂected by low spreading threshold (small η⋆) but very fast decay of interest (large λ).

For the class with activities concentrating before the peak, i.e. class B (before), we also observe two subgroups. One group shows long range behaviours in which the activities span over a long period of time reﬂected by long appearance before the peak but high spreading threshold (large η⋆). The other group shows short range behaviours in which the ac- tivities span over a very short period of time reﬂected by very short appearance before the peak but very low spread- ing threshold (small η⋆).

For the class with activities concentrating at the peak, i.e. class P (peak), the values of the parameters suggest two subgroups, both of which have very fast decay of interest (large λ). One group shows contagious behaviours in which the events appear very shortly before the peak but generate a lot of activities due to low spreading threshold (small η⋆). The other group shows inert behaviours due to very high spreading threshold (large η⋆).

The class with activities distributed symmetrically around the peak, i.e. class S (symmetric), generally has low spread- ing threshold (small η⋆) and slow decay of interest (small λ).

In Fig. 4, we show the diﬀerent proﬁles for each of the classes described above.

3.3 Content analysis After revealing the existence of the classes and subclasses of the hashtags, we turn to looking at content of each hash- tag and learn how it is related to the apparent classiﬁcation. In Appendix A, we have a table showing the hashtags to- gether with their corresponding type and class (and subclass, according to our results above). The table is organised in

such a way that the top rows contain the “simple” hashtag types, in the sense that the hashtags of those types generally belong to one class identiﬁed by our model. The rows further down at the bottom of the table contain more complicated hashtag types whose tweets fall into diﬀerent classes.

From the table, it could be seen that hashtags in the cat- egories of activism (#ie6, #pman) or technology (#safari, #safari4, #skype) indicate events that capture attention in a long period of time and make impact that keep peo- ple discussing. These events are called for attention on a particular matter, e.g. campaign or of great interest and im- pact to many people, e.g. technology products. The peak in these events are usually associated with a symbolised or iconic activities on that day, e.g. rally of people in a place or release of a product. The hashtags in the category of charity (#twestival, #protest) indicate events that generate activ- ities before a peak but soon decay after that. This is because these events usually call for people’s support to achieve a certain goal (e.g. fund raising, signature collection). And once the goal has been achieved, people are no longer in- terested in the follow-up. The hashtags in the category of marketing generally exhibit sudden appearance. That could be explained by the strategies of marketers releasing incen- tives to advertise their products. But our results show that it also depends on the type of product and how it is ad- vertised to determine the dynamical behaviours of people’s attention to it.

The hashtags in other categories generally spread across diﬀerent classes with no easy way of relating the content to the class. Nevertheless, content type like the Twitter (word) games spontaneously started by some user(s), which appear in all of the classes and subclasses identiﬁed in the work, could provide a very useful set-up to study what type of content would become popular in a social setting [4, 21]. Further analysis of the meaning of the hashtags and the content of the tweet messages containing the hashtags will be explored and reported elsewhere.

3.4 Discussions The classiﬁcation of hashtags allows us to identify their general features in terms of how people react to the informa- tion they receive and also possibly infer their content. Over- all, class S (symmetric) occupies the bottom left quadrant of the parameter space (λ, η⋆). In this quadrant, the threshold η⋆is low and the rate of decay λ is also low. They corre- spond to events that can easily spread (due to low threshold) and can last after a topic peaks in popularity (low rate of de- cay), e.g. movie (#watchmen), technology release (#safari, #skype) or activism (#pman). Our model in this study can reconstruct the data very well up to δ = 4 days before the peak but generally falls through beyond that. This suggests a diﬀerent pattern in people’s behaviour when spreading the information when the “sense of time” is relevant, i.e. before and near the event associated with the information.

On the other hand, class P (peak) occupies the right half of the parameter space, which corresponds to events that decay very quickly after the peak. They can further be cat- egorised into two groups: the upper one (high threshold η⋆) corresponds to events that capture immediate attention but decay immediately, e.g. unexpected and unpopular politi- cal events (#spectrial, #nsotu) or occasional media events (#grammys, #oscars); and the lower one (low threshold η⋆) corresponds to the events that spread very quickly (it ap-

#macheist (1.0,60)

#plurk (0.3,60)

#watchmen (0.4,25)

η⋆

#dbi (1.2,21)

#therescue (0.8,1)

#safari4 (0.8,1) #hoppusday (2.9,1)

#nsotu (4.0,31)

#poynterday (4.0,28)

#superbowlads (3.4,1)

λ 0 0.5 1 1.5 2 2.5 3 3.5 4

Figure 5: Fitted parameters η⋆and λ showing clustering patterns. The circles of larger size correspond to large value of δt. The colours (online) of the data points are determined by the classed identiﬁed in [14], red for S, black for P, blue for A and green for B.

pears one or two days before the peak) and also decay very quickly, e.g. sport events (#nfl, #superbowl). The remain- ing two classes A (after) and B (before) can both be divided into two groups: (1) low threshold, high decay rate; (2) high threshold, low decay rate. The diﬀerence between them is the time the users become aware of the events. Events in class A are sudden and people continue to discuss them due to either low decay rate (long last), e.g. lobbying marketing campaign (#macheist), or low threshold (easy to spread), e.g. honouring popular stars (#hoppusday). Events in class B depict anticipation where people already discuss the topics even before their popularities peak—this contributes to large amounts of activities before the peak, e.g. new feature of Twitter (#plurk) or anticipated show (#poynterday). The events in this class, however, display scattered pattern and in some rare cases make overlap with class S (#therescue).

It needs to be emphasised that the model proposed is straightforward and concise—carrying the heuristic and in- tuitive assumptions on the online behaviours of users, given the knowledge of their social network’s structure. Yet, the model produces the dynamical behaviours observed in real data and allows us to gain insights on the clustering of topics—telling us about the diﬀerent natures of the contents being circulated in the social media, and how these clusters relate to the classes presented in [14]. This signiﬁes that the three mechanisms included in the model are essential and suﬃcient in accurately describing the dynamics behind the collection attention of users on a Twitter network.

Knowing the relevant factors that inﬂuence the dynamics behind information spreading and trend setting is crucial for

various aspects of society which can range from governance to politics, and marketing. Everyday, we are overwhelmed with terabytes of information originating from various social media sources as people share news, comments, opinions, and updates in their blogs, microblogs, and homepages; and on Facebook, Twitter, and Instagram, among others. The key for the stakeholders is to know how to manipulate and strategize, if possible, their messages and campaigns such that theirs will stand out to attract attention and not get lost in the vast sea of online information.

What we have presented herewith so far is a model that recaptures the previous trends for certain issues and topics by describing certain attributes of the agents involved in the social network. The next important question is whether or not we can use this knowledge to reshape the trend pro- ﬁles of the diﬀerent information types. Our work hints on the importance of knowing the kind of audience on which a product, an idea, or a campaign has possible inﬂuence. That aspect to some extent is quantiﬁed in our model as the parameters λ and η∗.

CONCLUSIONS In this work, we proposed a model using three mechanisms that underlie the tweeting and retweeting behaviours of users on Twitter. These behaviours correspond to perceiving and propagating information in a social network. Despite the simplicity of the model, we are able to capture the general patterns of behaviours observed in real data. In particu- lar, we have not only illustrated the four dynamical classes

reported by Lehman et al. [14] but also demonstrated the existence of further subclasses in three of the classes.

ACKNOWLEDGMENTS We would like to acknowledge Bruno Gon¸calves and Yang Bo for meaningful and useful discussions. We thank Bruno for sharing with us the aggregated dataset for use in this study. HNH thanks Chew Lock Yue at the NTU Com- plexity Institute for his support. This research supported by Singapore ASTAR SERC “Complex Systems” Research Programme grant 1224504056.

6. REFERENCES

[1] V. Alﬁ, G. Parisi, and L. Pietronero. Conference

registration: how people react to a deadline. NATURE PHYSICS, 3(11):746, NOV 2007. [2] Y. Altshuler, W. Pan, and A. Pentland. Trends

Prediction Using Social Diﬀusion Models. In Social Computing Behavioral - Cultural Modeling and Prediction, pages 97–104. Springer Science+Business Media, 2012. [3] S. Asur, B. A. Huberman, G. Szabo, and C. Wang.

Trends in social media: Persistence and decay. In 5th International Conference on Weblogs and Social Media, page 434, 2011. [4] C. Castillo, M. El-Haddad, J. Pfeﬀer, and

M. Stempeck. Characterizing the life cycle of online news stories using social media reactions. In Proceedings of the 17th ACM Conference on Computer Supported Cooperative Work & Social Computing, CSCW ’14, pages 211–223, New York, NY, USA, 2014. ACM. [5] K. Chung, Y. Baek, D. Kim, M. Ha, and H. Jeong.

Generalized epidemic process on modular networks. Physical Review E, 89(5), may 2014. [6] D. Cosley, D. P. Huttenlocher, J. M. Kleinberg,

X. Lan, and S. Suri. Sequential inﬂuence models in social networks. In W. W. Cohen and S. Gosling, editors, ICWSM. The AAAI Press, 2010. [7] R. Crane and D. Sornette. Robust dynamic classes

revealed by measuring the response function of a social system. PNAS, 105(41):15649–15653, October 2008. [8] J. Goldenberg, B. Libai, and E. Muller. Talk of the

network: A complex systems look at the underlying process of word-of-mouth. Marketing Letters, 12(3):211–223, 2001. [9] M. Granovetter. Threshold Models of Collective

Behavior. American Journal of Sociology, 83(6):1420, may 1978. [10] N. O. Hodas and K. Lerman. The simple rules of

social contagion. Scientiﬁc Reports, 4:4343, 2014. [11] Y. Ikeda, T. Hasegawa, and K. Nemoto. Cascade

dynamics on clustered network. J. Phys.: Conf. Ser., 221:012005, apr 2010. [12] H. Kwak, C. Lee, H. Park, and S. Moon. What is

twitter, a social network or a news media? In Proceedings of the 19th International Conference on World Wide Web, WWW ’10, pages 591–600, New York, NY, USA, 2010. ACM. [13] E. F. Legara, C. Monterola, D. E. Juanico,

M. Litong-Palima, and C. Saloma. Earning potential

in multilevel marketing enterprises. Physica A: Statistical Mechanics and its Applications, 387(19-20):4889–4895, aug 2008. [14] J. Lehmann, B. Gon¸calves, J. J. Ramasco, and

C. Cattuto. Dynamical classes of collective attention in twitter. In Proceedings of the 21st International Conference on World Wide Web, WWW ’12, pages 251–260, 2012. [15] P. Lind, L. da Silva, J. Andrade, and H. Herrmann.

Spreading gossip in social networks. Physical Review E, 76(3), sep 2007. [16] P. G. Lind, L. R. da Silva, J. S. Andrade, and H. J.

Herrmann. The spread of gossip in American schools. Europhys. Lett., 78(6):68005, jun 2007. [17] A. Louni and K. P. Subbalakshmi. Diﬀusion of

Information in Social Networks. In Intelligent Systems Reference Library, pages 1–22. Springer Science+Business Media, 2014. [18] J. McAuley and J. Leskovec. Learning to Discover

Social Circles in Ego Networks. In Proceedings of the 2012 Neural Information Processing Systems Conference, 2012. [19] S. Myers, C. Zhu, and J. Leskovec. Information

diﬀusion and external inﬂuence on networks. In Proceedings of the 18th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD ’12, pages 33–41, New York, NY, USA, 2012. ACM. [20] J. Ratkiewicz, M. Conover, M. Meiss, B. Gon¸calves,

S. Patil, A. Flammini, and F. Menczer. Truthy. In Proceedings of the 20th international conference companion on World wide web, WWW ’11. ACM Press, 2011. [21] A. Rudat and J. Buder. Making retweeting social: The

inﬂuence of content and context information on sharing news in twitter. Computers in Human Behavior, 46(0):75–84, 2015. [22] Y. Sano, K. Yamada, H. Watanabe, H. Takayasu, and

M. Takayasu. Empirical analysis of collective human behavior for extraordinary events in the blogosphere. Phys. Rev. E, 87:012805, Jan 2013. [23] L. Shifman and M. Thelwall. Assessing global diﬀusion

with web memetics: The spread and evolution of a popular joke. Journal of the American Society for Information Science and Technology, 60(12):2567–2576, dec 2009. [24] T. Tassier. A model of fads fashions, and group

formation. Complexity, 9(5):51–61, 2004. [25] L. Weng, J. Ratkiewicz, N. Perra, B. Gon¸calves,

C. Castillo, F. Bonchi, R. Schifanella, F. Menczer, and A. Flammini. The role of information diﬀusion in the evolution of social networks. In Proceedings of the 19th ACM SIGKDD international conference on Knowledge discovery and data mining - KDD ’13. ACM Press, 2013.

APPENDIX A. HASHTAG TYPE vs. ITS CLASS The 88 hashtags used in this study. They belong to 13 types of event. Full description of the meaning of the hashtags could be found in [14].

Class −→ A B P S Hashtag type ↓

Activism (2) #ie6 #pman

Technology (3)

Charity (2) #twestival #protest

Sport (6)

Honour (3) #hoppusday #poynterday #asot400 Holiday (3) #aprilfool s

#safari #safari4 #skype

#masters #superbowlads #nfl #superads09 #nfldraft #superbowl

#easter #happy09

#rp09 #macworld #w2e #mix09 #ces #leweb #ces09 #drupalcon #cebit #25c3 Awareness (4)

#glmagic #skittles #evernote #free #macheist

Marketing (5)

#bsg #americanidol #grammys #watchmen #bachelor #starwarsday #oscars #phish #oscar

Political (10)

#amazonfa il

#googmayh arm

Disruption (14)

#g20 #rncchair #spectrial #budget #inaug09 #teaparty #nsotu #davos #coalition #hadopi

#gfail #snowmage ddon

#h1n1

#peace #winneden #gmail #mikeyy #influenza #swineflu #schiphol #bushfires #blackout

#yourtag #unfollow friday

Twitter (17)

#blogger #firstfol low

#socialmedia #plurk #3hotwords #oneword

#tweepme #iloveyou #nerdpick up

#crapname s

#dbi

#myfirstj ob

#oscarwil deday

#followme #politics

Source visual evidence

Figure 1: Rules of the model proposed in this work. t0 is the day of the peak and η is the amount of activities by the user’s leaders accumulated after his last tweet.

Figure 2: Behaviour patterns of diﬀerent types of users according to their number followers and lead- ers.

Figure 3: Distribution of diﬀerent types of event in the (λ, η⋆) parameter space.

Figure 4: Time series of activities (top) and users (bottom). Results from the model (blue) shown together with the data (red) presented in [14] for classes A, B, P, and S, respectively.

Figure 5: Fitted parameters η⋆and λ showing clustering patterns. The circles of larger size correspond to large value of δt. The colours (online) of the data points are determined by the classed identiﬁed in [14], red for S, black for P, blue for A and green for B.