Introduction

Massive data sets that comprehensively capture users’ behaviors in online social systems and their underlying network structures have reached an unprecedented scale, making it possible to develop computational methods to model complex patterns of human behavior at both individual and population levels [1–5]. Among various human-induced online processes, the study of social contagion—the spread of information, ideas, and behaviors through social networks—has attracted tremendous attention, especially in the fields of computational social science and network science [6, 7]. Many studies examine these peer-to-peer diffusion processes by focusing on a single piece of information and making assumptions about infectivity, recovery probabilities, and their intrinsic relations to network structures [4, 8–12]. We consider measuring the infectivity of information cascades to be the crux for predicting their ultimate virality.

Previous research has successfully advanced the modelling of information spread by studying memes in Twitter data, where a meme is defined by the use of a hashtag and includes all of the tweets with that hashtag [5, 13–16]. Gleeson et al. introduced a mathematical framework to examine the branching dynamics of this meme spread process [17]. Besides these theoretical efforts, many other studies try to explore this research topic with large scale empirical data and real world experiments. Vosoughi et al. analyzed over ten years of Twitter data on the dynamical diffusion of true and false news [12]. Bail et al. ran a field experiment on Twitter to study the spread of views and political opinion [18]. Del Vicario et al. studied emotional contagion and group polarization on another popular online social network platform: Facebook [19]. All these efforts provide a deeper understanding of social factors and behavioral patterns in online information spread.

Here, we reanalyze these data with an exclusive focus on modelling the direct transmission of information through a social network in the form of retweets. Our reason for focusing on retweets is that the transmission of a particular hashtag is more likely to occur not only from person to person through online social ties [15], but also through mass targeted broadcasting from other media sources outside the specific social network. As observed in Ref. [20], broadcasts contribute substantively to viral events, e.g., the World Cup Final attracts about 1 billion viewers worldwide, while news coverage from popular websites also reaches a similar number of Internet users. In such popular events, the discussion of a meme in broadcasting media (e.g. social network platforms, TV shows, radio and news reports) can greatly boost its spread. Retweets, by contrast, constitute an information cascade that originated from an identifiable individual user and is a contagion spread mostly through the links of the follower network (Fig 1).

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Fig 1. Schematic of social contagion information diffusion in Twitter.

(a) The Twitter user interface that displays three latest tweets with different degrees of interestingness from her friends. The first message was originally posted by someone with whom she does not have direct connection, but she is still able to see it after being retweeted by one of her friends. She chose to retweet the second tweet she found interesting, extending the information flow of that message to all her followers. If the “memory length” of this user is 3, she will not read or retweet messages posted more that 10 hours ago (the time of the third item in the display). (b), The online network environment of involved users and the flows of information cascades.

https://doi.org/10.1371/journal.pone.0214453.g001

Materials and methods

Decay factor and infectivity estimation

Previous studies have found that the attractiveness of online information does not remain constant over an indefinite period of time, but rather gradually declines as it grows older [23]. We adopt this observation of fading popularity by incorporating a decay factor α and assume that the infectivity of cascade i decays exponentially by λ_i(t) = λ_i0e^{−α(t−t_i0)}, where t_i0 is the time of the initial tweet. Among retweets for which we can identify at least one of the previous tweets in the same cascade posted by their neighbors, a fraction ψ = 0.69 of them occurred within one day after the tweet was last seen by the retweeted user. Using a mean-field approach that assumes the degree of all nodes to be equal to 〈k〉, we then express the average number of retweets of cascade i at time t as a_i,t = λ_i0e^−αt〈k〉a_i,t−1/ψ.

We define the number of total retweets of cascade i at time t as A_i,t, and derive the conditional expectation of A_i,t given that cascade i is retweeted at least once during its lifetime: (4) Here we make two assumptions about the retweet size and infectivity of cascades: first, the tweet will either be stifled by stochastic fluctuations at the beginning such that no followers retweet it, or will be retweeted with probability 〈k〉λ_i(t)ψ⁻¹ and reach the mean size determined by Eq (4) at time t; second, for fixed values of t and A_i,t, the infectivity λ_i0 calculated by Eq (4) is the minimum rate to reach a retweet size ≥ A_i,t. We further assume that the relation between the number of retweets S_i in the real Twitter data and A_i,t is S_i = A_i,t|_t→∞. Then we set t = 25 to fit the spread rate distribution in Eq (4). As such, we can obtain (λ_i0, S_i) pairs such that their probability distribution satisfies P(S ≥ S_i) = P(λ₀ ≥ λ_i0), which can be used to approximately estimate the distribution of λ₀ from empirical Twitter data.

The above analysis has taken the decay effect into account. We next approximate the distribution of initial infectivity λ_i0 for cascade i as a truncated lognormal form with an upper bound probability λ_max. Let p⁰(λ₀) be the lognormal distribution , where μ and σ are parameters, and the normalization factor for the infectivity distribution can be written as . Thus we have the probability distribution of infectivity p^infectivity(λ₀) = p⁰(λ₀)/P⁰(λ_max) in the truncated lognormal form with 0 < λ₀ < λ_max. If a random user tweets a cascade with initial infectivity λ₀, and it stays in the followers’ memory for an average lifetime 1/ψ, the probability that it is not retweeted by any follower is (1 − λ₀)^〈k〉/ψ. Therefore, the fraction of cascades being retweeted at least once is given by (5) This expression captures the fact that information cascades are likely to be stifled due to stochastic fluctuations at the initial stage, before it actually starts spreading. Assuming the infectivity is small such that [1 − (1 − λ₀)^〈k〉/ψ] ≃ λ₀〈k〉/ψ, we have (6) where erf(x) is the error function. We then estimate (λ_i0, S_i) pairs from empirical data with a pre-assumed decay factor α from Eq 4, and fit the outcome distribution with Eq 6 (see Fig 2a).

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Fig 2. Simulation parameter settings and results.

a, Truncated lognormal fit. Light and dark blue lines are fit with theoretical distribution function Eq (6), and the red and orange points are fit with distribution computed from real Twitter data with Eq (4). The parameters used in Eq (6) are as follows: when decay factor α = 0, μ = ln 0.0012, σ = ln 2.4, λ_max = 0.015; when α = 0.01, μ = ln 0.0012, σ = ln 2.4, λ_max = 0.017. b, Retweets at time t of cascades originated at time t₀ with different decay factors. c-e, Complementary cumulative distribution functions (CCDFs)—the fraction of cascades with more than n retweets for numerical simulations, compared with retweets from empirical Twitter data marked by green points. The model parameters are identical except for the network structure: c, The empirical Twitter follower network with N = 5.95 × 10⁵ and 〈k〉 = 47.94; d, Reconfiguration of the empirical Twitter network preserving the degree distribution; e, Scale-free network with N = 5 × 10⁵, 〈k〉 = 48 and exponent γ = 2.8.

https://doi.org/10.1371/journal.pone.0214453.g002

Results

The Twitter data we use contains a follower network with 6.0 × 10⁵ users, 1.7 × 10⁶ retweets and 1.2 × 10⁷ tweets generated by these users in 33 days [14, 15]. We estimate the probability distribution of the infectivity of cascades from real data, and simulate the process on the follower network (see Methods). A cascade consists of retweets that have the same hashtag and the same user who initially posted the tweet, together with the tweet that originated the cascade.

Previous studies have demonstrated that the topology of networks, especially the community structure, has pronounced effects on information diffusion [14, 26]. Communities could promote spread by homophily and social reinforcement, but may also hinder wider spread by trapping information, resulting in a high concentration of retweets within a community. To examine the influence of community structures, Weng et al. [14] introduced two statistical features of memes, which we modify for retweet cascades: the adoption dominance g computes the proportion of users retweeting the cascade in the community with the most adopters; and the retweet entropy H^r quantifies the distribution of retweets across different communities, as a measure of the concentration of the cascade across communities. We compute both measures based only on retweets in their early stages (first 50 tweets) to avoid bias from a cascade’s popularity.

Retweet cascades are very different from hashtag memes in that we can more realistically assume that social contagion through the follower network is the major mechanism by which the retweet cascade is propagating. To provide direct evidence of this, we sampled 10⁵ tweets and retweets, respectively, finding that for 23.8% of tweets we can find at least one earlier tweet with the same hashtag from the user’s friends, while 46.0% of retweets have at least one friend who previously retweeted in the same cascade. Importantly, these percentages are limited by the specific follower network available in the data set, which inherently undercounts the possibility of transmission through the online social network because the network in the data only includes the reciprocal following ties (to better reflect real social relations). We estimate the infectivity of a specific cascade assuming that all such identifications are the actual paths of information transmission, using only the first 50 retweets (see Methods). Despite the relatively high inaccuracies observed between the true and predicted infectivities in our simulated data (where we know the true imposed infectivity, cf. real Twitter data), we note the overall trends of the infectivity estimates are in the right direction, with a slope of 0.92, R² = 0.05 and p-value < 0.01 (Fig 3). The distribution of estimated infectivity is heavy tailed and not Gaussian, and the R² of the linear model is low. Interestingly though, as we proceed to consider predictive models for virality that include such estimates of cascade infectivity, we will see that it improves prediction despite relatively poor fit.

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Fig 3. Comparison between real and estimated infectivity in simulations.

Real infectivity λ₀ and estimated infectivity computed from simulation data according to Eq 2 without considering the decay effects. The solid line is the linear regression fit. Estimates are calculated from only the first 50 retweets of each tweet, so that they may be used to try to predict whether a given cascade “goes viral”.

https://doi.org/10.1371/journal.pone.0214453.g003

We now test whether this simple model of infectivity demonstrates predictive power for identifying viral retweet cascades in real Twitter data. In Ref. [14], Weng et al. used community concentration features to predict viral memes with three models: the random guess (RG) model randomly samples the cascade without any predictors; the null model (NM, referred to as the “community-blind model” in Ref. [14]) employs the number of distinct users and the total number of neighbors of early retweet users; the community-based (CB) model also incorporates three community-based features of the Twitter network: the number of infected communities, retweet entropy H^r, and the fraction of intra-community user interactions. We introduce two additional models adding features to the NM model to predict viral cascades with infectivity estimates: the infectivity-based (IB) model uses the estimated rate of infectivity from Eq (2), where 〈k〉 is the mean degree of early retweet users; and the community & infectivity based (C&I) model combines all of these infectivity and community-based features. Each of our classifiers includes only information about the first 50 retweets of each tweet, to try to predict whether the retweet cascade “goes viral”. We train random forest classifiers on 1, 272 real Twitter cascades and 20, 000 simulated cascades sampled from 20 replications, using 10-fold cross validation to predict viral cascades that attract more retweets than a certain percentile threshold θ of all cascades.

The results on the Twitter data suggest that in most cases our IB model performs better than the CB model (Fig 4a and 4b), indicating that estimated infectivity alone can improve the prediction even more than the community-based predictors. Moreover, the C&I model, incorporating both community and infectivity factors, reveals a striking increase of predictive power above the other models. Fig 4c and 4d shows random forest model prediction and recall rates on retweet data generated by our simulations, indicating patterns consistent with those observed in the Twitter data. The IB model, only adding infectivity to the NM model, is comparable to the CB model that includes three community features, and by considering all predictors the C&I model excels in both precision and recall rates. We note that replacing the estimated by the true λ₀ used in the simulations—a test we can obviously not reproduce in the real Twitter data—yields additional improvement in classification (S2 Table), suggesting substantial potential for a more refined estimate of to lead to even greater accuracy for predicting viral cascades.

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Fig 4. Random forest model predictions.

We predict whether a cascade will go viral or not; a cascade is viral if it produces more retweets than a certain percentile threshold (θ = 70, 80, 90) of cascades, using community-based features and infectivity estimates that are calculated based on the initial 50 retweets for each cascade. Random forests are trained on sets of features delineated by the labels RG, NM, CB, IB and C&I (see the main text). The classifier including estimated infectivity (IB) typically outperforms the community-based model (CB), while combining all of the community-based and infectivity features (C&I) gives the best predictions overall. a, Precision rates of Twitter data. b, Recall rates of Twitter data. c, Precision rates of retweet data from simulations. d, Recall rates of retweet data from simulations. The precision and recall rates reported in this figure are mean values of 100 randomizations of the random forest model.

https://doi.org/10.1371/journal.pone.0214453.g004

We further test our results using logistic regression with the same set of features as in the C&I model. We find that estimated infectivity is still a significant predictor in simulation data, but not in predicting virality in the real Twitter data (S3–S5 Tables). There may be multiple reasons for this apparent discrepancy between the random forest and logistic regression results. One possibility is that logistic regression is too specific in the functional form in which it estimates the probability of virality. In particular, we note the substantial noise in estimating infectivity we observe in our simulations; without any way to compare the estimated infectivities with “true” values in the real Twitter data, we cannot know whether the effect of this noise interacts poorly with the log-odds-shift assumptions of logistic regression.

Our simulations emulate the real-world diffusion process in Twitter by taking into consideration several human behavioral factors, such as a limited memory length and a gradual decrease in interest, in a simplified simulation model. We estimate a fixed memory length for all users from data and additionally incorporate a small but non-zero decay parameter to the infectivity of each cascade (see Methods). The initial infectivities of cascades are sampled from a probability distribution computed from empirical data (Fig 2a). The decay effect mainly affects the long time dynamics of viral cascades (Fig 2b). If we ignore the decay effect of infectivity, cascades with large infectivity will still keep spreading after long periods of time, even with fixed user memory length. With a small but non-zero decay parameter α, even the most popular cascades will diminish at some point, and the system quickly reaches equilibrium. We then use simulations on networks with different structural properties but otherwise identical parameter settings to calculate the distributions of cascade sizes.

Fig 2c shows that our simulations on the Twitter follower network replicate well the cascade distribution observed in the data. We also run a simulation on a configuration model network with the same degree distribution as the empirical Twitter network (Fig 2d). Simulation results on a synthetic network generated by the algorithm in Ref. [27] with the power-law exponent γ = 2.8, representing an analogous degree heterogeneity of the Twitter network (see S3 Fig), also recover the statistical features of Twitter data (Fig 2e). When we switch the decay parameter to 0.001 and 0.02, respectively, we still replicate the empirical retweet distribution fairly well by changing the corresponding λ_max parameter (S5 Fig).

Discussion

We have demonstrated the predictive power of infectivity for identifying viral retweet cascades in real-world Twitter data and in simulation. An important assumption of this study is that the spread of retweet cascades resembles the peer-to-peer social contagion through the Twitter follower network, which we argue is different from viral memes represented by hashtags that more heavily rely on transmission through broadcasting. We demonstrate that the early spread rate for retweet cascades can be a good indicator of the intrinsic interestingness of a tweet, and that the corresponding estimate of infectivity gives improved prediction of virality. But, importantly, the same scheme might not readily apply to some memes that need to be broadly broadcast before they become viral. This difference may help explain why the measure of early infectivity of a hashtag in Ref. [15] does little to improve the prediction of viral memes.

Our mean-field method to estimate infectivity from empirical data clearly leaves plenty of room for improvement. The predictive ability of machine learning methods improves further on simulation data when we include the true infectivity, demonstrating the importance of accurate estimations of the cascade infectivity. Apart from this indirect approach with strong assumptions, we could also design a more straightforward method. The biggest challenge for such a measurement is to gather large-scale, high-quality data with which it is possible to infer accurate retweet relations. Better data and more reliable methodology to estimate infectivity are key to improving the predictive power.

Our study shows that infectivity improves the prediction of viral cascades that are mostly induced by contagion along the links representing social network connections. Network community structure captures additional local environmental factors such as homophily, social reinforcement and a trapping effect that further affect the spread and likelihood of virality of retweet cascades. Nevertheless, the infectivity determines the internal attractiveness and seems to be one of the most important factors in driving the virality of a cascade. Said another way, we have successfully demonstrated that the inherent quality of content—in the sense of being sufficiently interesting to have high infectivity—is an essential element promoting the chances of a successful spread that might not otherwise be as plausible in light of the local environmental factors.

Supporting information

Acknowledgments

We thank the Ohio Supercomputer Center for their assistance. W.L. was supported by EPSRC Early Career Fellowship in Digital Economy (Grant No. EP/N006062/1). Z.Z. was supported by Program of National Natural Science Foundation of China Grant No. 11871004, Fundamental Research of Civil Aircraft Grant No. MJ-F-2012-04. P.J.M. was supported by the Eunice Kennedy Shriver National Institute of Child Health & Human Development of the National Institutes of Health under Award Number R01HD075712. The content is solely the responsibility of the authors and does not necessarily represent the official views of any of the agencies supporting this work.