 Research
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Impact of network density on cascade size and community growth
 Kazumasa Oida†^{1}Email authorView ORCID ID profile
 Received: 13 November 2018
 Accepted: 22 April 2019
 Published: 31 May 2019
Abstract
This paper addresses two critical questions related to (1) the coevolutionary dynamics between information diffusion and network topology and (2) the relationship between viral content and social reinforcement. A recurrence relation model is developed to formulate the growth dynamics of a community centered on a dominant user using the tweetretweetfollow (TRF) and exogenous linkcreation events. This model illustrates several fundamental relations among parameters and quantities that are critical to answering the questions. The model reproduces social reinforcement and structural trapping effects, including empirical evidence that viral content does not require strong social reinforcement. In addition, the model demonstrates that the community growth rate is influenced not only by the rate of retweets but also by the network density of the community.
Keywords
 Network density
 Community
 TRF events
 Recurrence relation
 Coevolutionary dynamics
 Information cascade
Introduction
Online social networks (OSNs) have become indispensable vehicles of communication, information access, and advertisement. Content on OSNs may have economically, politically, or socially strong influences when shared among a large number of OSN users. Numerous papers in the literature have documented the manner in which such phenomena emerge; however, the mechanism that causes the phenomena may not be completely understood. This paper presents new insights into this field of study, shedding light on certain mechanisms that work in OSNs.
Every once in a while, largescale information cascades emerge. Because of their rare occurrence, statistical or machine learning approaches present risks associated with overdependence on small samples. Therefore, analytical approaches must verify pieces of empirical evidence and derive unknown causal relationships based on the evidence. This paper contributes to the literature in the field of information diffusion by answering two important questions.
Question 1
(coevolution) Does the information diffusion coevolve with the network topology?
It has been established that the underlying network topology affects information diffusion. Conversely, can information diffusion significantly alter the topology? To answer this question, this paper evaluates the community growth rate while a diffusion process is underway and demonstrates that the growth rate is greatly influenced by the network density of the community.
Question 2
(viral content) Why does viral content not require strong social reinforcement?
The seminal paper by Weng et al. (2013) presented a discussion on several communityrelated features that exhibit significant influences on information diffusion, including social reinforcement, which indicates that each additional exposure significantly increases the chance of adoption. One of the key findings of the paper was that nonviral content requires strong social reinforcement, whereas viral content does not. From an analytical perspective, this paper illustrates that largescale cascades emerge only if the network density of the community is low.
These questions are directly related to improvements in future cascadesize prediction. Therefore, answering these questions can help, for example, develop more effective viral marketing strategies and realize more efficient content delivery network (CDN) resource deployment for viral content.
The model in this paper presents a community structure for revealing the mechanisms that yield largescale information cascades because the communityrelated features are considered to exhibit a strong influence on future content popularity (Weng et al. 2014). An important factor that must be considered while designing an analytical model is that deviations from the empirical evidence must be small. Based on the evidence that a large number of adoptions occurred within only one degree of a few dominant individuals (Goel et al. 2012), this paper presents a scenario in which one dominant user, who has a significant influence on some online application or some social issues, is followed by a large number of Twitter users. This paper incorporates linkcreation events, called tweetretweetfollow (TRF), into the model because TRF linkcreation events are orders of magnitude more likely than linkcreation events caused by exogenous reasons (Antoniades and Dovrolis 2015). In addition, the paper also presents a discussion on the effects of exogenous linkcreation events.
This paper focuses on the two Twitter features (a dominant user and TRF events) mentioned previously and emphasizes their importance by reproducing the communityrelated effects (social reinforcement and structural trapping) and the evidence describing that nonviral (viral) content (does not) requires strong social reinforcement (Question 2). After that, the paper presents a discussion on the manner in which the parameters in the model, particularly the network density, relate to the two questions.
The paper is organized as follows. “Related work” section presents the related work. “The model” section defines the retweet diffusion model, and “Results” section derives the fundamental relations among parameters and quantities based on the computational results, and presents some insights into Questions 1 and 2. “The extended model” section extends the model by detailing user behavior. “Discussion” section presents a discussion on the implications of the findings and the limitations of this approach. “Conclusions” section concludes the paper.
Related work
Information diffusion on OSNs has been a subject of intensive study. It has been widely recognized that information diffusion is a more complex contagion than the spread of infectious diseases. Several empirical studies indicate that connectivity distributions among online users are highly inhomogeneous (Goel et al. 2012; Goel et al. 2015; Bose and Shin 2008). Prior studies have examined the influence of underlying network properties, such as scalefree, smallworld, and community structures, on the process of information diffusion under the assumption that the networks are static (Weng et al. 2013; Moore and Newman 2000; Watts 2002; Moreno et al. 2004). The influence maximization problem in social networks, under a variety of influence mechanisms and network properties, is an ongoing focus of research (Kempe et al. 2003; 2005; Aral and Dhillon 2018).
Current OSNs include a variety of communities such as language communities (Jin 2017); therefore, many studies have examined or applied the characteristics of information diffusion processes as information is transmitted within/across OSN communities. For example, the authors in (Li et al. 2015) categorized nodes into six communityaware types and evaluated their diffusion capabilities. This paper investigates the dynamics of a single Twitter community, assuming all community members are homogeneous and uniformly connected with each other.
Recent work (Tur et al. 2018) evaluated the effect of social reinforcement using a threshold model that assumes that additional contacts are not redundant but reinforce the probability of contagion. The model in this paper also assumes that additional retweet arrivals are not redundant; the retweet rate is constant independent of the number of arrivals. A fundamental difference is that the model in Tur et al. (2018) considers static networks. Another similar study (Banisch and Olbrich 2019) introduced positive and negative social feedback, and discussed the role of network density in the emergence of opinion polarization. The model in Banisch and Olbrich (2019) also assumes static underlying networks.
There are numerous studies that focus on predicting largescale information diffusion (Weng et al. 2013; Szabo and Huberman 2010; Cheng et al. 2014; Zhao et al. 2015; Cheung et al. 2017; Krishnan et al. 2016). Incorporating the community structure into diffusion models is expected to improve model accuracy (Bao et al. 2017). Several studies utilize community properties for prediction, such as predicting the future popularity of a meme (Weng et al. 2014) and the virality timing of content (Junus et al. 2015). Contrary to these results, this paper discusses the intrinsic difficulty associated with the prediction.
Moreover, there are studies on the dynamic nature of the OSNs (Leskovec et al. 2008; Myers and Leskovec 2014). The work in (Myers and Leskovec 2014) empirically demonstrated that large information diffusion has the potential to generate sudden influxes of edge creation or deletion events. In contrast to the assumption that the network topology is static while a diffusion process is ongoing, some papers have attempted to understand the diffusion phenomena on the basis of the coevolutionary dynamics, where network topology and information diffusion influence each other reciprocally. In (Weng et al. 2013), several linkcreation mechanisms were introduced and their roles in the diffusion process were examined. The authors in Farajtabar et al. (2017) formulated a stochastic process expressing the joint dynamics of information diffusion and network evolution. In Antoniades and Dovrolis (2015), the authors studied the importance of a linkcreation event called tweetretweetfollow (TRF). They demonstrated that TRF events can create dense user communities. There have been several studies that focused on linkcreation events such as triadic closure (Leskovec et al. 2008), shortcuts (Weng et al. 2013), homophily (Hours et al. 2016; Aragón et al. 2017), and social influence (Aragón et al. 2017). This paper formulates a retweet diffusion model based on TRF events because the events express general content sharing and link creation dynamics on Twitter. Instead of introducing stochastic processes or differential equations, this paper uses recurrence relations to express TRF dynamics because the recurrence relations derived are simple and ensure extensibility.
The model
The model includes an initiator, a large number of Twitter followers of the initiator (referred to as supporters), and Twitter users who are not supporters (referred to as nonsupporters). The model assumes that supporters are closely related to each other on the basis of the followerfollowee relationship and form a community. Therefore, the model is a community centered on the initiator. This section introduces recurrence relations to express the retweet propagation phenomena that originate from the initiator to supporters and nonsupporters according to the TRF events.
Parameter values in the model
Parameter  Description  Default value 

N _{ i}  Number of supporters at step 0  10,000 
N _{ f}  Number of followers a user has  500 
α  Ratio of supporters in N_{f} followers  (0,1] 
λ _{ s}  Retweet rate of supporters  0.01 
λ _{ n}  Retweet rate of nonsupporters  0.0015 
The model is designed under the following assumptions:
Asm 1

Supporters (nonsupporters) retweet with rate λ_{s} (λ_{n}). λ_{s}≥λ_{n}.

Supporters and nonsupporters have the same N_{f} followers.

Supporters have αN_{f} followers who are supporters (0<α≤1), while the followers of nonsupporters are all nonsupporters.
Asm 2
Users synchronously post retweets after receiving (re)tweets. The synchronization points are called steps.
Asm 3
The community does not have any specific network topology. At each step, αN_{f} followers of each supporter are uniformly selected from all supporters.
Asm 4
All nonsupporters who post retweets become supporters (i.e., follow the initiator).
Asm 4 is relaxed in “The extended model” section; however, the assumption provides most of the essential results in this paper. The model is simplified to focus on two Twitter properties: TRF events and locally scalefree structure (one dominant user and other users with a relatively small number of followers). The importance of the properties is confirmed by reproducing the communityrelated effects (social reinforcement and structural trapping) and (non)viral content behavior.
where the numerator denotes the number of supporters who have not posted any retweets, and the denominator denotes the number of supporters who have the possibility to receive the kth retweet. In (3), r^{′} is the number of supporters who have retweeted at step i. In the denominator, “ − 1” indicates that retweeters do not receive their own retweets, and “ −(k−1)” indicates that a user does not receive multiple retweets from the same user. Because P_{unret} is also the mean of the binomial distribution B(1,P_{unret}), the mean value of the number of supporters who retweet after receiving the kth retweet is given by λ_{s}P_{unret}. For all \((r(i1)+\bar q(i1)) \times \alpha N_{f}\) retweets, λ_{s}P_{unret} values are computed and added to r^{′} in line 5. r(i) is then derived in line 8.
which indicates that the number of users who post retweets at step T+1 is less than one. This paper defines T (steps) as the duration of retweet propagation.
Results
Sparse community effect
This paper terms this phenomenon the sparse community effect. The effect implies that sparse communities, in which supporters have many (few) links to nonsupporters (supporters), yield the largest cascades (R+Q) and the most longlived cascades (T) when the rates of retweet are high. The sparse community effect presents a case in which the network structure affects content virality.
Note that in Fig. 2(lower right), R and Q do not change with N_{i} (these are quantities relative to N_{i}).
Structural trapping and social reinforcement
Answering two questions
Figure 3 illustrates the case where λ_{s} increases from 0.01 to 0.05. From the figure, Q(0.05)/T(0.05) at λ_{s}=0.05 is approximately 30 times greater than that at λ_{s}=0.01. This implies that an attractive tweet from the initiator has the potential to greatly enlarge the growth rate. Note from Fig. 3 that Q(0.05)/T(0.05) at N_{i}=10^{4} and λ_{s}=0.05 is approximately 0.06, and this implies that after 10 steps, the average community size increases by 60%.
In other words, communities with a large number of supporters grow slowly. Figure 3 illustrates that Q/T at N_{i}=10^{5} is slightly smaller that at N_{i}=10^{4}. Accordingly, the growth rate is particularly sensitive to α and λ_{s}.
where α_{W} is the ratio α that maximizes W. Namely, a large number of users post retweets (i.e., viral content emerges) at small α values where the social reinforcement effect is considered to be small.
Let us intuitively explain why (16) occurs. Community members (i.e., supporters) exhibit a greater tendency to post retweets than nonmembers (i.e., nonsupporters) because of the social reinforcement effect and λ_{s}>λ_{n}. When the value of λ_{s} is high, for example, members easily retweet; therefore, by reducing α, more retweets must be sent outside of the community, providing nonmembers more chances to retweet; otherwise, the structural trapping effect becomes more pronounced. In other words, to maximize W, the ratio α_{W} controls the numbers of retweet messages flowing inside and outside the community such that fewer (more) messages flow inside (outside) the community (i.e., α_{W} is reduced) when the value of λ_{s} or λ_{n} increases (the sparse community effect).
The extended model
Detailed user behavior
Asm 4 in “Results” section assumes that all nonsupporters who post retweets become supporters. This section relaxes the assumption and introduces exogenous linkcreation, which states that nonsupporters become supporters without receiving retweets, by replacing Asm 4 with the following assumptions:
Asm 5
satisfies 0<β_{r}≤1, where S_{n} is the set of nonsupporters.
Asm 6
satisfies 0≤β_{n}<1.
Asm 7
is fixed and 0≤c≤β_{n}.

In line 2 in Algorithm 1, \(r(i1)+\bar q(i1)\) (the number of supporters who retweet at step i−1) is replaced with \(r(i1)+\beta _{r} \bar q(i1)\).

In line 4, P_{unret} is revised as$$\begin{array}{@{}rcl@{}} P_{unret} &=&\frac{N_{i}  r'  \sum_{1\le \ell\le i1} r(\ell) + \beta_{r}\frac{\beta_{n}}{1\beta_{n}} \sum_{1\le \ell\le i1} \bar q(\ell)} {N_{i}1(k1)+\beta_{r} \left(1+\frac{\beta_{n}}{1\beta_{n}}\right) \sum_{1\le \ell\le i1} \bar q(\ell)}. \end{array} $$(20)
Therefore, compared with (3), the numerator in (20) includes a new term \(\beta _{r}\frac {\beta _{n}}{1\beta _{n}} \sum _{1\le \ell \le i1} \bar q(\ell)\), and in the denominator, \(\sum _{1\le \ell \le i1} \bar q(\ell)\) is replaced by \(\beta _{r} \left (1+\frac {\beta _{n}}{1\beta _{n}}\right) \sum _{1\le \ell \le i1} \bar q(\ell)\).
which indicates that β_{e} linearly increases with β_{n}. Therefore, β_{n} is referred to as the exogenous linkcreation indicator.
where R is normalized such that it satisfies 0≤R≤1, and Q and W are quantities relative to N_{i}. Note that Q and \(\bar Q/N_{i}\) are different; \(\bar Q/N_{i}\) denotes the relative number of nonsupporters who post retweets until T. Note also that the definition of W is the same as before.
Relaxing sparse community effect
From (32), an increase in β_{r} strengthens the sparse community effect, whereas β_{n} weakens the effect. The following partly explains (32). An increase in β_{r} or a decrease in β_{n} indicates that a percentage of retweeters in the community increases; therefore, to enlarge R and Q, more retweets must flow outside the community, which implies that α_{Q} and α_{R} decrease. black
The extended model also satisfies the fundamental relations (5)(9) in “Results” section and exhibits the sparse community, social reinforcement, and structural trapping effects. Therefore, the extended model provides the same discussion on Questions 1 and 2 as in “Answering two questions” subsection.
It is worth noting that in Fig. 6(upper left), Q(0.1)/T(0.1) at β_{r}=1 is approximately 50 times greater than that at β_{r}=0.1; i.e., a low retweeter’s follow rate makes networks static. In addition, from Fig. 6(lower right), the high peak of W at β_{n}=0.4 suggests another mechanism that creates largescale cascades.
Large cascade creation mechanisms
It must be noted that the cascade size is determined not by λ_{s} (content infectivity) but by α (network structure) when α>0.1.
which also coincides with (16). This result implies that large α values in (38) enlarge small and middlescale cascades; i.e., nonviral content requires social reinforcement. The result also demonstrates that largescale cascades require not only a large λ_{s} but also a large λ_{n}.
which indicates that compared with (36), this mechanism requires somewhat weak social reinforcement. This result agrees with (32).
Figure 7(lower right) illustrates contour lines of growth rate Q/T obtained using the same parameter values as Fig. 7(upper left). It can be observed that the α−λ_{s} plane is separated into dynamic (high Q/T) and static (low Q/T) areas. The static area is located in a small area where λ_{s} and α are both small. This result agrees with (13), Figs. 3, 6(upper left), and 6(upper right). From Fig. 7(upper left) and (lower right), W and Q/T are not correlated and they have the maximum values at different (α,λ_{s}) points.
Discussion
Implications of the findings
There are many studies that focus on the prediction of largescale information cascades. Figure 7(upper left) demonstrates the potential difficulty in predicting such information cascades. From the figure, it can be observed that not only largescale cascades but also small and middlescale cascades appear at low ratios and their future sizes are highly sensitive to α,λ_{s}, and λ_{n}. The same result can be seen in Figs. 2(upper left) and 2(upper right), where R and Q vary significantly near ratio zero when λ_{s} and λ_{n} are high. It is not easy to measure these rates accurately at the initial stage of a cascade because the sample size at that stage is small. Even if the rates are accurately measured, the future size will, in general, have a certain probability distribution. For example, W at (α,λ_{s})=(0.01,0.01) in Fig. 7(upper left) is not deterministically given but will take values such as 4, 6, and 8. Furthermore, ratio α, a network density indicator, itself may vary with time (Antoniades and Dovrolis 2015; Oida and Okubo 2019). This paper considers the ratio as timeinvariant; however, if TRF events affect the network densities of Twitter communities (Antoniades and Dovrolis 2015), the prediction of viral content becomes even more difficult.
This paper incorporated exogenous linkcreation events into the model and demonstrated that these events have the potential to create largescale cascades. According to (Antoniades and Dovrolis 2015), their occurrence probability is significantly low (two or three orders of magnitude lower than the occurrence probability of TRF events). However, big data analytical services, such as personalized recommendation in social networking sites, may boost exogenous link creation in the future.
This paper presents a simplified model, focusing on two Twitter properties: the TRF events and the existence of a dominant user with a large number of followers; nevertheless, the model reproduces social reinforcement and structural trapping effects and presents many interesting causal relations related to Questions 1 and 2. This outcome suggests that these properties are important factors for understanding real network dynamics.
Limitations of this approach
The results in the paper are only hypotheses and must be validated by detailed eventdriven simulations or empirical evidence. Because of the uniformlyconnected homogeneous agent and synchronous retweet assumptions (Asms 1–3), the model presents the average behavior of the retweet diffusion phenomena. To measure the effects of variations in agent behavior or network structure, one approach is to introduce a heterogeneous agent model with some network topological features. Another approach is to extend the model such that communities or subcommunities inside a community have different attribute values. The latter approach minimizes the complexity of the model and is therefore suited to simulate largescale diffusion processes. Using the recurrence relations and Algorithm 1 in the paper, the heterogeneous (sub)community model can be easily derived.
Conclusions
This paper addressed the following questions: (1) Can the underlying network be considered static while a diffusion process is underway? (2) Why does viral content not require strong social reinforcement? The paper introduced a recurrence relation model that describes the retweet diffusion and community growth dynamics, in which one dominant user (initiator) with a large number of followers posts a tweet, then retweets by followers propagate the tweet, and TRF and exogenous linkcreation events generate new followers. Consequently, the community comprising the followers of the initiator grows. The paper focused on the ratio of community members in the followers of each community member (α), which is also a network density indicator of the community. blackThe model not only illustrates the structural trapping and social reinforcement effects but also presents several important relations among three ratios α_{T},α_{Q}, and α_{R}, three quantities R, Q, and T, and four parameters λ_{s},λ_{n},β_{r}, and β_{n}. Above all, the magnitude relation (α_{T}<α_{Q}<α_{R}), the sparse community effect, and the result that Q(α_{Q}) grows and T(α_{T}) falls as λ_{s} increases were directly associated with the two questions.
The computational results derived from the model answered the two questions as follows: (1) The criterion for network variability was the community growth rate. Except for the cases in which the retweeter’s follow rate β_{r} is significantly small or the community size is extremely large, the growth rate is small when retweet rate λ_{s} and ratio α are both small. If the condition does not hold, the growth rate may rise by more than 10 times; i.e., the networks are potentially dynamic. Therefore, depending on the evaluation conditions of diffusion models, the effect of topological changes must be considered. (2) There are two mechanisms that create largescale cascades: the endogenous and exogenous mechanisms. Currently, only the endogenous mechanism is functional in the Twitter network. Both mechanisms demonstrated that the cascade size is maximized at a low ratio α (i.e., at a low network density), at which the social reinforcement effect is considered to be low. Thus, the model reproduced the evidence that viral content does not require strong social reinforcement. This result is closely related to the sparse community effect, which indicates that as the retweet rates increase, the α value that maximizes the cascade size decreases. Therefore, largescale cascades emerge only if the ratio is low.
Notes
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