2.1 Structural contagion models

Most contagion models focus on simple contagion and complex contagion [21, 22]. Simple contagion describes a controlled process with a contagion probability being independent of the number of exposures [22]. Complex contagion describes a process that requires multiple exposures to a contagious entity [22] and describes better than simple contagion the spread of ideas, technologies, or a behavior [22, 23]. Most information-spread studies, whether devoted to simple or complex contagion, focus on local contagion as a result of user-to-user exposure [1, 2, 24].

Addressing local contagion with structural modeling, Leskovec et al. [25] studied a recommendation graph and measured the extent to which a user’s activity in recommending a product is contagious and affects the purchase decisions of adjacent neighbors. Similarly, Sun at al. [26] studied the contagion effect on the participation of users in fan pages after some of their neighboring friends have done so, and Bakshy et al. [27] examined contagion in adopting the use of gestures among friends in the Second Life platform. The Linear Threshold Model [28] allows a user’s transition from a non-active to an active state following a similar transition in the participation of a network neighbor. Kleinberg et al. [29] used a structural transition model with a similar contagion definition of user participation and analyzed the propagation of local contagion by allowing a user to activate her/his inactive neighbors. Two-step diffusion models [15, 30], however, posit that a piece of information first spreads globally from the mainstream media to opinion leaders and, only afterward, propagates locally in a node-to-node manner from opinion leaders to a broader population.

The local neighbor-to-neighbor mechanism at the basis of the structural contagion models discussed so far, miss the global mechanisms behind exposure of OSN users to varied content by non-neighbors, beyond network structure, that can lead to global contagion [4, 6]. For example, a contagion event like retweeting (RT) a message on Twitter by re-posting someone else’s tweet [1, 3] and passing interesting pieces of information to followers, is limited neither to viral information that a user is exposed to nor to a local user-to-user mechanism only. Since contagion can occur beyond pairwise interaction, through higher-order structures than neighbor-to-neighbor [4, 6], it is important to consider contagion by all network users upon modeling contagion.

Detecting both local and global contagion mechanisms while considering network structure is crucial to better understanding human behavior online, as manifested by user interactions [31]. In the case of Twitter, users are exposed to information posted by non-neighbors via exposure to hashtags [32] as well as to promoted content on a user’s Timeline feed [13], which includes tweets of accounts that a user follows as well as content tweeted by non-neighbors either by advertisers purchase or tweets ranked as having a large engagement potential [33]. Similarly, Facebook and Reddit allow global exposure via trending topics that appear on a user’s front page [34]. This facilitation of global exposure in OSNs raises the need to consider non-structural contagion modeling, beyond network structure, as discussed next.

2.2 Non-structural contagion models

Non-structural contagion models (e.g. [35]) do not rely on the structure of the network to infer contagion. The Susceptible-Infected-Resistant (SIR) model [36] and the Susceptible-Infectious-Susceptible (SIS) model [36], for example, assume that every individual has the same probability to be infected. In SIR and SIS models, all users thus have the same contact rate, which is indicated by an edge formation in a network. However, contagion in OSNs is not evenly distributed among users [37] and is likely to depend on exposure rates [6], as modeled by the Linear Influence Model [6], which assumes a static network structure in which the infection likelihood of a user is affected by the number of contagious network users [38].

Aiming to better explain contagion spread with non-structural modeling, Wang et al. [39] modeled contagion by mainly focusing on temporal and topological dynamics while integrating a single topological feature that represents the distance between the infecting and the infected users. Global contagion can also result from homophily-driven diffusion by a peer-to-peer influence model [39], where the user interface has a substantial effect on the contagion process [1]. Other studies [12, 40] focus on the linguistic properties of textual information in predicting contagion spread. In recent years, evidence of global contagion due to exposure to external sources like mainstream media was found by [6, 7, 11] among others.

Since this study aims to investigate not only local versus global contagion but also the spread of viral and non-viral information, structural contagion models have been reviewed in the previous sub-section and non-structural contagion models have been reviewed in the present sub-section. Another goal of this study is to go beyond most studies [18, 20, 41–47] which, as discussed next, focus on viral contagion, largely ignoring contagion spread of non-viral information.

2.3 Viral versus non-viral contagion

Analyzing viral contagion, Romero et al. [2] modeled complex contagion of the 500 most frequent hashtags in a Twitter dataset of three billion messages with a median hashtag count of over 93,000 occurrences. Dow et al., [17] studied the spread of 1-million images among Facebook users, restricting their analysis to viral images shared at least 100 times. The “Yes We Can” slogan used in the 2008 U.S. elections reached over 20 million views, demonstrating yet another analysis of viral contagion [48]. The number of contagion events that make a piece of information viral differs among studies. Gleeson et al. [14] found that the structure of the network and temporal dynamics can explain sub-critical (non-viral) cascades.

One of the lowest contagion events was defined by Myers et al. [6] who restricted their analysis to messages shared at least 50 times, while Dow et al. [17] restricted their viral-contagion analysis to images shared at least 100 times. Liben-Nowell et al. [42] measured cascades with hundreds of steps, showing that re-sharing contagion activities have the potential of information spreading to millions of users [20].

The reach distribution of contagious information is long-tailed since most information nuggets are shared only a small number of times if shared at all [18, 49]. Cheng et al. [18], who found that the cascade size distribution of photos posted by users follow a power-law curve with an exponent of α = 2.2, concluded that most information nuggets are scarcely shared further. It is reasonable, therefore, to assume that there exists a contagion mechanism in the few cases in which an information nugget is shared enough times to create a viral contagion.

Several studies [41, 45] have addressed the important task of detecting whether a piece of information will go viral [20] by focusing on information cascades as well as by modeling the contagion spread of hashtags [43], behavioral dynamics [46] or, for example, YouTube views [50]. Other studies tried to predict the size of information cascades by using network features [18, 47]. For instance, Cui et al. [44] applied a logistic model that considers the relative importance of each node given the list of previously infected nodes. Most of these approaches, however, lack the ability to detect virality early.

The scarcity of research regarding contagion spread of non-viral content and regarding global effects on the contagion mechanism has motivated us to distinguish between local versus global contagion and viral versus non-viral information. Complementing and contributing to the body of OSN scholarship about contagion spread of information, we focus on non-viral contagion as a mean to better understand both local and global contagion mechanisms and the nature of human behavior online.

3.1 Defining information virality

According to Dow et al. [17], viral events are manifested by resharing of photos on Facebook at least 100 times. According to Goel et al. [15], rare events are those manifested by resharing tweets on Twitter—0.025% of viral events identified by diffusion trees with at least 100 nodes. Following these authors, we define an information nugget as viral if it is shared at least 100 times and as non-viral if it is shared 10 to 99 times, overlooking information nuggets shared less than 10 times since their contagion spread signature is too low to allow studying contagion spread.

3.2 Detecting contagion spread of information

Consider a directed social network G = (V, E) as defined in Table 1. A post of User v_j ∈ V can result in local contagion of User v_i ∈ V who Follows v_j, starting a cascade of local contagions. Since contagion is time-dependent, we define it more formally while considering the temporal activities of users.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 1. A summary of variables definitions.

https://doi.org/10.1371/journal.pone.0230811.t001

Let w denote an information nugget, like an original tweet, posted at Time t₀ by User v₀ ∈ V. Contagion spread, like retweeting, of w at times t₁…t_k by users v₁, …, v_j ∈ V, along with v₀, could be thought of as a temporal activity network G_Tw = (V_Tw, E_Tw) as defined in Table 1. The social network G and the activity network G_Tw allow us to define next, as summarized in Table 1, local and global contagion events.

Global contagion event.

A contagion event of User v_i is global if e_ji ∉ E and e_ij ∈ E_Tw. In other words, v_i has shared a post w before any of the users s/he follows has shared or posted w.

Fig 1 illustrates an example of local and global contagion spread. Consider a network (Fig 1) in which User v₀ posts a tweet w at Time t₀, exposing Users v₂ and v₄ who Follow her/him. At Time t₁, v₂ retweeted w, demonstrating local contagion since s/he follows v₀, while v₁ retweeted w, demonstrating global contagion since s/he does not follow v₀. The retweets of v₃ at t₂, v₄ at t₃, and v₅ at t₄ all demonstrate local contagion events since their opposite edges in E_Tw exist in E.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 1. An activity network with a social network.

Distances are measured on the social network.

https://doi.org/10.1371/journal.pone.0230811.g001

To detect the reach depth of an information nugget w, the distance d, is defined (1):

1. 1.. d—The distance on G from the source who originated w to a newly infected user.

To detect whether a contagion is local/global, distance d_ca is defined in (2).

1. 2.. d_ca—The distance on G to a newly infected user from the closest infected user who adopted w or from the user who originated w.

The definitions of the variables are summarized in Table 1.

Returning to Fig 1, consider an example where the retweeting source User v₀, and her Follower v₃ are the only members of V_Tw at Time t_k−1. Next, global contagion spreads to User v₅ at Time t_k. The minimal distance from any user in V_Tw to User v₅ before t_k is d_ca(v₃, v₅) = 2, or d_ca(v₀, v₅) = 2.

Several posts that describe the same subject can be grouped into a topic, and analyzing the contagion spread of a topic instead of the spread of a more specific nugget can uncover more realistic patterns of contagion spread [51].

3.3 Detecting contagion spread of a topic

A topic contagion event occurs when a single User v_i posts several original messages about the same subject, with each original message potentially shared by different users.

For example, assume that User v₀ (Fig 1) tweeted twice about the same specific topic—the discovery of the Higgs boson particle—described further in Section 4. The First tweet (i) was: #CERN scientists inexplicably present #Higgs #boson findings, and the second tweet (ii) was: New result from #LHC reinforces belief that the particle has been found. User v₂ who follows v₀, retweeted (i) exposing to the topic a follower—User v₃—who does not follow v₀. Next, v₃ retweeted (ii). This contagion sequence facilities an event of local topic contagion: v₃ was exposed to a topic and to v₀, upon being infected by v₂ whom s/he follows.

Our definitions of viral topic contagion and non-viral topic contagion are similar to the definitions of particular viral and non-viral tweets suggested in Section 3.2. A topic is viral if its set of original messages were shared at least 100 times and is non-viral if the set of its original messages were shared 10 to 99 times. There are two differences between contagion of a particular tweet and contagion of a topic. First, the number of retweets of each original message within a topic that was posted by the same User v_i are summed. Second, while users who share a particular message whose origin is User v_i are considered infected, a user infected by a topic is one who shares any of v_i’s original messages on the same topic. Stated differently, in the case of topic contagion, a user can be directly exposed to one message whose origin is v_i but share another message on the same topic whose origin is the same user v_i.

Topical contagion is a more realistic scenario as users are infected by a concept rather than by a fixed sequence of characters.

For a more formal definition of topic contagion, the social network G is defined as suggested in Section 3.2, and a topical contagion is then defined: A set of original tweets w_1p, …, w_np, posted within a selected time interval by User v₀ ∈ V about a topic p (denoted by w_μp, μ = 1, …, n). The retweeting spread of w_μp ∈ W at times t₁…t_k by Users v₁, …, v_j ∈ V, along with User v₀ who originated w_μp, form a temporal activity network G_TW, which is laid over G. In G_TW, Nodes V_TW depict v₀ as well as users who retweet any w_μp ∈ W at Times t_i ≤ t_k and Edges E_TW represent retweet relations. These definitions are also summarized in Table 1.

3.4 Detecting tweet virality in early stages

Our goal is to detect whether a tweet will become viral at its early stages—before it becomes viral. To achieve this goal, we developed the Back-in-time (BIT) approach that allows us to study the spreading patterns of viral tweets before they became viral and compare them to the spreading patterns of non-viral tweets. To analyze contagion spread of viral information in its early stages before it became viral, we roll a viral tweet back in time to a point when it was still a non-viral tweet (i.e., having 10 to 99 retweets). At the end of this stage, both rolled-back tweets (BIT-tweets) and non-viral tweets have 10 to 99 retweets. Aiming to show that the spreading patterns of BIT-tweets and non-viral tweets differ significantly, we compare the spreading patterns of BIT-tweets with the spreading patterns of non-viral tweets. Under the BIT approach, a viral tweet is sent back in time and the number of retweets (retweet-count) it had back then is determined, via three steps: (i) Learn the retweet count distribution of non-viral tweets by creating a kernel density estimate (KDE) of their retweet-counts. Accounting for non-viral tweets, the left-most and right-most points of the grid at which the density is estimated are 10 and 99 respectively; (ii) Randomly sample from the KDE a number y that represents a retweet-count and assign it to an original viral tweet. For that purpose, we draw a point x_i from the set of points x₁, …, x_n included in the KDE. Then, we draw a value y associated with x_i to ensure that more probable values of non-viral retweet-counts in the distribution are more likely to be randomly sampled; (iii) Roll a viral tweet back in time by keeping the oldest retweets, such that its retweet-count is equal to y.

Next, we describe the three datasets used in this study.

6.1 Analysis of contagion reach depth

This section describes how we detect a contagion and measure its reach. Given a network, local contagion begins at some node and then spreads over the edges. Typically, local contagions are measured on the activity network [56]. However, the social network plays a fundamental role in the dynamics of spreading [53, 57, 58].

We detect a contagion using the activity network (G_Tw) and measure the depth it reaches on the social network (G). The depth is defined as the largest distance (d) that the information (w) spreads from the user originating the information (Table 1).

To detect the reach depth of an original tweet (w), each user retweeting w was assigned a distance (d) on G and in the absence of a path on G, an infinite distance (INF) was assigned. Instead of INF, a path might exist at larger distances by collecting wider circles of Following-lists but contagion at such larger distances is global and, thus, does not affect interpreting the results.

Most contagion events of both viral and non-viral information are local (d = 1). Viral information (Fig 3a) has a contagion reach depth d ≤ 18, whereas non-viral information (Fig 3b), has a shorter contagion reach with depth d ≤ 7.

For both viral information (when d ≤ 9, d ∈ [13, 18]), and non-viral information (when d ≤ 7) in DS1, if a path exists on G, the more distant a user is from the source who originated the information (User v₀), the less likely s/he is to retweet w. We observed a trend shift for viral information when d ∈ [10, 12], and found that the more distant a user is from the source User v₀, the more likely s/he is to retweet w.

For non-viral information, DS2 presents similar trends to DS1 (Table 3) for d ≤ 9, d ≠ 8 (Fig 3c), if a path exists on G, the more distant a user is from the source User v₀, the less likely s/he is to retweet w. For d = 8, users are more likely to be infected than d = 7. This finding might be attributed to global contagion (e.g. mass media or the Timeline). Similar to DS1, most contagion events in DS2 are local (d = 1).

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 3. Contagion spread: The larger the distance the less likely a user will retweet.

https://doi.org/10.1371/journal.pone.0230811.t003

To better understand contagion spread of viral and non-viral information, we analyze next, contagion spread by type.

6.2 Analysis of contagion types

This section explains how we detect local and global contagion. As explained in Section 3.2 and demonstrated in Section 6.1, local contagion results when d = 1. However, the mechanism for global contagion is different. Fig 2 demonstrates global contagion where User v₀ is the source of w and User v₃ retweets w at Time t₁ before User v₂ retweets w at Time t₂. This order of events implies global contagion of User v₃.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 2. An activity network with a social network illustrating global contagion.

https://doi.org/10.1371/journal.pone.0230811.g002

To account for such cases, we assigned to each infected node a local or global contagion type as described in Section 3.2. Table 2 shows the percentage of local and global contagions in each dataset. Next, we computed d_ca (Table 1). Analyzing DS1 for viral information, Fig 3a depicts two types of distances and, by analyzing DS1 for non-viral information Fig 3b also presents two types of distances. By analyzing DS2, which contains non-viral information only, Fig 3c and 3d depict two types of distances as well. Similarly, by analyzing DS3 for viral topic spread, Fig 3e depicts two types of distances and, by analyzing DS3 for non-viral topic spread, Fig 3f depicts two types of distances.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 3. Distributions of distances, measured on the social network (G) for each of the three analyzed datasets, DS1 to DS3 (Table 2).

https://doi.org/10.1371/journal.pone.0230811.g003

In Fig 3, which presents a distribution of d and d_ca distances as measured on the social network G for each of the analyzed datasets D1 to D3, d_ca = 1 indicates the existence of a Following relationship among the closest adopter and an infected user, thus demonstrating local contagion. The most frequent contagion mechanism is depicted by the highest bar. On the other hand, d_ca > 1 indicates no Following relationship among the closest adopter and an infected user, thus demonstrating global contagion.

Table 3 summarizes these indications. In both DS1 and DS2, local contagion is the most frequent contagion mechanism (d_ca = 1), and the closer a user is to an adopter, the more likely s/he is to retweet. It is likely that a user who is retweeting at d_ca < 8 was exposed to content via network neighbors by manually crawling through the feeds of neighbors and neighbors of neighbors, or through exposure to the Twitter Timeline that presents tweets of accounts that a user follows. These results are in line with findings of previous studies on neighbor-to-neighbor contagion [1, 2, 24] that network members have a significant influence on their neighbors.

For global contagion spread of viral and non-viral information, in both DS1 and DS2, the larger the distance the less likely a user is to retweet w. In DS1 for viral information when d_ca > 8 (Fig 3a) and for non-viral information when d_ca > 7 (Fig 3b), as well as in DS2 when d_ca > 6 (Fig 3d): the larger the distance, the more likely is a user to retweet w. These finding can be attributed to the global contagion, resulting from global exposure to external sources. Since non-viral tweets are less likely to be covered by the mainstream media yet their global contagion distances (Fig 3b and 3d) are similar to those of viral tweets (Fig 3a), it is reasonable to conclude that the Twitter Timeline algorithm exposed users to non-viral information. Due to the low available time and limited attention of users [51], information pushed on one’s Timeline significantly increases the contagion rate [1, 19, 33]. Thus, it is more likely that the global contagion observed in Fig 3b and 3d occurred due to content promoted by the Timeline algorithm rather than due to local neighbors.

Each added local contagion expands the circles of exposed users. Hence, the number of globally exposed users at an arbitrary distance of d_ca also increases and might lead to global contagion. Aiming to learn the effect of each added local contagion on global contagion spread, we examine next the explanatory power of local contagion on the number of global contagions.

6.3 Explaining contagion spread

Our goal is to explain contagion spread by type, whether local or global while differentiating between viral and non-viral information. We developed one model for viral information and another model for non-viral information. Given contagion events, we aim to predict whether a user will be infected by a local or by a global contagion.

We measured the following explanatory terms for each individual user at the time of adoption, when s/he retweeted the information.

1. ΔT_o—Time difference from the posting of the original tweet.
   Captures bursty user interactions [59, 60], which can explain contagion spread [61].
2. GC—Total number of global contagions.
   Captures contagion spread similar to non-structural models (Section 2.2).
3. d—Distance from the tweeting source (Table 1).
   Accounts for our findings about the likelihood of a user to retweet (Fig 3a, 3b and 3c). Note, a local contagion can occur even when d > 1.

Table 4 presents the results of a logistic regression model fit with least squares, separated by tweet virality, where the outcome variable is local or global contagion, and local contagion is the base parameter. The ΔT_o variable yields the greatest explanatory power on the contagion type of users for both viral, and non-viral information.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 4. Logistic regression results.

https://doi.org/10.1371/journal.pone.0230811.t004

Non-viral information. A significant negative coefficient of ΔT_o indicates that the larger the time difference from the posting of an original tweet, the less likely a user to be infected via global contagion and, thus, local contagion is more likely.

Viral information. The opposite is found: the larger ΔT_o, the more likely a user to be infected by a global contagion. Viral information is persistent in the sense that repeated exposures to information are more likely to have contagion effects [2].

For both models, a significant negative coefficient of the term GC indicates that the more global contagions occur in a network, the less likely another global contagion will occur. This finding can be attributed to one’s reluctance to retweet information of whom s/he does not know [62]. A single global contagion, e.g. of a user within a clique, can start a cascade of local contagions. Thus, local contagion minimizes the chance of a global contagion by minimizing the number of non-adopters if it spreads faster than global contagion. We test this assumption in Section 6.4. Finally, a significant positive coefficient of the d Term in both models indicates that the larger the distance from the user originating the tweet, the higher the probability of global contagion. To estimate model performance, we used for training in the viral group and the non-viral group 70% of the data and for testing 30%. The performance of each model is consolidated into the F₁-score measure. Since an F₁-score of 1.0 means perfect precision and recall, an F₁-score close to 1.0 in Table 4 implies better model performance.

Since time difference has the largest explanatory power (Table 4), we present next, a temporal analysis of contagion spread of information.

6.4 Temporal analysis of contagion spread

To analyze the patterns of contagion spread, we calculate the time difference between every two consecutive retweets, i.e. inter-retweet times of viral and non-viral tweets. For DS1, Fig 4a presents four empirical cumulative distribution functions (ECDFs) of inter-retweet times that correspond to each contagion type, local versus global, and virality: viral tweets versus non-viral tweets.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 4. ECDFs of inter-retweet times.

The time axis scale is transformed into a log scale.

https://doi.org/10.1371/journal.pone.0230811.g004

For DS1, the two ECDF curves (Fig 4a) for local and global contagion of viral tweets are located above the two ECDFs of non-viral tweets, showing that users are more likely to retweet viral information. Regardless the of tweet vitality, the local contagion curve is located above the global contagion curve (Fig 4a), showing for DS1, that a user is more likely to retweet information tweeted or retweeted by users whom s/he follows. Unlike DS1, however, the ECDF of global contagion for DS2 (Fig 4b) is located above the ECDF of the local contagion up to Point P₁, indicating that in DS2 local contagion spreads more slowly than global contagion. DS2 users retweeting due to global contagion, respond quicker than users infected by local contagion. At P₁, 88% of the inter-retweet times of users were under 2 hours and afterward, the spread of global contagion is slightly slower than the spread of local contagion.

Since DS2 contains non-viral information, which is less likely covered by mainstream media, the Twitter Timeline algorithm likely promoted information up to Point P₁. Since the algorithm constantly changes [33] and affects the spread of global contagion, however, the algorithm likely stopped promoting the information at Point P₁.

Our findings show that Twitter users interact in bursts in the sense that short periods during which they send several tweets are separated by long periods of reduced activity [60, 63]. For DS1, the results show that local contagion spreads faster than global contagion regardless of tweet virality. Furthermore, regardless of it is local or global, contagion of non-viral information spreads faster in DS2 than in DS1 (Table 5).

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 5. Time when 80% of inter-retweet times of local and global contagion are reached, considering non-viral information.

https://doi.org/10.1371/journal.pone.0230811.t005

The nature of local contagion may explain our findings. Local contagion brings information to a user through neighbors whom s/he trusts, whereas global contagion can be promoted by algorithms. Another explanation involves the nature of the datasets. DS2 contains tweets written in English. Thus, many users worldwide can understand and retweet as, for example, when external regimes interfere with U.S. politics [64] or in discussions about movies, TV shows, and sports [2]. The high engagement of users about different topics is expressed by short inter-retweet times and call for a deeper analysis of topic-contagion spread.

6.5 Contagion spread of a topic

This section aims to detect and measure for viral and non-viral topics the reach depth of local and global contagion. Each topic contagion event W in DS3 can contain several original tweets w_μp ∈ W about the Higgs boson particle.

Viral events. The percentage of global topic-contagion of viral events (16.9%) in DS3 is lower than the percentage of global contagion of viral original tweets (21.75%) in DS1 (Table 2). In terms of the contagion spreading distance from the tweeting source (d) of viral events (Fig 3e), DS3 presents similar trends to the contagion spread of original tweets in DS1 (Fig 3a), as summarized in Table 3. In DS3, for d ≤ 17, d ≠ 10, if a path exists on G, the more distant a user is from the user originating the information (source User v₀), the less likely s/he is to retweet w_μp ∈ W.

Both DS1 and DS3 present a shift in trend for d = 10. In DS1, a shift is observed for d ∈ [10, 12], while in DS3 a shift is observed only for d = 10. This trend shift might be attributed to global contagion. Also, local topic-contagion spread of viral events (DS3), and local contagion spread of viral original tweets (DS1), are the most frequent (d = 1).

Considering the distance from the closest adopter (d_ca in Table 1), viral global topic-contagions in DS3 (Fig 3e), present similar trends (for d_ca ≤ 9) as viral global events in DS1 for d_ca ≤ 8 (Fig 3a). The larger d_ca, the less likely a user is to retweet w_μp ∈ W in DS3 and, similarly, the less likely a user is to retweet an original tweet w in DS1. This trend is reversed for d_ca > 9 in DS3 and for d_ca > 8 in DS1.

Non-viral events. For non-viral events, the percentage of global topic-contagion (Table 2) in DS3 (12.8%) is similar to that in DS1 (12.14%), but smaller than in DS2 (18.29%). The trends of the contagion-spreading distances (d) for non-viral events in DS1 (Fig 3b), in DS2 (Fig 3d), and in DS3 (Fig 3f) are similar. For d ≤ 13 in DS3, similarly to DS1 and DS2 (Table 3), if a path exists on G, the more distant a user is from the source User v₀, the less likely s/he is to retweet w_μp ∈ W. We also observe that local contagion spread of non-viral events in DS1, DS2, and DS3 are the most frequent ones (d = 1). Considering the distance from the closest adopter (d_ca), in all three datasets (Fig 3b, 3d and 3f), global contagion spread presents similar trends. In DS1 and DS3 for d_ca ≤ 7, the closer a user to an adaptor, the more likely that user will retweet, while in DS2 the findings are similar for d_ca ≤ 6.

In all three datasets, local contagion is the most frequent contagion mechanism. The results show similar contagion spreading trends of (i) Viral information and viral topics, and (ii) Non-viral information and non-viral topics. In Fig 3, a contagion at d > 1 does not necessarily indicate node-to-node contagion spread and might be attributed to global contagion as well.

Based on the results of the data analysis in the present section, we test the three hypotheses of this study (Section 5).

7.1 H1 testing

To test H1 and reveal whether local and global contagion has different temporal spreading patterns, we compare the distribution of the ECDFs in Fig 4 above by using the Kolmogorov-Smirnov (KS) D-statistic test [18, 65]. The D-statistic is defined as the maximum distance: D = max(|F₁(x) − F₂(x)|), where x represents the range of the random variable, and F1 and F2 represent the empirical cumulative distributions functions. The smaller the distance, the more similar the distribution curves and, hence, the more likely are the two samples to come from the same distribution. In the KS-test, a p_value < 0.05 indicates that the samples are not drawn from the same distribution.

For DS1, we found support for H1 via four KS-tests for pairs of ECDFs (Fig 4a) and found, as elaborated upon in Table 6, significant differences between the ECDFs (with varying D-statistics and every P_value ≤ 2.2 × 10⁻¹⁶). Thus, we found the temporal spreading patterns of local and global contagion to be significantly different. Also, for DS2, similar to DS1, we found support for H1 via a single KS-test between the ECDFs of local and global contagions (Fig 4b). The KS-test revealed that the two ECDFs are significantly different (D-statistic 0.56, P_value ≤ 2.2 × 10⁻¹⁶), thus uncovering different patterns of user behavior. In sum, the KS-tests for both DS1 and DS2 support H1 by revealing that local and global contagion have significantly different temporal spreading patterns.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 6. Kolmogorov-Smirnov results.

https://doi.org/10.1371/journal.pone.0230811.t006

Aiming to go beyond examining and explaining local versus global contagions, we apply next, the Back-in-time (BIT) approach for early detection of viral posts before becoming viral.

7.2 H2 testing

Next, to test H2 and reveal if viral and non-viral information have different local and global contagions spreading patterns, we use the BIT approach for DS1 since DS1 contains both viral information, needed for applying the BIT approach, and non-viral information.

Applying the developed BIT approach to DS1 in Step i, we created a KDE to estimate the probability density function of the retweet-count based on 1,950 non-viral tweets (Table 2). Having found in Step ii (Table 2) that the number of original viral tweets was 127, we sampled therefore 127 numbers from the KDE and randomly assigned a BIT retweet-count y to each of the 127 viral tweets. In Step iii, we rolled back in time each of the 127 viral tweets to a point where its retweet-count was y. We repeated Steps ii to iii 100 times. Fig 5 presents for DS1 the retweet-count distribution of non-viral information, the KDE, and the distribution of the KDE samples, i.e. the distribution of the retweet-count of BIT tweets obtained in Step ii.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 5. Retweet-count distribution of BIT and non-viral tweets.

https://doi.org/10.1371/journal.pone.0230811.g005

The BIT-tweets are viral tweets that were rolled back in time to a point when their retweet-count was 10 to 99, the same as the retweet-count for non-viral tweets. We then assigned to each BIT tweet a contagion type. Fig 4c above presents two local and global curves for the BIT tweets and two local and global curves for the non-viral tweets. As depicted in Fig 4c, although the BIT tweets have a smaller retweet count compared to viral tweets, their contagion spread patterns are similar to those of viral tweets and the ECDFs of BIT tweets are located above the ECDFs of non-viral tweets. In addition, local contagion is located above global contagion for both non-viral tweets and BIT tweets.

Support for H2 was found via four KS-tests by analyzing the four curves in Fig 4c based on DS1 for: local non-viral versus local BIT tweets, global non-viral versus global BIT tweets, global non-viral tweets versus non-viral local tweets, and global BIT tweets versus local BIT tweets. As elaborated upon in Table 6, all ECDFs are significantly different from one another (with varying D-statistics and every P_value ≤ 2.2 × 10⁻¹⁶). Finding support for H2 suggests that both the global and the local contagion spreading patterns of the BIT tweets significantly differ from the patterns of non-viral tweets.

7.3 H3 testing

Based on DS1 and DS2 analyses, H1 and H2 testing focused on contagion spread of a single original tweet. However, contagion of an idea or a topic is not limited to a single tweet and can spread by a sequence of tweets discussing the same topic, as in the DS3 dataset, which is devoted to the announcement about the discovery of the Higgs boson particle.

To test H3 and reveal if topic contagion has similar temporal spreading patterns to information contagion spread, we studied the inter-retweet time of topic contagion spread. Fig 4d presents four ECDFs of inter-retweet times that correspond to each contagion type—local versus global and each event virality—viral versus non-viral. Similar to the inter-retweet patterns of viral original tweets in DS1 (Fig 4a), Fig 4d shows for DS3 that the two ECDFs for local and global viral events are located above the respective ECDFs for non-viral events. Similar to the inter-retweet patterns of non-viral original tweets in DS1 (Fig 4a), Fig 4d also shows for DS3 that the global contagion curves are located below the local contagion curves, up to point P₂. After P₂, the global contagion curves slightly exceed the local contagion curves. In DS3, user interaction occurs before, during, and after the announcement of the Higgs boson discovery and different time-varying dynamics of user activities were found in these three periods [53]. The popular media’s coverage of the events toward the last period of the data collection, led most likely to global contagion [53]. In both DS3 and DS2, as depicted in in Fig 4d and 4b respectively, the ECDFs for local and global contagion of non-viral events alternate their relative location at P₁ and P₂. This alternation implies for non-viral events the sensitivity of global contagion spread to exposure by global contagion sources like the mainstream media. Like in DS1 and as elaborated in Table 6 for DS3, we found support for H3 by conducting four KS-tests. The results revealed that the four ECDFs curves are significantly different (with varying D-statistics and every P_value ≤ 2.2 × 10⁻²²).