The Anatomy of a Scientific Rumor

M. De Domenico,1 A. Lima,1 P. Mougel,1, 2 and M. Musolesi1

1School of Computer Science, University of Birmingham, United Kingdom2Universite de Lyon, INSA-Lyon, Villeurbanne, France

The announcement of the discovery of a Higgs boson-like particle at CERN will be remembered asone of the milestones of the scientific endeavor of the 21st century. In this paper we present a studyof information spreading processes on Twitter before, during and after the announcement of thediscovery of a new particle with the features of the elusive Higgs boson on 4th July 2012. We reportevidence for non-trivial spatio-temporal patterns in user activities at individual and global level,such as tweeting, re-tweeting and replying to existing tweets. We provide a possible explanation forthe observed time-varying dynamics of user activities during the spreading of this scientific “rumor”.We model the information spreading in the corresponding network of individuals who posted a tweetrelated to the Higgs boson discovery. Finally, we show that we are able to reproduce the globalbehavior of about 500,000 individuals with remarkable accuracy.

The Higgs boson, whose existence has been hypothe-sized in 1964 [1], has gained the title of the most elusiveparticle in modern science. The search for its existencehas been among the top research priorities of the parti-cle physics community for nearly 50 years. 2012 will beprobably remembered as one of the most important yearsin this century for physics: on 4th July 2012 the ATLASand CMS collaborations, two international experimentsinvolved in the search for the Higgs boson, announced theresults of the discovery of a new particle with the featuresof the elusive Higgs boson, the missing component of theStandard Model.

The elusive nature of the Higgs boson required the de-velopment of a new generation of large-scale experimen-tal facilities, resulting in the construction of the LargeHadron Collider (LHC) at CERN, in Geneve (Switzer-land), the largest and most powerful particle acceler-ator ever built. The other detector able to find hintsabout the existence of the Higgs boson is the Tevatronat Batavia, IL (USA). The association of the Higgs bo-son to the idea of the final understanding of our Universeand the possibility of the Grand Unified Theory [2–5] islikely to be responsible for the huge popularity of thisresearch project in both academic and non-academic cir-cles. Indeed, the interest from both specialized and pop-ular media increased after the “God particle” nicknamewas assigned to the Higgs boson [6].

The announcement of this discovery was the first ofthis kind in the era of global online social media, such asTwitter: the entire world followed and discussed the newsand updates through them, commenting and providingpersonal views about the event. All this information ispublicly generated online and represents an extremely in-teresting source of data for analyzing the global dynamicsof this scientific rumor around the world.

On 2nd July, initial results were presented by the Teva-tron team, but they were not sufficient to claim a sci-entific discovery. The statistical significance of all thecombined analyses was 2.9 sigma, equivalent to a 1-in-550 chance that the signal was due to a statistical fluc-tuation [7]. Although of remarkable importance for thescientific community, such an announcement had a weak

impact on the general public. Following this, there was astrong expectation, accompanied by rumors, for the cor-responding results from the CERN teams. An unofficialvideo was even leaked during those days [8]. The spread-ing of these rumors about a possible discovery attractedthe interest of media, also outside the academic com-munity, until the official day of the announcement on 4th

July during the International Conference on High-EnergyPhysics 2012 in Melbourne, Australia.

We can summarize the events before and after the dis-covery of the boson, dividing them into 4 different peri-ods:

• Period I: Before the announcement on 2nd July,there were some rumors about the discovery of aHiggs-like boson at Tevatron;

• Period II: On 2nd July at 1 PM GMT, scientistsfrom CDF and D0 experiments, based at Tevatron,presented results indicating that the Higgs particleshould have a mass between 115 and 135 GeV/c2

(corresponding to about 123-144 times the mass ofthe proton) [7];

• Period III: After 2nd July and before 4th of Julythere were many rumors about the Higgs boson dis-covery at LHC [8];

• Period IV: The main event was the announce-ment on 4th July at 8 AM GMT by the scientistsfrom the ATLAS and CMS experiments, based atCERN, presenting results indicating the existenceof a new particle, compatible with the Higgs bo-son, with mass around 125 GeV/c2 [9, 10]. After4th July, popular media covered the event.

In this paper, we present the anatomy of the spread-ing of this scientific rumor by following and analyzingthe related Twitter user activity during and after theannouncement. More specifically, we consider the mes-sages posted in Twitter about this discovery between 1st

and 7th July 2012. We report evidence for non-trivialspatio-temporal patterns in user activities at individualand global level, as tweeting, re-tweeting or replying to

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existing tweets. Well-defined trends can be associated todifferent periods of time. Abrupt changes can be linkedto the key events around the announcement. We ana-lyze the activity patterns of the individuals that tweetedabout this discovery over the period taken into consider-ation. We propose a model for the information spreadingover the Twitter network, assuming memoryless individ-uals where the activation process is driven by social rein-forcement at neighborhood level. Finally, we show thatwe are able to reproduce the global behavior of more than500,000 individuals with remarkable accuracy.

Results

Overview of the Dataset

Our dataset consists of messages posted on the Twit-ter social network, crawled by means of the Applica-tion Programming Interface (API) made available by theservice itself. We collected tweets sent between 00:00AM, 1st July 2012 and 11:59 PM, 7th July 2012 con-taining at least one of the following keywords or hash-tags: lhc, cern, boson, higgs (see Methods). The finalamount of tweets we analyzed was 985, 590. Hence, webuilt the corresponding social network of the authors ofthe tweets: the resulting graph is composed of 456,631nodes and 14,855,875 directed edges. Nodes correspondto the authors of the tweets and edges represent the fol-lowee/follower relationships between them. We discarded70,838 users from the original dataset containing 527,469users because of the non accessibility of the list of theirfollowees and followers due to privacy settings. Twit-ter users can specify their location by filling the Locationfield of their profile, on optional basis and at different lev-els of granularity (e.g., United States, New York, Chelsea,etc.). We use this information, when available, to assigna geographic position to each tweet: the resulting numberof geo-located tweets is 632,027 (see Methods).

In Fig. 1 we show the distributions of the in-degree,out-degree and total degree of the users that tweetedabout the Higgs boson. Intriguingly, the underlyingtopology is not trivial. The out-degree distribution showsa power-law scaling with two different regimes: P (kout) ∝k−1.4out and P (kout) ∝ k−3.9

out , with crossover for kout ≈ 200,which indicates that very few users follow more than afew hundred users. Conversely, the in-degree distributionshows a different behavior: for in-degree smaller thankin ≈ 100 the scaling relation is not satisfied, whereasabove this threshold the network exhibits a power-lawscaling P (kin) ∝ k−2.2

in . A standard method to uncoverthe presence of correlations in the network is to investi-gate the assortative mixing of its nodes [11, 12]. In fact,the nodes in the network with a large number of links maytend to be connected to other nodes with many connec-tions (assortative mixing with positive assortative index)or to other nodes with a few connections (disassortativemixing with negative assortative index). In both cases,

Figure 1: Probability density of in-degree, out-degree andtotal degree of the nodes that tweeted about the Higgs boson.The corresponding distributions have been shifted along they axis to put in evidence their structure. Dashed lines areshown for guidance only.

Figure 2: Number of tweets per second as a function of timeduring the period of data collection. The curves correspondto tweets containing only the CERN, Higgs, LHC keywords andat least one of them, respectively.

the network shows degree correlations resulting in an as-sortative index different from zero, at variance with anuncorrelated network where this index is close to zero.In our case study we find a value of about -0.14, indi-cating the presence of correlations in the network, withdisassortative mixing of users.

Spatio-temporal Analysis

In this section, we investigate both spatial and tem-poral features of the observed data, i.e., user activity onTwitter before, during and after the main event on 4th

July 2012. More specifically, we focus our attention onthe study of user behavior by considering two differentanalyses: the first one is performed at a global (macro-scopic) level, while the second one is performed at anindividual (microscopic) level.

Macroscopic Level. We consider the entire set of in-dividuals as a large-scale complex system of interactingentities, and we analyze the dynamics at a macroscopic

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level of such a system by inspecting spatio-temporal pat-terns of consecutive tweets. The inter-tweets time (space)is defined by the temporal delay (spatial distance) be-tween two consecutive tweets posted by any user in thenetwork.

In Fig. 2 we show the evolution of the rate of tweetscontaining the CERN, Higgs, and LHC keywords. The rateshows a rapidly increasing trend up to the day of theannouncement of the CERN teams, after which it slowlydecreases. It is worth noting that, when all the keywordsare considered, the rate of tweets increases from approx-imately 36 tweets/hour at the beginning of Period I upto about 36,000 tweets/hour at the beginning of PeriodIV. The rumors anticipating the presentation of results atTevatron caused the initial spreading of tweets about theHiggs boson. This was further sustained by the subse-quent comments to these initial postings and the rumorsabout the results to be presented by the scientists belong-ing to the ATLAS and CMS experiments. During a fewhours after the announcement of the discovery, the rateincreased by more than one order of magnitude, while itslowly decreased in the following days.

In the top panels of Fig. 3 we show the density of tweetsbefore (left panel), during (middle panel) and after (rightpanel) the main event, on 4th July 2012. In the bottompanels in the same figure we show the corresponding net-works of users built from re-tweets.

The impact of the announcement on 4th was trulyglobal. Instead, before and after this main event thecountries with a significant number of tweets were Euro-pean, probably due to the fact that CERN is in Switzer-land and the largest number of scientists working thereare from Europe. A large number of tweets were also ob-served from the United States, which hosts a very largecommunity of scientists.

Our first goal is to gain insights into the spatial andtemporal patterns of this complex geographic social net-work: in order to do so, we estimate the distance in timeand space of tweets posted in the network. In Fig. 4 weshow the number of tweets as a function of the inter-tweets time (first panel) and the inter-tweets space (sec-ond panel) between two consecutive messages. In bothcases, we show the distributions corresponding to the pe-riod before, during and after the main event on 4th July,respectively. While the distribution of inter-tweets spacesis the same regardless of the time window taken into con-sideration, the distribution of inter-tweets times in thethree windows is very different. From a global point ofview, this Twitter activity exhibits long tails before andafter the main event, with a large number of tweets sentwithin a few seconds, and a small number sent within afew minutes. On the other hand, the dynamics of the pro-cess changes dramatically during the main event, whenthe inter-tweets time between consecutive tweets is likelyto be less than two seconds and no more than six seconds,indicating a frenetic user activity. A deeper investigationat an individual level of this bursty behavior is presentedin the next section.

In order to unveil the presence of spatio-temporal pat-terns of individuals with non-trivial relationships, in thelast three panels of Fig. 4 we show the joint probabilitydensity of inter-tweets times and inter-tweets spaces be-fore (third panel), during (fourth panel) and after (fifthpanel) the main event. Before the main event, consec-utive tweets are mainly sent at local scales within lessthan half a minute interval, generally within 8 seconds:consecutive tweets are more likely to be sent by users liv-ing within 20 km, although a significant number of tweetsis still posted on larger inter-tweets spatial scales. Thesystem dynamics changes dramatically during the mainevent: tweets from any part of the world are likely to besent within 2 seconds without a specific spatial pattern.User activity now is frenetic and information is quicklyspreading at any spatial scale. After the main event,the spatio-temporal dynamics tends to become similarto the activity before the main event, even if, in thiscase, users from any part of the world are still involvedin the process, with no apparent prevalence of small orlarge inter-tweets spaces.

Microscopic Level. We now analyze the dynamicsat a microscopic level (i.e., treating individuals sepa-rately) by inspecting inter-arrival times of activities astweeting, replying and re-tweeting. In the following,the inter-activity time for user u is defined by τu(i) =tu(i + 1) − tu(i), where tu(i) and tu(i + 1) indicate thetimes when user u sent the i−th and the i+ 1-th tweets,respectively.

In the first two panels of Fig. 5 we show the distri-bution of inter-tweets times τ (i.e., between consecutivetweets) during (first panel), before and after the mainevent (second panel). Intriguingly, before and after themain event, such distributions show power-law scaling oftype P (τ) ∝ τ−α, with α ≈ 1, over three decades ofinter-tweets times, from the scale of one minute to thescale of one day.

Timing of user activities is usually modeled using aPoisson distribution. However, there is evidence thatinter-tweets times between subsequent user actions fol-low a non-Poisson statistics, characterized by bursts ofrapidly occurring events separated by long periods of in-activity [13–15]. The bursty nature of user behavior hasbeen recently attributed [16] to decision-based queuingprocesses [17], where individuals tend to act in responseto some perceived priority. According to this model, thetiming of tasks to be executed is heavy-tailed, with rapidresponses in the majority of cases and a few responseswith very long waiting times [16]. Moreover, it has beenshown that bursty user activity patterns might have aremarkable impact on the spreading dynamics over com-plex networks: this dynamics might be related to thewaiting time distribution but it is not sensitive to thenetwork topology [18]. A similar dynamics is observedin our data: the distribution of inter-tweets times shownin the first panel of Fig. 5 reflects the bursty nature ofuser activities on online social networks, where individ-

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Figure 3: Top: heatmap for the density of tweets before (left panel), during (middle panel) and after (right panel) the mainevent on 4th July 2012. Bottom: corresponding networks of re-tweets between users. During the announcement, the Twitteractivity is truly global, whereas before and after the announcement, the most active countries were European and American,due to the large presence of scientists in these geographic areas. Map data: c© OpenStreetMap contributors, available underthe Open Database License.

Figure 4: First and second panels: Global spatio-temporal activities of any user in the social network. Number of entriesfor inter-tweets times (first panel) and inter-tweets spaces (second panel) between consecutive tweets, before, during and afterthe main event on 4th July. The dashed line indicates a power law ∼ τ−2 and is for guidance only. Third, fourth and fifthpanels: Joint probability density of inter-tweets times and inter-tweets spaces between consecutive tweets before (third panel),during (fourth panel) and after (fifth panel) the main event. In both cases, only the sub-set of geo-located tweets is considered.

uals are more likely to send several tweets in quick suc-cession within a few minutes, followed by long periods ofno or reduced activity, up to one day.

Inter-tweets times distribution during the main eventshows a very different behavior, more compatible witha log-normal law instead of a power-law scaling relation-

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Figure 5: Activity inter-tweets times of users in the social network. First panel: Number of entries for inter-arrival timesbetween consecutive tweets, before and after the main event on 4th July. Power scaling behavior is visible for certain ragnges ofvalues, dashed lines are for guidance only. Second panel: Number of entries for inter-arrival times between consecutive tweetsafter the main event. The dashed curve corresponds to a lognormal fit. Third panel: Number of entries for inter-arrival timesbetween replies. Fourth panel: Number of entries for inter-arrival times between re-tweets.

ship. In this case, if τ is the random variable representingthe time to the next tweet, the random variable log τ isnormally distributed with mean µ and standard deviationσ. If a user starts to tweet because he or she is triggeredby tweets of users in his or her social neighborhood, thetotal number of tweets can pass a threshold value abovewhich a cascading effect may occur in the network. Ifthis is the case, we observe a random multiplicative [19]spreading of information, whose inter-tweets times aredistributed following a log-normal law if the number ofvertices involved in the cascade is large enough. The log-normal law with µ = 5.627±0.008 and σ = 1.742±0.006describes the observed activities during the main eventwith remarkable accuracy.

In the last two panels of Fig. 5 we show the distribu-tions of inter-arrival times for replies (third panel) andre-tweets (fourth panel) during the entire data collectionperiod. Intriguingly, user activities are still character-ized by bursty behavior. Inter-arrival times for repliesfollow a power-law P (τ) ∝ τ−1.2 from a few minutes upto one day: such a scaling can be explained with the ex-istence of bursty behavior in the timing of user actions,previously discussed in the case of inter-tweets times be-fore and after the main event. It is worth noting thatthe scaling exponent is larger for intervals between theoriginal tweets and replies than for re-tweets. For tem-poral scales larger than one day, the power-law scalingis not present; an exponential cut-off does not model theobserved decay.

The case of re-tweets deserves particular attention.From the time scale of a few minutes up to the time scaleof a few hours, we observe the power-law scaling relation-ship P (τ) ∝ τ−0.8, with a cut-off on the time scale of oneday. We model the data with a power-law with an ex-ponential cut-off P (τ) ∝ τ−0.8 exp(−τ/τ0), with cut-offscale τ0 ≈ 11 hours. It is worth remarking that power-law scaling relationships with exponent α ≤ 1 cannot benormalized and do not occur in nature unless the scalingdeviates from power law after some threshold value, the

cut-off scale, above which the distribution rapidly falls tozero. Even in such cases, phenomena exhibiting scalingexponents smaller than unity are very rare [20].

Rumor Spreading

In this section, we investigate the dynamics of informa-tion spreading in the social network of users who twittedabout the Higgs boson. Despite the fact that informa-tion spreading shares some general dynamical featureswith the spreading of diseases, their nature is deeply dif-ferent. For instance, disease epidemics depends on thephysical contacts between individuals and the differentbiological characteristics of both the infectious agent andthe carrier, as well as many other factors [21], whereasinformation can also be spread through non-physical con-tacts making use of communication infrastructures suchas telephone, television and Internet [22]. Informationis very volatile and it is not subject to incubation pe-riods: it is only worth spreading or not and this deci-sion is made by individuals, unlike the case of diseasespreading. In the last decade, the study of contagiondynamics, involving either information or disease trans-mission, has greatly benefited from key results in complexnetworks modeling [23–30]: in fact, the structure of so-cial relationships plays a fundamental role for any typeof spreading dynamics [31–33]. If the underlying topol-ogy of the network is homogenous, the dynamics can bestudied by adopting a mean-field approximation and thespreading occurs only if the rate of transmission of in-formation exceeds an epidemic threshold. Conversely,heterogeneous structures like scale-free networks requireheterogeneous mean-field approximation [23, 24], involv-ing the single-site equation governing the time evolutionof the relative density of “infected” vertices with givenconnectivity k, i.e., the probability that a vertex with de-gree k is infected. Moreover, such networks have the pe-culiar property of facilitating the spreading of infections:

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Figure 6: Visualisation of the social network of active users,based on k-core decomposition and components analysis. Thesize of each vertex is proportional to its degree, whereas colorcodes the k-coreness. A sample of 10% of the whole networkhas been used for this visualisation.

in fact, if the corresponding degree distribution shows di-verging second moment, then the epidemic threshold iszero independently from the degree correlations [34]. Al-though mean-field approximations are fundamental toolsto capture the main features of the spreading dynamics,particularly in the early stage, the models are less effi-cient when the finite size of the population becomes asignificant factor. More recent approaches focus on theprobability of transmission of individual vertices [35] andnon-perturbative formulation of the heterogeneous mean-field approach [36].

In our analysis, we will distinguish between two differ-ent states for users in the social network: “active” and“non-active” vertices. We will indicate with “tweetingactivation” or “rumor spreading” the user-to-user inter-action process for transmitting information related to aparticular topic. In the following, we will indicate withA(t) and D(t) the number of active and non-active usersat time t, respectively, with A(t) + D(t) = N , where Nis total number of users considered in the social network.The observed social network of active users is shown inFig. 6, where a visualization based on k-core decomposi-tion and component analysis is presented [37, 38]. Thek-core of a graph is defined as the maximal connectedsubgraph in which all vertices have degree at least k. Inpractice, a k-core is obtained by recursively removing allvertices with degree smaller than k, until the degree ofall remaining vertices is larger than or equal to k. The k-coreness of a vertex is the index of the highest k-core con-taining that vertex. Vertices with the highest k-corenessact as the most influential spreader of information in thenetwork. In fact, it has been recently shown that in someplausible circumstances the best spreaders are not themost highly connected or the most central people but

Figure 7: Points indicate the fraction of users who are activeat least once (see the text for more detail) with respect tothe total number of users in the dataset at the end of theperiod taken into consideration, i.e., A?(t = 8 July 2012), asa function of time. Lines indicate the fitting results obtainedseparately for each temporal range by adopting the modelgiven by Eq. (3). The rate of activation λ? for each period isreported at the bottom of the figure.

those with higher k-coreness [39], and there is evidenceof a positive correlation between k-coreness and the sizeof cascades of messages, suggesting that users at the coreof the network are more likely to be the seeds of globalchains of information diffusion [40].

The k-core decomposition allows to identify somesalient features of the observed social network of activeusers, uncovering structural properties due to its specifictopology. In Fig. 6, the presence of an inhomogeneousdistribution of vertices in the shells is a signature of non-trivial correlations. Moreover, the presence of verticeswith high degree in any k-shell, i.e., a very low correla-tion between degree and shell-index, indicates that hubsare likely to be found also in external shells, a behaviortypical of networks without an apparent global hierarchi-cal structure like the World Wide Web [37, 38].

Modeling the Dynamics of User Activation with-out De-activation. As the first step, we do not considerthe influence of the structure of the network on the pro-cess. We define a node as active at time t if he or she hastweeted at least once about the Higgs boson within thatinstant of time. In the following, we indicate with A?(t)and a?(t) the number and the fraction of active users attime t, respectively.

Hence, the number A?(t) of active users is expected tobe a monotonic increasing function of time. We dividethe whole period of data taking into four temporal rangesof interest, corresponding to periods I, II, III and IV,previously described. In Fig. 7 we show for each periodthe observed evolution of the fraction a?(t) = A?(t)/N ofinfected users versus time, where N is the total numberof users in the dataset in the data collection period.

In order to model the evolution of the active usersover time, we firstly tried to exploit classic susceptible-infected (SI) models in an unstructured population [41],

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but this led to a very poor fit of the data. For this reason,we developed a new model starting from the observationof the specific characteristics of our dataset. In general,once a user has twitted, we observe that he or she willnot probably tweet significantly about the Higgs boson inthe near future, according to the bursty behavior shownpreviously. Therefore, we make the simplifying assump-tion that he or she will not tweet again after the tweetingactivation. In this case, the number of newly active ver-tices at time t is proportional to the number of users whohave not been active before:

A?(t+ ∆t) = A?(t) + λ? [N −A?(t)] ∆t, (1)

where λ? is a constant activation rate. In the limit ofsmall ∆t, we obtain the following ordinary differentialequation:

da?(t)

dt= λ? [1− a?(t)] , (2)

corresponding to our model for the fraction of userstweeting at least once about the Higgs boson. The evo-lution over time of a?(t) is the solution of Eq. (2), givenby

a?(t) = 1− [1− a?(tk)] e−λ?(t−tk), (3)

where k = I, II, III and IV indicates the period of in-terest, tk is the starting date of period k and a?(tk) isthe corresponding initial fraction of users active at leastonce.

We fit the evolution function given by Eq. (3) to theobserved data for each period of interest: the resultingmodel for each case is shown in Fig. (7), demonstratingthe agreement with the data. The activation rate in-creases during the four intervals of time taken into con-sideration, from about one user per minute on 1st July2012, up to about 519 users per minute in the last period.

Modeling the Dynamics of User Activation withDe-activation. In this subsection, we will focus on thepropagation of interest on the event through social cas-cading. A user is considered non-active in a given timewindow ∆t if he or she has not tweeted in that time in-terval. In other words, in this refined model, an activeuser can become non-active again (de-activated) if he orshe does not keep tweeting about the Higgs boson. Inthe time interval between t and t + ∆t active users canbecome non-active after a certain amount of time for anyreason: we indicate with β(t) the probability per unit oftime for the transition from active to non-active state.Hence, the number of users that becomes non-active inthe interval ∆t is given by β(t)A(t)∆t. By introducingde-activation we also account for the limited visibilityof tweets on the timelines of Twitter clients, i.e., newertweets replace older ones. Moreover, we observe that thenumber of non-active users at time t that will become

active at time t+ ∆t is a function of both their in-goingdegree and the out-going degree of active users at time t.

A non-active user connected to more than one activeuser at the same time is more likely to become activewith respect to non-active users connected to only oneactive user. Let us indicate with jA the number of ac-tive users connected to a non-active user. If λ(t) indi-cates the activation probability per unit of time per link,for a non-active user with degree kin the probability perunit of time of changing from non-active to active state

is given by pλ(t; jA) = 1 − [1− λ(t)]jA . In general, the

probability that such a non-active user is connected to jAactive users at the same time depends on the out-goingdegree of active users, i.e., on network vertex-vertex cor-relations. More specifically, such a probability dependson the conditional probability of observing a vertex without-going degree kout connected to a vertex with in-goingdegree kin.

It has been shown that a pure scale-free degree dis-tribution with exponent between 2 and 3 is a sufficientcondition for the absence of an epidemic threshold inunstructured networks with arbitrary two-point degreecorrelation function [34], i.e., correlations at neighbor-hood level do not affect the spreading dynamics. We usethis result as a simplifying assumption for modeling thespreading in our network, exhibiting a scale-free degreedistribution with exponent 2.5 for k > 200. Therefore,we neglect correlations and we estimate the probabilitythat a non-active user, with in-going degree kin, is con-nected to jA active users, with any out-going degree, by

p(t; jA, kin) =

(A(t)jA

)(N−A(t)−1kin−jA

)(N−1kin

) , (4)

accounting for all the possible ways to arrange A(t) ac-tivations within jA users from the total number of pos-sible combinations of the remaining N − 1 users withinkin users.

Hence, the probability that a non-active user with in-going degree kin is activated by at least one active userin its neighborhood is given by

Pλ,kin(D −→ A) =

kin∑jA=1

p(t; jA, kin)pλ(t; jA). (5)

It follows that the total probability that non-active userswill become active per unit of time is given by

Θλ(t) =∑kin

P(kin)Pλ,kin(D −→ A), (6)

being P(kin) the probability density of the in-going de-gree. It follows that (N−A(t))P(kin) indicates the num-ber of non-active users with in-going degree kin at timet. We model the dynamics of the number of active usersin the time interval ∆t by

A(t+ ∆t) = A(t) +[−β(t)A(t) + (N −A(t))Θλ(t)(t)

]∆t.

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Therefore, by choosing ∆t = 1, i.e., equal to the timeunit of observation, we obtain the general discrete model

A(t+ 1) =(

1− β(t))A(t) + (N −A(t))Θλ(t)(t), (7)

valid for the general case of activation and de-activationrates that change over time.

In Eq. (7) the parameters β = β∆t and λ = λ∆t indi-cate probability instead of probability rates. However, inthe particular case of ∆t = 1 it is possible to mix ratesand probabilities because both will have the same values,even though their units are different [36]: for sake of sim-

plicity, in the following we use the notation β = β andλ = λ. Eq. (7) represents the balance equation indicat-ing that the number of active users at a certain instantis given by the number of vertices that at the previousinstant did not change from active to non-active stateplus the number of newly active users. In the followingwe will consider the density of active users defined byρ(t) = A(t)/N , leading to the evolution equation

ρ(t+ 1) = (1− β(t)) ρ(t) + (1− ρ(t))Θλ(t)(t). (8)

In general, the solution of Eq. (7) and Eq. (8) cannotbe obtained analytically because of the complexity ofΘλ(t)(t): therefore, some simplifying assumptions or nu-merical methods should be adopted instead.

Let us focus only on Period IV, i.e., during and afterthe main event, from 03:00 AM on 4th July to the end ofthe data collection period. The initial fraction of activeusers is approximately ρ(0) = 0.1% of the total numberof users in our dataset.

In order to assess the validity of our analytical model,we perform large-scale Monte Carlo simulations of thespreading dynamics through the network of observed con-nections among users. More specifically, we considerthe case where activation and de-activation rates do notchange over time: we vary their values from 0 to 1, inde-pendently; for each possible configuration correspondingto the pair (β, λ) we perform 200 random independentrealizations of rumor spreading and we calculate the en-semble average at each time step t to obtain an estima-tion of the expected value of the density ρ(t). The resultsfor the case with β = 1 and several different values of λare shown in the first two panels of Fig. 8. In the firstpanel, we show the evolution of ρ(t) versus time: forλ < 6 × 10−3 the density ρ(t) tends to decrease to zerofor increasing time, while for λ ≥ 6 × 10−3 the densityρ(t) tends to reach a stationary state, indicating that thespreading becomes endemic. We indicate with ρ? the sta-tionary value reached by ρ(t) after the transient time. Inthe second panel, we show the value of ρ? versus the ac-tivation rate: the endemic state, where ρ? > 0, is quicklyreached for small values of λ. This result is qualitativelyconfirmed by our analytical model (see Eq. (8)) and itis in agreement with the result reported in [34], statingthat the epidemic threshold of an endemic state tends tozero for increasing network size with a scale-free topol-ogy. However, such results do not reproduce the observed

spreading dynamics, whose density ρ(t) is shown in thethird panel of Fig. 8. The data show a quickly increasingnumber of users within a few hours, with a maximumvalue reached at the beginning of the International Con-ference on High-Energy Physics. Such a fast increasingbehavior can be explained by tweets related to the ex-citement for a possible announcement of the discovery ofthe Higgs boson. In fact, the number of active users inthe following hour rapidly decreases by about 40%, stay-ing stable for the subsequent 2 hours and then decreasingagain.

In case of epidemics with constant activation rate inscale-free networks with a large number of nodes we ex-pect the appearance of an endemic state. However, thisis not the case in our dataset. For this reason, we modifythe model by introducing a variable activation rate λ, ac-counting for the decreasing interest on a tweet over time,according to recent studies suggesting the existence of anatural time scale over which attention fades [42]. Wemodel the evolution of λ as follows:

λ(t+ 1) = (1− ξ)λ(t), 0 < ξ < 1, (9)

which is the discrete counterpart of the continuous equa-tion whose solution is the exponential decay λ(t) =λ(t0)e−ξ(t−t0). Here, we interpret ξ as the inverse of acharacteristic scale τ regulating the decay dynamics. Weuse the coupled equations (8) and (9) to model the ob-served spreading dynamics.

During the whole Period IV, we identify five sub-periods, each one characterized by an increasing numberof active users followed by a decreasing one. We thenvary the parameters β, λ and τ in order to try to repro-duce the data in each sub-period. The solid curves in thethird panel of Fig. 8 correspond to our model (Eq. (8)and Eq. (9)) using the set of parameters minimizing χ2.The rapid increase of active users in the first sub-periodof Period IV is followed by a fast decrease, with timescale τ ≈ 1.13 hours, initial activation rate λ0 = 1 andβ = 0.17. Such a fast decreasing trend is slowed afterabout 9 hours, approximately at the time when the ru-mor has reached the other side of the world in the earlymorning: from this time instant up to the end of theobservation, the values of de-activation probability andthe initial value of the activation probability are almostconstant (ranging from 0.31 to 0.38, and from 0.40 to0.45, respectively). In the following sub-periods only thedecay time scale τ significantly varies from 3 to 17 hours.

Discussion

On 4th July 2012, the ATLAS and CMS collabora-tions announced the discovery of a new particle, with thesame features of the elusive Higgs boson. Such a findingrepresents a milestone in particle physics and a uniqueoccasion to study the dynamics of information spread-ing on a global scale. In this study, we have monitoreduser activities on Twitter before, during and after the

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Figure 8: First panel: Evolution of the density of active users versus time obtained from simulations of spreading dynamics.The de-activation rate is β = 1 and the different curves correspond to different values of the activation rate λ. Each curvecorresponds to the ensemble average of 200 random independent realizations. Second panel: Average value of the density ofactive users in the stationary state, as a function of λ. Third panel: Observed evolution of the density of active users versustime (points) in Period IV, i.e., during and after the main event, from 03:00 AM, 4th July. Curves indicate the predictionsobtained from the model defined by Eq. 8 coupled to Eq. 9, where the values of the corresponding parameters are reported inthe figure for different sub-periods. The reported λ refers to the initial value of the activation rate.

announcement of the discovery of this Higgs boson-likeparticle.

The joint analysis of spatial and temporal user activitypatterns unveiled specific dynamics in different periodsof the rumor spreading. Before the announcement of theteams based at CERN, tweets were more likely to be sentwithin a few seconds by users living within 20 km. Dur-ing the main event the activity became frenetic and itstime scale reduced to 2 seconds without a specific spatialpattern. After the main event a less frenetic activity hasbeen observed while users from any part of the world werestill involved in the process, with no apparent prevalenceof small or large inter-tweets spaces.

Finally, we have focused our attention on the networkof individuals who posted at least one message about thediscovery (tweets, re-tweets or reply to a tweet). The ob-served network of users exhibits a non-trivial structurewith a typically bursty tweeting process happening overit. We have proposed a model for information spreadingwith variable activation rate in heterogenous networksshowing that we are able to reproduce the collective be-havior of about 500,000 users with remarkable accuracy.Our model assumes memoryless individuals where theactivation process is driven by social reinforcement atneighborhood level. The active (or non-active) states ofhis or her social neighbors at each time step act as a“topological memory”, causing the individual to be acti-vated with larger probability if most of his or her friendsare tweeting the rumor repeatedly in time.

Even if the proposed models have been developed forthis specific series of events, we believe that the pro-posed framework can be applied and fitted to many otherspreading processes on online and physical social net-works.

Methods

For each search, Twitter provides information aboutthe number of missing tweets, which is usually negligi-ble. In any case, it worth noting that during this datacollection no messages related to missing tweets were re-ceived: for this reason, to the best of our knowledge,we might claim that this dataset includes all the tweetssatisfying our search criteria.

Our original list of relevant keywords was larger, in-cluding terms as alice and cms. However, the amountof tweets retrieved using these keywords but not relatedto the Higgs boson was not negligible and, for this rea-son, we decided to avoid considering such terms in ouranalysis.

The geographic names contained in the locationtextfield of Twitter user profiles were converted using theGoogle Geocoder API.

Recent studies make use of the retweet network (i.e.,who retweets whom) and the mentions network (i.e., whomentions whom) to investigate, for instance, the patternsof sentiment expression [43]. However, it is worth re-marking that the main source of information consump-tion on Twitter website and clients is the home timeline,which contains messages from all social contacts, regard-less of the number of reciprocal interactions. Therefore,users activity can be triggered (or not) according to theirinterests. In our study, we prefer to use the follower net-work to investigate the temporal dynamics of the fractionof people involved in the process of spreading the infor-mation about the Higgs boson discovery as a function oftime.

The social graph was obtained by retrieving the fol-lowing list for each user participating in the process. Wemake the reasonable assumption that the time requiredto collect the data is much shorter than the time requiredto observe significant changes in the full social graph.

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Acknowledgments

The authors thank J.I. Alvarez-Hamelin and A. Mam-brini for useful and fruitful discussions. This work was

supported through the EPSRC Grant “The Uncertaintyof Identity: Linking Spatiotemporal Information Be-tween Virtual and Real Worlds” (EP/J005266/1).

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