On the statistical properties of viral misinformation in online social media

On the statistical prop erties of viral misinformation in online so cial media Alessandro Bessi ∗ a University of Southern California, Information Scienc es Institute, Marina del R ey, L os Angeles, CA, USA b IUSS Institute for A dvanc e d Study, Pavia, IT AL Y c IMT Institute for A dvanc e d Studies, Luc c a, IT AL Y Abstract The massiv e diffusion of online so cial media allo ws for the rapid and uncon- trolled spreading of conspiracy theories, hoaxes, unsubstan tiated claims, and false news. Suc h an impressive amoun t of misinformation can influence pol- icy preferences and encourage b eha viors strongly divergen t from recommended practices. In this pap er, w e study the statistical prop erties of viral misinfor- mation in online so cial media. By m eans of metho ds b elonging to Extreme V alue Theory , we sho w that the num ber of extremely viral p osts ov er time fol- lo ws a homogeneous Poisson process, and that the in terarriv al times betw een suc h p osts are indep enden t and iden tically distributed, follo wing an exp onen tial distribution. Moreo ver, we c haracterize the uncertaint y around the rate param- eter of the Poisson pro cess through Bay esian metho ds. Finally , we are able to deriv e the predictiv e p osterior probability distribution of the num b er of p osts exceeding a certain threshold of shares o ver a finite interv al of time. Keywor ds: misinformation, online so cial media, extreme v alue theory 1. In tro duction The wide a v ailability of user-pro vided con tents in online social media encour- ages the aggregation of p eople around common interests and narrativ es. The direct path from pro ducers to consumers of conten ts driv es the emergence of a disin termediated enviromen t that is changing the wa y p eople b ecome informed, in terpret facts, and form their opinions [1, 2, 3, 4, 5]. Unfortunately , suc h a disin termediation can facilitate the spreading of ru- mors, hoaxes, fake news, and conspiracy theories, that often arouse naive and a wkward so cial resp onses on different topics, suc h as health, environmen t, na- tional security , and p olitics [6, 7, 8, 9, 10, 11]. In particular, conspiracy theories simplify causation, reduce the complexit y of realit y , and are form ulated in a w a y ∗ Corresponding author Email addr ess: bessi@isi.edu (Alessandro Bessi) Pr eprint submitte d to Elsevier August 6, 2018 that is able to contain a certain level of uncertaint y [12, 13, 14, 15]. Ho w ever, suc h an impressive amount of misinformation can influence policy preferences and encourage b eha viors strongly divergen t from recommended practices. Since the W orld Economic F orum listed massiv e digital misinformation as one of the main threats to our so ciet y [16], comm unity-driv en [17] and algorithmic- driv en [18, 19, 20, 21, 22, 23, 24] solutions ha ve been proposed to coun teract the p erv as iv eness of online misinformation. Ho wev er, a part of the scientific comm unity is sk eptical ab out the real effectiveness of such solutions. Indeed, the communit y-driven approach prop osed by F aceb o ok – where users can flag false conten ts to correct the newsfeed algorithm – is con trov ersial, b ecause it raises fears that the free circulation of ideas may b e threatened. Moreov er, algorithmic-driv en approaches ma y not b e effective, since the acceptance of a claim (either substantiated or not) is heavily influenced by so cial norms and in- dividual cognitive factors [25, 26, 27, 28, 29, 30]. Indeed, recent w orks p oint out b oth the inefficacy of correcting false b eliefs and the concrete risk of a bac kfire effect [31, 32, 33] from the usual and most committed consumers of conspir- acy theories. In fact, false b eliefs, once adopted b y an individual, are rarely corrected [34, 35, 36, 37]. More sp ecifically , b oth the formation and the revision of b eliefs are strongly affected b y the communities wherein ideas and facts are debated [38, 39]. Such a phenomenon is emphasized in online so cial net works, where users pro cess infor- mation through a shared system of meaning [40, 41] inside their echo cham bers [42, 43, 44, 45], making sense of facts in wa ys that are often biased tow ard self- confirmation. Indeed, recent studies show that increasing the exp osure of users to unsubstan tiated rumors increases their tendency to b e credulous [26, 46], and that the con tent-selectiv e exp osure is the primary driv er of conten t diffusion, and generates the formation of ec ho cham b ers [42]. In this work, w e study the statistical prop erties of viral misinformation in online so cial media. In particular, we apply metho ds of Extreme V alue The- ory – a branc h of statistics dealing with extreme deviations from the median of probabilit y distributions – to analyze a large dataset of p osts published by F aceb ook pages supp orting conspiracy theories and m yth narratives. By means of an in-depth statistical analysis of the shares distribution and the application of the P eaks Ov er Threshold (POT) approac h, we show that the n umber of extremely viral posts (e.g. > 250 K shares) ov er time follo ws a homogeneous P oisson pro cess, and that the in terarriv al times b et ween suc h posts are indep en- den t and identically distributed, following an exp onen tial distribution. F urther, w e characterize the uncertaint y around the rate parameter of the Poisson pro- cess through Ba yesian metho ds. Finally , we are able to derive the predictiv e p osterior probabilit y distribution of the num b er of p osts exceeding a certain threshold of shares o ver a finite interv al of time. The relev ance of our results is not necessarily limited to the field of computa- tional social science coping with misinformation [47, 48, 49]. Indeed, despite the prediction of extremely viral posts and rare ev en ts remains an hard task [50, 51], w e b elieve that both our findings and the metho dology used herein may b e of in terest to the broader field of computational so cial science dealing with forecast- 2 ing and tracking of viral conten ts and ev en ts [52, 53, 54, 55, 56, 57, 58, 59, 60, 61]. 2. Metho ds 2.1. Ethics Statement The entire data collection pro cess has b een carried out exclusively through the F aceb ook Graph API, which is publicly a v ailable. W e used only public a v ailable data. The pages from which we downloaded data are public F aceb ook en tities. 2.2. Data Col le ction W e analyzed 328 US public F acebo ok pages diffusing conspiratorial b eliefs, m yth narratives, and contro versial information, usually lacking supp orting evi- dence and most often contradictory of the official news. Suc h a space of inv es- tigation is defined with the same approac h as in [26, 42], with the supp ort of differen t F aceb o ok groups v ery active in monitoring the conspiracy narratives. F or each page, w e downloaded all the posts (and their resp ectiv e metadata) in a timespan of 5 years (Jan 1, 2010 to Dec 31, 2014). The dataset is comp osed by 345 , 054 p osts. T o our knowledge, the dataset is the complete set of conspiracy- lik e information sources activ e in the US F aceb o ok scenario up to December 31, 2014. 2.3. F undamentals of Extr eme V alue The ory Extreme v alue theory (EVT) is a branch of statistics dealing with the ex- treme deviations from the median of probability distributions. In particular, it aims at assessing the probabilit y of even ts that are more extreme than any pre- viously observed. Extreme v alue theory is widely used in many fields of science where pow er laws play a role in mo deling [62], such as structural and geological engineering, finance and risk managemen t, earth sciences, traffic prediction, etc. In this section, we briefly review some fundamental results of extreme v alue theory . F or extended discussion, pro ofs, and theorems see [63, 64, 65]. 2.3.1. Extr eme V alue The ory Supp ose X 1 , X 2 , . . . are indep enden t and iden tically distributed (iid) random v ariables with common cumulativ e distribution function (cdf ) F . Let M n = max { X 1 , . . . , X n } denote the maximum of the first n random v ariables (partial maxima) and let u ( F ) = sup { x : F ( x ) < 1 } denote the upp er endpoint of F . Since Pr ( M n ≤ x ) = Pr ( X 1 ≤ x, . . . , X n ≤ x ) = F n ( x ) , M n con verges almost surely to u ( F ) whether it is finite or infinite. Extreme v alue theory se eks norming constants a n > 0, b n ∈ R , and some nondegenerate distribution function G such that the cdf of the normalized M n con verges to G , i.e. Pr  M n − b n a n ≤ x  = F n ( a n x + b n ) → d G ( x ) . 3 If this holds for suitable choices of a n and b n , then w e say that G is an extreme v alue distribution function, and F b elongs to the maximum domain of attraction of G , i.e. F ∈ M DA ( G ). The Extremal T yp es Theorem c haracterizes the limit distribution function G as of the t yp e of one of the following three classes: • Gum b el: Λ( x ) = exp (exp( − x )) , x ∈ R • F r´ ec het: Φ α ( x ) = ( 0 , if x ≤ 0 exp( − x − α ) , if x > 0 • W eibull Ψ α ( x ) = ( exp( − ( − x ) α ) , if x ≤ 0 1 , if x > 0 for some α > 0. The three extreme v alue distributions can b e represen ted using the general- ized extreme v alue (GEV) distribution (family). Let H ξ ( x ) = ( exp  − (1 + ξ x ) − 1 ξ  , if ξ 6 = 0 exp( − exp( − x )) , if ξ = 0 where 1 + ξ x > 0. Then, • ξ = α − 1 > 0 ← → Φ α • ξ = − α − 1 < 0 ← → Ψ α • ξ = 0 ← → Λ F rom a mo deling p oin t of view, the three extreme v alue distributions are v ery differen t, esp ecially for what concerns the b eha vior of the tails, i.e. the part of the distribution more relev an t when dealing with extreme even ts. Here, we fo cus on the F r´ echet case, Φ α with α > 0. If we consider the tail of Φ α ( x ), a T aylor expansion shows that 1 − Φ α ( x ) = 1 − exp( − x − α ) ∼ x − α , x → ∞ . Hence, Φ α ( x ) tends to decrease as a p o wer law. Moreo ver, ev ery distribution function that b elongs to MD A(Φ α ) has necessarily and infinite righ t endp oin t, i.e. it is defined for x ∈ [0 , ∞ ). It follo ws that all the distribution functions b elonging to MD A(Φ α ) are appropriate for modeling phenomena with extremely large maxima. 4 Consider a random v ariable X with unknown distribution function G and righ t endp oin t x G = sup { x ∈ R : G ( x ) < 1 } . The exceedance distribution function of X ab o ve a given threshold t is defined as G t ( x ) = Pr ( X ≤ x | X > t ) = G ( x ) − G ( t ) 1 − G ( t ) , x ≥ t. F or a large class of distribution functions G and a high threshold t → x G , G t can b e approximated by a Generalized Pareto Distribution, i.e. G t = GP D ( x ; ξ , β , t ) =    1 −  1 + ξ x − t β  − 1 ξ , if ξ 6 = 0 1 − exp  − x − t β  , if ξ = 0 where x ≥ t for ξ ≥ 0, t ≤ x ≤ t − β /ξ for ξ < 0, t ∈ R , ξ ∈ R , and β > 0. The shap e parameter, ξ , gov erns the fatness of the tails, and thus the existence of the moments. The moment of order p of a Generalized P areto distributed random v ariable only exists if and only if ξ < 1 /p . 2.3.2. Extr eme V alue Analysis Tw o approaches exist for practical extreme v alue analysis. The Blo c k Max- ima (BM) approac h consists on splitting the observ ation p erio d into a certain n umber of non-ov erlapping p erio ds of equal size – e.g. w eeks, months, y ears – and then considering only the maximal v alue within eac h p erio d. Such maximal v alues follow approximately a Generalized Extreme V alue (GEV) distribution. The P eaks Ov er Threshold (POT) approach relies on considering only the v alues exceeding a certain high threshold. The probability distribution of those selected observ ations is approximately a Generalized Pareto Distribution (GPD). Both approaches hav e some limitations. The POT approac h picks up all relev an t high observ ations, and thus seems to make b etter use of the av ailable information. Con versely , the BM approac h misses some of these high observ a- tions and retains some low er observ ations. How ever, there ma y b e reason for using the BM metho d: the only av ailable information ma y b e blo c k maxima (e.g. daily , weekly , mon thly , or yearly maxima) and the BM approach may b e preferable when the observ ations are not exactly iid. Ho wev er, if the BM approach ma y b e easier to apply when the blo ck perio ds app ear naturally , some problems arise when this do es not happen. In suc h a case, the choice of the block size for the BM approach may b e as difficult as the c hoice of the threshold for the POT approach. Figure 1 pro vides a graphical representation of the tw o approaches. 3. Results and Discussion In this pap er, we aim at inv estigating the statistical prop erties of viral mis- information on F aceb ook by means of extreme v alue theory . More sp ecifically , the ob ject of the analysis is the num b er of times that p osts supporting — in 5 Figure 1: Blo ck Maxima (BM) vs. Peaks Over Threshold (POT). this study , we are assuming that eac h share of conspiracy p osts represents the will to supp ort a given conspiracy narrativ e — conspiracy theories hav e been shared, whic h can b e considered as a random v ariable X following a generic distribution function F with supp ort [0 , ∞ ). Suc h an infinite right endp oin t is justified b y the fact that users can share a p ost how many times they desire. 3.1. Explor atory Data Analysis Since for eac h p ost we know the time of creation, we hav e a temp orally or- dered collection of observ ations. Such a time series is irregularly spaced, in the sense that it is characterized b y v arying in terarriv al times b et ween observ ations. A common approach to analyze irregularly spaced time series consists in trans- forming the data into equally spaced observ ations using interpolation metho ds, and then apply standard metho ds for equally spaced data. Ho wev er, suc h a transformation can introduce a num b er of significant and hard to quantify bias, esp ecially when the in terarriv al times b et ween observ ations are highly irregular. Since in our case the spacing of observ ations v aries from seconds to da ys, we a void to transform data. Moreo ver, despite observ ations are temporally ordered, it is difficult to assume some kind of time dep endence b et ween the num b er of shares receiv ed b y p osts — i.e. the num ber of shares receiv ed by a post do es not affect the n umber of shares receiv ed by following p osts. Rather, if we can conceiv e that some external ev ents (e.g. breaking news, top stories, scandals, etc.) can cause an un usual n umber of p osts in a restricted temp oral window (clustering), we can safely assume that the num ber of shares receiv ed by each of those p osts is indep enden t and identically distributed (iid). Figure 2 shows the cumulativ e n umber of weekly p ost ( top p anel ) and the n umber of weekly p osts ( b ottom p anel ). Despite it looks like there is a clear gro wth trend — which is likely due b y the increase of users on F aceb ook o c- curring from 2010 to 2014 — and some form of seasonality , w e are not able to iden tify an y meaningful seasonalit y pattern. Indeed, we can assume that the activit y of this kind of pages is primarily driven b y external even ts, suc h as breaking news, top stories, scandals, etc. 6 Figure 2: W eekly p osts. Cumulativ e num b er of weekly p ost ( top p anel ) and num b er of weekly p osts ( b ottom p anel ). The solid red line indicates the fitted linear trend. Ho wev er, an increase in the num ber of p osts published by pages in a giv en temp oral window ma y reflect an increase in the users’ excitement and activ- it y . W e accoun t for such a possible c haracteristic of the phenomenon under in vestigation by rescaling raw data by a factor defined as R i = w i max( w ) , i ∈ { 1 , . . . , 261 } where w i represen ts the num b er of p osts published b y pages in week i . Suc h a rescaling factor inflates the num b er of shares of p osts published in weeks c haracterized b y an ov erall low activity . Different rescaling strategies hav e b een considered — e.g. rescaling by the mean or the median —, and similar results ha ve b een obtained. Figure 3: Shares time series. Time series of the original num b er of shares ( r aw data ) and the rescaled one ( r esc ale d data ). Figure 3 shows the time series of the original num ber of shares ( r aw data ) and the rescaled one ( r esc ale d data ). The rescaling pro cedure should hav e remov ed or at least reduced p ossible clustering phenomena that would hav e led to a violation of the iid assumption. W e chec k the iid assumption b y means of the 7 records plot, a simple and intuitiv e exploratory to ol widely used in extreme v alue analysis whic h exploits the fact that successive records for iid data should b ecome more and more rare as time go es by . Since a record x n for the random v ariable X o ccurs if x n > max { x 1 , . . . , x n − 1 } , it is intuitiv e that if data are iid it b ecomes more difficult to exceed all past observ ations, and thus the n umber of records should follow a logarithmic pattern [63]. Records plots in Figure 4 sho w that the iid assumption for ra w data is violated, but still v alid for rescaled data, where records are distributed around their exp ected v alue and within the 95% confidence in terv als. Figure 4: Records plots. The iid assumption for raw data is violated, but still v alid for rescaled data, where records are distributed around their exp ected value and within the 95% confidence interv als. Figure 5 sho ws the empirical complementary cumulativ e distribution func- tions of ra w data and rescaled data. The double log scale of the figures highlights the fatness of the righ t tail in b oth the tw o empirical distributions. Beyond re- mo ving an y form of dep endence in the ra w data, we observe that the rescaling pro cedure slightly exacerbates the pow er la w behavior of the tail without in- fluencing the b ody of the distribution. Thus, rescaled data will be used for successiv e analysis. 3.2. Statistic al Pr op erties of Vir al Misinformation W e use the distribution function of rescaled data to characterize the statisti- cal prop erties of viral misinformation b y means of EVT to ols. First, we analyze the limit b eha vior of the Maximum/Sum ratio R n ( p ) = M n ( p ) S n ( p ) , n ≥ 1 , p > 0 , where S n ( p ) = P n i =1 ( X p i ) and M n ( p ) = max( X p i ). The momen t of order p of the distribution exists, i.e. E [ X p ] < ∞ , if and only if R n ( p ) conv erges to zero for n → ∞ . Con versely , an erratic limit behavior of R n ( p ) indicates the infiniteness of the p -th momen t of the distribution, i.e. E [ X p ] = ∞ . Figure 6 shows that only the first moment of the distribution exists, i.e. E [ X ] < ∞ , whereas moments of order greater than p = 2 are infinite, i.e. 8 Figure 5: Empirical Complementary Cumulativ e Distribution F unction. The rescal- ing procedure slightly exacerbates the pow er law behavior of tail without influencing the bo dy of the distribution. E [ X p ] = ∞ for p ≥ 2. Iden tical results hold for raw data. Our distribution function b elongs to the maximum domain of attraction of F r´ echet, i.e. F ∈ M D A (Φ α ). The existence of the first momen t of the distribution function allo ws us to compute a r e asonable (in a sample one can compute basically anything, ev en meaningless quan tities) estimate of the conditional tail mean ab o ve a given threshold. Indeed, by the law of total exp ectation E [ X | X > t ] = E [ X ] − Pr ( X ≤ t ) E [ X | X ≤ t ] Pr ( X > t ) , where E [ X | X ≤ t ] is finite since bounded from ab ov e b y t , and the finiteness of E [ X ] implies that the conditional tail mean, E [ X | X > t ], is finite. Suc h a measure is known in finance as the exp ected shortfall of a loss distri- bution, and it let us answer to question such as “What is the exp e cte d numb er of shar es for a p ost onc e it has exc e e de d the 250 K shar es thr eshold?” . Indeed, E [ X | X > t ] = P N i =1 x i I ( x i > t ) P N i =1 I ( x i > t ) = P N i =1 x i I ( x i > 250 K ) P N i =1 I ( x i > 250 K ) ≈ 467 K. Since we show ed that the moments of order greater than 1 do not exist, one should prefer the mean absolute deviation o ver the v ariance as a measure of disp ersion around the conditional tail mean. Recall that the momen t of order p of a Generalized Pareto distributed ran- dom v ariable only exists if and only if ξ < 1 /p , and th us the shape parameter w e are going to estimate can not b e smaller than 1 / 2. Suc h an observ ation has the 9 Figure 6: Maxim um/Sum ratio plot. Only the first moment of the distribution exists, i.e. E [ X ] < ∞ , whereas moments of order greater than p = 2 are infinite, i.e. E [ X p ] = ∞ for p ≥ 2. main implication that we can safely use the maximum likelihoo d (ML) approach to estimate ξ , since the ML estimates are consistent only when ξ > − 1 / 2. Since our time series is highly irregularly spaced, with interarriv al times ranging b et ween seconds and days, we prefer the Peaks Over Threshold (POT) approac h ov er the Block Maxima (BM) approac h to estimate the shape parame- ter ξ . Indeed, when the blo c k p erio ds used in the BM approac h do es not app ear naturally , the choice of a threshold for the POT approach ma y b e easier. Before fitting the distribution function to a Generalized Pareto Distribu- tion, w e hav e to identify a feasible threshold. T o accomplish such a task, we rely on the Mean Excess F unction (MEF). The empirical MEF of a sample of observ ations x 1 , . . . , x n is defined as e n ( t ) = P n i =1 ( x i − t ) P n i =1 I ( x i > t ) , that is the ratio b et ween the sum and the n umber of the exceedances ov er the threshold t . Figure 7 sho ws the MEF plot for the rescaled data. W e observe that the empirical MEF b egins to linearly increase in the threshold at t ≈ 10 4 . Suc h a b eha vior characterizes pow er law distribution functions [66], and thus w e choose t = 10 4 as threshold. Giv en the heavy-tailed b eha vior of the rescaled data distribution function, the shap e parameter ξ is likely to b e p ositiv e. The left panel of Figure 8 shows the Pic k ands plot, based on the nonparametric Pic k ands estimator for ξ , defined as ˜ ξ ( P ) τ ,n = 1 log 2 log X τ ,n − X 2 τ ,n X 2 τ ,n − X 4 τ ,n , τ = 1 , . . . , b n/ 4 c where X τ ,n is the τ -th upp er order statistics out of a sample of n observ a- tions. The Pick ands plot shows a more or less stable b eha vior of the Pick ands estimates for different v alues of τ , suggesting that the true v alue of ξ lies in the in terv al (0 . 5 , 1). 10 Figure 7: Empirical Mean Excess F unction plot. The righ t panel of Figure 8 shows the Hill plot, based on the nonparametric Hill estimator for ξ , defined as ˜ ξ ( H ) τ ,n = 1 τ τ X j =1 ln( X j,n ) − ln( X τ ,n ) , where X j,n is the j -th upp er order statistics out of a sample of n observ ations. The Hill plot outp erforms the Pic k ands plot in stability , suggesting a true v alue of ξ around 0 . 75. Figure 8: Pick ands and Hill nonparametric estimators for the shap e parameter ξ . The main implication is that, as already anticipated by the analysis of the Maxim um/Sum ratio plot, the true v alue of ξ is greater than − 1 / 2, and thus w e can obtain consistent estimates via a maximum likelihoo d (ML) approac h. T able 1 shows ML estimates of ξ and β for differen t thresholds. W e observ e a stable v alue of ˜ ξ M L for increasing v alues of the threshold. W e obtain similar results for ra w data (i.e. ˜ ξ M L = 0 . 769(0 . 0198) with t = 2 . 5 K ). 11 T able 1: Maxim um Likelihoo d estimates, standard errors, and num b er of ex- ceedances for different thresholds. threshold ˜ ξ ( M L ) ˜ β ( M L ) # exceedances 10 K 0 . 770 8 , 750 6 , 408 (0 . 0220) (205) 25 K 0 . 800 19 , 500 2 , 153 (0 . 0391) (805) 50 K 0 . 737 43 , 170 884 (0 . 059) (2 , 730) 100 K 0 . 746 74 , 380 399 (0 . 0869) (6 , 923) 150 K 0 . 726 120 , 460 223 (0 . 120) (15 , 500) 3.3. F r e quency of Vir al Misinformation The Peaks Over Threshold (POT) metho d has t wo main implications: the exceedances o ver a high threshold follo w a Generalized P areto Distribution, and the num b er of excesses ov er time follows a homogeneous Poisson pro cess. In a homogeneous P oisson pro cess the num ber of even ts, N ( θ ), in a finite interv al of time of length θ follo ws the Poisson distribution, i.e. Pr ( N ( θ ) = n ) = ( λθ ) n n ! exp( − λθ ) . Moreo ver, the interarriv al times b et ween even ts are indep enden t and follow the exp onen tial distribution, i.e. Pr (in terarriv al time > θ ) = exp( − λθ ) . Figure 9: Exp onential Quan tiles vs. Interarriv al Times. Figure 9 shows that the interarriv al times of pos ts shared more than 750 K times (rescaled data) follow approximately an exp onen tial distribution. More- o ver, the auto correlogram function (ACF) plot — i.e. a plot showing the simi- larit y b et ween observ ations as a function of the time lag betw een them [67] — in 12 Figure 10 sho ws that the interarriv al times b et ween those p osts are independent, supp orting the i.i.d. h yp othesis suggested by the records plot in Figure 4. Simi- lar results approximately hold for raw data when considering p osts shared more than 250 K (the bo otstrap test of fit for the Generalized P areto Distribution [68] giv es a p-v alue equal to 0 . 2). W e conclude that the num b er of extremely viral p osts ov er time follows a homogeneous P oisson pro cess. Figure 10: Auto correlogram for Interarriv al Times. W e find no correlation as a function of the time lag b etw een them. Suc h a conclusion allows us to exploit some useful prop erties of the homo- geneous Poisson pro cesses to quan tify the frequency of rare viral con tents on online so cial media. Indeed, the exp ected v alue of the n umber of ev ents, N ( θ ), in a finite in terv al of time of length θ is defined as E [ N ( θ )] = λθ , where λ > 0 is known as the rate parameter of the Poisson pro cess. The recipro cal of such a parameter, i.e. 1 /λ , is kno wn as the surviv al parameter of the exponential distribution follo w ed by the interarriv al times b et ween the N ( θ ) ev ents. Given a sample z 1 , . . . , z n of in terarriv al times, the surviv al parameter is estimated through the sample mean 1 λ = P n i =1 z i n . Essen tially , we can estimate the surviv al parameter, 1 /λ , of the exp onen tial distribution describing the interarriv al times b etw een rare ev ents exceeding a certain threshold, and then use the rate parameter, λ , of the Poisson process to estimate the exp ected num b er of even ts exceeding that threshold in a finite time of length θ . F or instance, the surviv al parameter of the interarriv al times distribution of p osts exceeding 250 K shares (raw data) is 1 /λ = 18 . 5. It follows that λ = 1 / 18 . 5 = 0 . 0541. Basically , if by means of the surviv al parameter w e can answ er to questions suc h as “What is the me an waiting time b etwe en p osts exc e e ding 250 K shar es?” , through the rate parameter we can answer to questions such as 13 “What is the exp e cte d numb er of p osts exc e e ding 250 K shar es in the futur e 365 days?” . Indeed, E [ N ( θ )] = λθ = 0 . 0541 × 365 = 19 . 8 ≈ 20 . A con venien t wa y to assess the uncertain ty around the rate parameter, λ , consists in using a standard Bay esian probability updating metho d. Indeed, the conjugate prior distribution for a Poisson distribution is the Gamma distribu- tion, and w e can express the prior distribution of λ as Pr ( λ ) = Gamma( α, β ) . Since the exp ected v alue (mean) of a Gamma distribution is defined as α/β , w e ma y w ant to choose the h yp erparameters, α and β , of the prior distribution Pr ( λ ) so that 1 λ = β α = P n i =1 z i n , where z 1 , . . . , z n are the observed interarriv al times. Then, the posterior distribution of the rate parameter is defined as Pr ( λ | z ) = Gamma( α + k , β + k X i =1 z i ) , where z 1 , . . . , z k represen t k new observed interarriv al times. F or instance, w e ma y define the prior distribution of the rate parameter of the interarriv al times distribution of p osts exceeding 250 K shares (raw data) as Pr ( λ ) = Gamma( α, β ) = Gamma( n, n X i =1 z i ) = Gamma(38 , 702) , with mean equal to α/β = 38 / 702 = 0 . 0541, and v ariance equal to α/β 2 = 38 / 702 2 = 7 . 71 × 10 − 5 . Then, if after 60 da ys w e observe a p ost exceeding the 250 K shares threshold, the p osterior probability distribution of the rate parameter is Pr ( λ | z ) = Gamma( α + k , β + k X i =1 z i ) = Gamma(38 + 1 , 702 + 60) , with mean equal to ( α + k ) / ( β + P k i =1 z i ) = (38 + 1) / (702 + 60) = 0 . 0512, and v ariance equal to 38 + 1 / (702 + 60) 2 = 6 . 72 × 10 − 5 . Figure 11 shows b oth the prior and the p osterior distributions of the rate parameter in the aforemen tioned example. After such an up date, the exp ected v alue of the num ber of p osts exceeding the 250 K threshold in the next 365 days is defined as E [ N ( θ )] = E [ λ | z ] θ = 0 . 0512 × 365 = 18 . 7 ≈ 19 , 14 Figure 11: Prior and Posterior Distributions of the rate parameter. The grey and red dashed lines indicate, resp ectiv ely , the mean of the prior and the mean of the posterior distribution of the rate parameter. Figure 12: Posterior predictive probability distribution. P osterior predictive probabil- ity distribution of the num b er of p osts exceeding the 250 K shares threshold in the next 365 days. The red dashed line indicates the mean, i.e. 18 . 7. and the mean w aiting time b etw een p osts exceeding the 250 K shares is 1 / 0 . 0512 = 19 . 5 days. Moreo ver, the full probability assessment of the uncer- tain ty around the rate parameter, λ , allo ws us to express the predictive p osterior probabilit y distribution of the num b er of p osts exceeding a certain threshold o ver a finite interv al of time Pr ( N ( θ ) | λ ) = Pr ( λ | z ) θ. Figure 12 sho ws the p osterior predictive probability distribution function of the num b er of p osts exceeding the 250 K shares threshold (raw data) in the finite in terv al time of length 365 days. 3.4. Concluding R emarks In this pap er, we study the statistical prop erties of viral misinformation in online so cial media. In particular, we fo cus our atten tion on F aceb o ok p osts 15 spreading false news, hoaxes and unsubstantiated claims. By means of an Ex- treme V alue Theory approach, we show that the n um b er of extremely viral posts o ver time follo ws a homogeneous P oisson process, and that the interarriv al times b et ween such posts are independent and identically distributed, following an ex- p onen tial distribution. Moreo ver, we c haracterize the uncertain ty around the rate parameter of the P oisson pro cess through Ba yesian metho ds. Finally , we are able to derive the predictive p osterior probabilit y distribution of the n umber of p osts exceeding a certain threshold of shares ov er a finite interv al of time. The relev ance of our results is not necessarily limited to the field of com- putational so cial science coping with misinformation. 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