From Strangers to Friends: Tie Formations and Online Activities in an Evolving Social Network

Network Coevolution

Recent literature on network coevolution typically adopts a continuous-time framework in which network users decide whether to alter their network ties or to perform other actions at random instants in time (Greenan 2015; Lewis, Gonzalez, and Kaufman 2012; Snijders, Steglich, and Schweinberger 2007; Steglich, Snijders, and Pearson 2010). For example, Snijders, Steglich, and Schweinberger (2007) develop a continuous-time first-order Markov model that describes both the formation of a network from an individual's perspective and the incidences of individuals’ other action(s) within the network. In the model, both the network structure and individuals’ actions evolve in a dynamic fashion: individuals are selected at random rates, and each selected individual decides whether to make a change in friendship ties, to perform an action, or to do neither. The randomness in the timing of friendship and activity decisions is necessary in this class of models because of the data limitation of observing the network only at discrete moments (i.e., a few snapshots of the network). As a result, the timing and sequence of tie formations and other actions of users that take place between two observed discrete moments are unavailable to researchers. In an empirical context more closely related to ours, Bhattacharya et al. (2019) extend the work of Snijders, Steglich, and Schweinberger and apply the model to study the coevolution of users’ social network structure and content posting behaviors using monthly data from a major social networking site.

There are several limitations to the modeling approach introduced in Snijders, Steglich, and Schweinberger (2007) and adopted in a number of later studies (e.g., Bhattacharya et al. 2019; Greenan 2015; Lewis, Gonzalez, and Kaufman 2012; Steglich, Snijders, and Pearson 2010). First, since individuals cannot both change their ties and perform other actions at the same time, simultaneous incidences of tie formation and other actions cannot be accommodated in this framework. Second, because of the randomness in the decision timing within a time period (between two sequential snapshots of the network observed by researchers), the effects of time-varying behaviors that happen continuously within the time period cannot be identified, and, as a result, their effects cannot be properly assessed. Third, although Snijders, Steglich, and Schweinberger capture homophily by accounting for observed similarities among users when modeling tie formations, latent homophily (arising from the similarity among friends in their unobserved intrinsic preferences) remains a confounding factor that may bias the effect of friends’ influence.

In this article, we overcome these three limitations by proposing a coevolution model of individuals’ concurrent decisions to both form friendship ties and perform online activities (such as product adoptions and content generation) at each point in time. We are able to do so because we observe the continuous development of tie formations and all concurrent online activities in our novel data. In other words, our data set records the occurrence and timing of all tie formations and other online activities performed within the network. This allows us to capture the effects of any time-varying behaviors while studying simultaneous incidences of these decisions. Furthermore, we are able to account for the latent homophily by explicitly estimating individuals’ unobserved intrinsic preferences for actions absent of their friends’ influence and therefore provide a cleaner identification of peer effects. We are able to do so because we observe users’ actions both before and after they make friends in our data.

Friendship Formation

More broadly, this research is related to the network formation literature. Researchers study network formation using three main modeling approaches: nodal attribute models, exponential graph models, and strategic network formation models. ⁴ The first category of models explains the existence of ties and the resulting network structure via similarities among pairs of individuals (e.g., Boguñá et al. 2004; Hoff, Raftery, and Handcock 2002). However, this modeling approach explains the status quo of a network, that is, who is or is not friends with whom, rather than its evolution. Exponential graph models explain the network development on the basis of structural patterns such as triangular connections or transitivity, but do not provide insights into the mechanisms that drive individuals’ tie formation decisions (e.g., Katona and Móri 2006; Mele 2017). These models are well suited for predictions, but not for causal inferences, and therefore not suited for counterfactual analysis. To overcome these shortcomings, strategic network formation models, the most recently developed modeling approach among the three, take the perspective of an individual actor's utility maximization when explaining the evolution of a network, and allow it to depend on the existing state of the network (e.g., Hanaki et al. 2007). ⁵ This article falls into the last category of modeling approaches. Note that unlike the network coevolution models discussed in the previous section, strategic network formation models describe only friendship tie formation and not any other activities.

Two notable studies in this stream of literature are Christakis et al. (2010) and Snijders, Koskinen, and Schweinberger (2010). In both, the authors model future states of a network on the basis of characteristics of the existing network state (current friendship ties) and similarity in terms of demographic traits. However, the models proposed in these two studies are not able to capture the effects of users’ changing behaviors and activities on tie formation outcomes because of the simulated network states between (a limited number of) observed snapshots of the network. Christakis et al. (2010) apply their model to study a network of 669 students with 1,541 friendship ties and find that, while having common friends is important for friendship formation, people are less likely to become friends with popular individuals. They also find that people prefer to become friends with people of the same gender and age. Snijders, Koskinen, and Schweinberger (2010) use a data set that contains six snapshots of a network of 32 students from the same class. They find that network characteristics such as transitivity matter in friendship formation. In terms of demographic characteristics, they find that male students are more popular as friends, but having the same gender does not increase one's chance of making a friend.

Studies in the social psychology literature find that similarity (homophily) is an important driver in individuals’ friending decisions in social networks. In this literature, similarity is defined more broadly as a match between two persons’ interests, experiences, backgrounds, or personalities and is not limited to similarity in demographic traits. Similarity between two individuals increases the chance of them becoming friends (e.g., Berscheid and Reis 1998; McPherson, Smith-Lovin, and Cook 2001; Sun and Taylor 2020). To put it differently, “birds of a feather flock together.” In addition, expertise or knowledge in the subject domain increases a person’s desirability as a potential friend. The increased desirability is due to the potential gain from information sharing and learning from a friend who is knowledgeable about the subject domain (Brandtzæg and Heim 2009; Watson and Johnson 1972).

Building on the extant research in the strategic network formation and social psychology literature, we identify three main drivers of users’ friendship decisions in an online social network: (1) network properties, such as out-degrees (i.e., number of friendship ties) and transitivity (i.e., number of common ties); (2) homophily between two users, including similarity in their demographic traits as well as similarity in their interests, such as animes both users have watched; (3) expertise of a user, which is reflected in the current level of activities and captures domain expertise, such as the number of anime-related posts the user has published. Although the first two types of drivers are studied in the existing research, to the best of our knowledge no studies in the extant strategic network formation literature are able to quantify the impact of the last driver on network formation. This constraint is largely due to the data limitation that only one or a few snapshots of the network are available to researchers; therefore it is almost impossible to capture the effects of time-varying expertise variables.

Our research complements and enriches the strategic network formation literature in two important ways. First, by observing the continuous evolution of the network, we do not need to rely on assumptions to simulate the current network state. Second and more importantly, we observe each individual's entire activity history, enabling us to explicitly account for the effects of individuals’ time-varying behaviors in their friendship formation decisions. The latter improvement is especially important because we study network formation in an online environment. In such an environment, individual users’ behaviors and opinions as observed by other individuals are likely to be among the most important drivers of their friending decisions.

Peer Effects

This research is also related to the literature on the identification of peer effects. Making a causal inference of friends’ influence is a challenging task (Manski 1993). Multiple social phenomena can confound the identification of social influence (see Hartmann et al. 2008; Shalizi and Thomas 2011, and discussion in the “Identification” section). Among these phenomena, homophily is probably the most challenging one to account for. Because of similarity, friends may exhibit the same behavior without one necessarily influencing the other. Different approaches have been proposed to account for homophily, including the use of instrumental variables (e.g., Bramoullé, Djebbari, and Fortin 2009; Claussen, Engelstätter, and Ward 2014; De Giorgi, Pellizzari, and Redaelli 2010), the inclusion of correlated group effects (e.g., Lee 2007; Lee, Liu, and Lin 2010; Ma, Krishnan, and Montgomery 2014), the use of exogenous shocks to peers (e.g., Tucker 2008) or exogenous randomness (e.g., Sacerdote 2001), experiments (e.g., Aral and Walker 2011), the incorporation of individual-specific unobserved tastes/preferences (e.g., Ameri, Honka, and Xie 2019; Nair, Manchanda, and Bhatia 2010; Trusov, Bodapati, and Bucklin 2010), modeling the coincidence of network and outcome formation (Badev 2021; Snijders, Steglich, and Schweinberger 2007), and explicitly accounting for the selection bias that arises from endogenous network formation (Hsieh and Lee 2016).

Our approach is similar to that of Hsieh and Lee (2016), who simultaneously model the network formation and social interaction processes. By allowing the unobserved components in both processes to be correlated, they explicitly account for the selection bias that arises from endogenous network formation, and thereby correct biases on estimated peer effects in the social interaction model. In our research, we take a similar approach by allowing the stochastic error term in our network formation equation to be correlated with the error terms in the other two online activity equations in which peer effects are estimated. However, the employed models differ. In particular, Hsieh and Lee model unidirectional tie formation, which results in a simpler model than ours since mutual agreement is not necessary. Furthermore, in contrast to Hsieh and Lee, we are able to account for latent homophily by explicitly estimating individuals’ unobserved intrinsic preferences for actions absent of their friends’ influence. Therefore, we provide a cleaner identification of peer effects. We can do so because we observe users’ actions both before and after they make friends.

Seeding

This research is also related to the literature on seeding. Seeding refers to the determination of whom to target for motivational stimulation with the goal of triggering large information cascades, adoptions, or other types of actions. The most commonly studied seeding strategies are based on network metrics such as “in/out-degree centrality” ⁶ or “betweenness centrality.” ⁷ For example, Hinz et al. (2011) compare three seeding strategies—stimulating high-degree, low-degree, and high-betweenness individuals with random seeding—in terms of their effectiveness in increasing adoption. They find that seeding to well-connected individuals is the most successful strategy in increasing participation in viral marketing campaigns. Similarly, Aral, Muchnik, and Sundararajan (2013) examine the effectiveness of seeding to high-degree individuals, dense regions of the network, and hubs unlikely to adopt against the effectiveness of random seeding and find that high-degree and dense region targeting generally perform better. Katona (2013) studies seeding strategies in a theoretical framework and shows that highly connected influencers are valuable only if they cover consumers who are not connected to many other influencers. In a more recent empirical study of a music network, Lanz et al. (2019) also find that most music creators do not benefit from seeding to influencers.

Many of the previous studies focus on (one-time) adoption behavior. However, for the growing number of online social platforms, users’ continuous engagement with the website may be more important than one-time adoption behavior. Trusov, Bodapati, and Bucklin (2010) focus on the activity levels of users within an online platform instead of adoption behavior. The authors develop an approach to determine which of a focal user's friends have a significant influence on the focal user's activity level. They find that, on average, only 20% of a focal user's friends influence the focal user and suggest these friends for seeding.

Similar to Trusov, Bodapati, and Bucklin (2010), we add to the existing knowledge on the effectiveness of seeding strategies in increasing user engagement on online social platforms. Going beyond the work of Trusov, Bodapati, and Bucklin, we examine different seeding strategies that not only are based on network metrics, as is common in this literature, but also can depend on users’ activity levels on the website. Furthermore, by modeling the coevolution of friendship formations and users’ actions over time, we not only capture the immediate effect of friends’ influence on each other but also capture how that effect propagates into the development of future friendship ties and into future actions of users.

Estimation Sample

The website was started in 2004 as a private domain. In April 2006, it was moved to a public domain and began to take its current shape. At that time, the website had about 300 users. After its transfer to a public domain, the number of members started to grow quickly (see Figure 1). About a year later, on March 16, 2007, the function of forming friendships was added. At that time, the website had about 450 members, and this number grew rapidly to 2,700 at the beginning of July 2007 and to about 10,000 by the end of 2007.

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Figure 1.

Dates Users Joined MyAnimeList.net.

We focus on users who joined the website in the second half of 2007, mainly for two reasons. First, users who joined the website before March 2007 are likely to add each other as friends on the basis of past interactions. To put it differently, had they had the option of adding friends before, they would have done so. Second, existing members might have taken some time to learn about this new feature. Therefore, we start our study period about three months after the introduction of the friending function.

Studying daily friendship formation among all users who joined between July and December 2007 is, however, computationally impossible because the data set would include over 7 billion pair-day observations. One potential solution is to shorten the observation period. However, this approach would result in insufficient variation in the dependent variables. Figure 2 shows the distributions of the number of days between activities of each type. In about 50% of the cases, users add a friend and publish a post more than a month after their last action of the same kind. In 40% of the cases, users watch an anime more than a month after the last watched anime.

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Figure 2.

Number of Days Between Activities.

Our solution is to sample from the network. ¹¹ We implement this solution via egocentric sampling: ¹² We first draw a random sample of 400 users (“egos”) out of 7,419 users who joined the website in the second half of 2007. We then include all of the egos’ friends in our estimation sample. We term the users who are not egos themselves but are friends with an ego “alters” (986 users). Thus, our estimation sample contains 1,386 users (400 egos and 986 alters). Note that egos’ friends can either be egos or alters and that all friends of all egos are included in the estimation sample. Figure 3 illustrates our sampling strategy. As an example, user 2, who is an ego, has three friends: user 1, who is another ego, and users 3 and 4, who are alters.

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Figure 3.

User Sampling Strategy.

In the estimation, we model all anime adoptions and all UGC production activities for both egos and alters. For friendship formation, we model all potential ties among egos (e.g., between users 1 and 2 in Figure 3), all potential ties among egos and alters (e.g., between users 1 and 3 in Figure 3), and all potential ties among alters (e.g., between users 3 and 4 in Figure 3). We do not model the friendship formation decisions between alters and users outside our estimation sample, but we do take their friendships into account when creating related explanatory variables.

Given that activities of users in the three areas can be correlated, missing a portion of the network formation for the 986 alters could lead to bias in our estimates related to friendship formation decisions. To alleviate this concern, we estimate separate coefficients for the 400 egos (for whom we model all their tie formations) and for the 986 alters (for whom we do not model the portion of the friendship network that includes users outside of our sample) in all three utility functions (i.e., friendship formation, anime adoption, and UGC production). We only use the coefficient estimates for egos to make inferences. ¹³

The observation period is 184 days between July 1, 2007, and December 31, 2007. We have fewer observations for users who joined after July 1, 2007. On average, we observe each user for 140 days.

Representativeness

To evaluate whether our random sample of 400 egos is representative of the entire network of 7,419 users who joined the community in the second half of 2007, we calculate summary statistics of four network structure measures (degree centrality, betweenness centrality, eigenvalue centrality, and transitivity/clustering coefficient) for both groups. We then test whether the average network structure measures for the 400 egos and the 7,419 users are significantly different. The summary statistics and t-values of the mean comparisons are presented in Table 1. We find none of the differences to be significant at p < .05 . ¹⁴

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Table 1.

Network Representativeness.

	Complete Network	Estimation Sample	t-Value
Average degree	4.37	3.81	1.12
Average betweenness centrality	8,986.13	5,234.28	1.50
Average eigen centrality	.0073	.0068	.94
Transitivity	.1986	.1779	1.32

To further validate the representativeness of our estimation sample, we examine the degree distributions among the 400 egos and compare it with the degree distribution in the entire network of 7,419 users who joined the community in the second half of 2007. The distribution graphs are shown in Figure 4. The degree distributions for both groups follow a similar power law distribution with the majority of users having fewer than 20 friends.

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Figure 4.

Degree Distributions.

Data Description

Within our sample of 1,386 users, we observe 5,038 ties out of 947,155 possible ties being formed during the observation period and about 68 million daily observations of possible pairs. Figure 5 shows the states of the network for snapshots taken at Days 1, 60, 120, and 184. The nodes represent individual users in the network, and the links between nodes represent friendships ties. Furthermore, the color of a node reflects the quantity of a user's UGC production, and the size of a node reflects the number of animes a user watched. The nodes become darker as users publish more posts, and the size of the nodes increases as users watch more animes. As expected, the nodes become darker, bigger, and more connected over time.

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Figure 5.

Network Coevolution over Time.

Table 2 summarizes key statistics of our data. In terms of demographics, 78% of users report their age and are, on average, 19 years of age, and 93% of users report their gender, with 39% being female and 54% being male.

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Table 2.

Descriptive Statistics.

	Mean	SD	Min	First Quartile	Median	Third Quartile	Max	N
Age	19.3	5.2	12	17	18.5	23	78	1,088
Gender (% female)	38.5
Gender (% male)	54.0
Gender (% not specified)	7.5
Egos
Number of active days	12.62	15.43	0	3	7	17	101	400
Number of friend adding days	2.55	4.69	0	0	1	3	59	400
Number of anime watching days	9.19	10.37	0	2	5	13	68	400
Number of content generating days	2.47	9.17	0	0	0	1	95	400
Percentage of active days	17.76	17.07	0	5.04	12.90	24.24	100	400
Percentage of friend adding days	3.80	6.19	0	0	1.63	4.60	40	400
Percentage of anime watching days	13.55	14.11	0	3.23	9.70	19.44	85.71	400
Percentage of content generating days	2.56	7.72	0	0	0	.82	61.69	400
Friend adding interval in days	44.26	36.05	1	15	34	64	156	12,414
Anime adding interval in days	25.64	28.20	1	5	14	37	138	20,680
Post adding interval in days	32.07	29.33	1	7	25	48	109	6,330
Alters
Number of active days	24.06	24.42	1	7	16	34	181	986
Number of friend adding days	5.93	6.40	1	2	4	7	48	986
Number of anime watching days	15.55	16.65	0	3	10	22	121	986
Number of content generating days	6.31	17.57	0	0	0	3	181	986
Percentage of friend adding days	22.49	18.36	.54	9.26	17.95	30.43	100	986
Percentage of anime watching days	6.85	8.31	.54	2.17	4.23	8.15	80	986
Percentage of content generating days	14.46	13.38	0	4.35	10.87	21.20	82.22	986
Percentage of content generating days	4.72	12.01	0	0	0	2.78	98.37	986
Friend adding interval in days	48.41	40.36	1	16	37	71	181	78,996
Anime adding interval in days	24.82	29.86	1	5	13	32	163	84,133
Post adding interval in days	47.98	48.96	1	7	28	78	182	40,287
Egos with friends
Number of friends on last day	4.72	7.83	1	1	2	6	94	320
Number of animes on last day	76.49	83.02	0	20	50.5	100.5	522	320
Number of posts on last day	6.13	22.50	0	0	0	2	228	320
Egos without friends
Number of friends on last day	0
Number of animes on last day	46.34	57.22	0	8	24	70.5	313	80
Number of posts on last day	.68	3.19	0	0	0	0	23	80

Figure 6 demonstrates how average activity levels of users who joined the website in the second half of 2007 change over time from the day they joined the website. Figure 6, Panel A, shows a decreasing trend in making new friendship ties. Because one benefit of having friends is the reduced costs associated with learning about the website and new animes, users are more likely to add friends shortly after joining the website. Motivated by this data pattern, we assume that users’ intrinsic propensity to make friends contains two components: a constant intrinsic propensity component and a second component that decreases with membership length. Panels B and C of Figure 6 show the activity trends for anime watching and content generation. Both graphs reveal a rather constant trend over time. ¹⁵ These two data patterns prompt us to assume that the intrinsic propensities in anime watching and content generation are constant over time.

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Figure 6.

Average Activity Levels of New Users over Time Since Joining.

Users can engage in multiple types of activities simultaneously. On average, egos have 2.6 active days in terms of friend adding, 9.2 active days in terms of anime watching, and 2.5 active days in terms of post writing (see Table 2). In total, they have 12.6 days in which they participate in at least one type of activity. To put it differently, on average, egos are active on about 18% of the days during the study period. We observe a similar pattern for alters albeit with higher average levels of all three activities. As shown on the bottom of Table 2, 20% of egos have no friends at the end of our study period. Egos with friends are more active in both anime watching and UGC production than egos without friends, suggesting the presence of peer influence from friends. Figure 7 shows a Venn diagram of the joint probabilities of each type of activity conditional on engaging in at least one type of activity. Users are active in only one area in 84.16% of the cases. Users are active in two and three of the areas of interest in 14.79% and 1.05% of cases, respectively. Figure 8 plots the percentages of egos engaging in zero, one, two, or three activities every calendar day and reveals constant levels during our study period. ¹⁶

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Figure 7.

Percentage of Observations with Certain Activities Conditional on Performing at Least One Activity.

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Figure 8.

Percentages of Egos Engaging in Zero, One, Two, or Three Activities per Day.

Lastly, Figure 9 shows histograms of individual users’ daily activity intensities conditional on them being active. In more than 80% of the cases, users add only one friend on an active day. Similarly, in about 60% of the cases, users watch only one anime per active day. However, the content generation intensity is higher: users publish one post per active day in about half of the cases and publish two or three posts per active day in about 20% and 10% of the cases, respectively. From these data patterns, we make the simplifying assumption to model anime watching and content generation as binary indicator variables; that is, we model whether a user watches an anime or publishes a post, but not the number of animes watched or posts published by a user, in a given day. ¹⁷

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Figure 9.

Number of Activities in Each Area per Day.

Tie Formation

We start by describing how we model tie formations among users over time. In each time period (day), a user makes decisions whether to become friends with any other user with whom the user is not friends yet. ¹⁸ Because there are usually many users with whom the individual is not friends yet, at each point in time, a user can become friends with multiple users. Note that we model a user's tie formation decisions for each possible friendship pair in each time period and not whether a user makes a friend in a time period.

Suppose that the website has i = 1 , … , N users and these users can become friends with other users during t = 1 , … , T time periods. ¹⁹ Let M denote the adjacency matrix of the network, which shows the status of ties between each pair of individuals i and j with j = 1 , … , N and i ≠ j . In the matrix, m ijt equals 1 if i and j are friends at time t and 0 otherwise. Ties are bidirectional and symmetric; that is, m ijt = m jit . Furthermore, both users have to agree to become friends. In our data, we do not observe users’ requests for friendship with other users, only the formation of ties on mutual agreement. Therefore our model describes the decision of both users to become friends regardless of who first initiated the friendship (see, e.g., Christakis et al. 2010). ²⁰ In our estimation sample, N = 1 , 386 ( 400 egos and 986 alters), T = 184 , and M is a 1,386 × 1,386 matrix. Note that the elements of M change over time as ties are formed.

The decision of two users to become friends depends on the utilities both individuals derive from becoming friends (see, e.g., Christakis et al. 2010). User i ’s utility of becoming friends with individual j in time period t , U ijt m , is given by the following: ²¹

U ijt m = f ( N m , R m , K m , C m , ϵ m ) ,

(1)

We assume that individuals have myopic utilities, that is, individuals do not anticipate future states of the network and only care about the current state of the network when deciding whether to form a tie (Christakis et al. 2010; Snijders, Steglich, and Schweinberger 2007). They do not take future links of themselves or the other party into consideration when making the decision to become friends. The assumption of myopic utility is appropriate for large networks in which individuals can meet numerous other individuals at each point in time and the number of future states of the network increases exponentially. Furthermore, users are not limited in the number of ties they can make in an online friendship network. As a result, users independently and nonstrategically form ties if the utility of such ties is positive.

We model the tie formation between two users as a noncooperative decision (e.g., Boucher 2016; Christakis et al. 2010); that is, each pair's decision to become friends depends only on their own friendship utilities and is conditionally independent from friendship decisions of other pairs of users. A tie between user i and user j is formed if and only if both parties decide that it is beneficial for them to become friends; that is,

m ijt = 1 iff U ijt m > 0 and U jit m > 0.

(2)

Product Adoption and Content Generation

Next, we describe how we model a user's activities on the website. We study the incidence of users’ activities in two broad areas, namely, product adoption and content generation. Let U it pa denote user i ’s utility from watching an anime at time t and U it cg denote user i ’s utility from producing content at time t . Then U it pa and U it cg are given by

U it pa = g ( F pa , A pa , C pa , ϵ pa ) and U it cg = h ( F cg , A cg , C cg , ϵ cg ) ,

(3)

Integrating All Actions

We now present the full model integrating user i ’s actions in all three areas:

U ijt m = f ( N m , R m , K m , C m , ϵ m ) ∀ j = 1 … N , i ≠ j U it pa = g ( A pa , F pa , C pa , ϵ pa ) U it cg = h ( A cg , F cg , C cg , ϵ cg ) .

(4)

G = [ 1 ρ m , pa ρ m , cg ρ pa , m 1 ρ pa , cg ρ cg , m ρ cg , pa 1 ] .

(5)

Utility Specifications

In this section, we present detailed utility specifications for the specific context of our data. We model the utility user i receives from forming a tie with user j as

U ijt m = α ~ it m + δ m N j , t − 1 m + R ij , t − 1 m + γ m K j , t − 1 m + λ m C ijt m + ϵ it m .

(6)

α ~ it m = α i m + κ 1 m W it ,

Next, N j , t − 1 m captures the properties of the network concerning user j , including the out-degree, operationalized as the cumulative number of friends user j has, and transitivity, operationalized as the proportion of friends that user j and user i have in common. ²³

The variable R ij , t − 1 m is tie-specific and captures the similarity (homophily) between individual i and individual j . We model R ij , t − 1 m as follows:

R ij , t − 1 m = κ 2 m R ij , t − 1 m + ζ ij m ,

The term C ijt m contains several variables whose effects we control for. First, we include a weekend dummy. Second, we control for the cumulative number of animes that user j has adopted. Third, we also include a dummy variable indicating whether user j was active on the platform during the previous week. This variable captures user j ’s visibility. ²⁵ Fourth, as mentioned previously, we include dummy variables indicating whether user i provided demographic information. Fifth, we include time fixed effects to address common shocks that might result in correlated unobservables affecting friending decisions across users. We operationalize these time fixed effects as week dummies. ²⁶ And, sixth, we include a dummy variable indicating whether user i joined the website before July 2007. Lastly, we assume that ϵ it m follows a normal distribution with a correlation matrix as specified in Equation 5.

User i ’s utility from watching an anime, U it pa , is given by

U it pa = α i pa + β pa F i , t − 1 pa + γ pa A i , t − 1 pa + λ pa C t pa + ϵ it pa ,

(7)

The term A i , t − 1 pa captures state dependence in anime watching and is operationalized as a dummy variable that equals 1 if user i watched an anime at t − 1 and 0 otherwise. ²⁹ Furthermore, C t pa contains several controls, including a weekend dummy and week fixed effects. And, lastly, we assume that ϵ it pa is normally distributed with a correlation matrix as specified in Equation 5.

Similarly, user i ’s utility from writing a post, U it cg , is given by

U it cg = α i cg + β cg F i , t − 1 cg + γ cg A i , t − 1 cg + λ cg C t cg + ϵ it cg ,

(8)

Then, A i , t − 1 cg represents user i ’s past activities and contains two variables: a dummy variable capturing state dependence in UGC posting behavior and the (cumulative) number of animes watched by user i by time t − 1 . User i ’s past anime watching behavior may influence user’s posting decisions because a user who watches more animes is likely to have more things to write about. As in the previous equation, C t cg contains a weekend dummy and week fixed effects. And, lastly, ϵ it cg follows a normal distribution with a correlation matrix as specified in Equation 5.

Friendship Formation

We start by discussing the parameter estimates for the friendship formation utility. As expected, network properties matter. The effect of the number of friends user j has is positive and significant (marginal effect of .0003 ). Given the bidirectional nature of friendships in our data and the operationalization of the out-degree variable as user j ’s number of friends, the direction of the effect is not identified in our model; that is, it is possible that either (1) users with more friends are more attractive as potential friends or (2) users with more friends seek to form more friendships. However, institutional details and descriptive data patterns, which we discuss in detail in Web Appendix D, suggest that (1) is the more appropriate interpretation; that is, well-connected users are desirable candidates for friendship, in our empirical context. The most important pieces of evidence presented in Web Appendix D are that users had no monetary incentives to add more friends (such as influencers have nowadays) and that users mostly form friendships during the first 12 months of platform membership. Users with longer membership duration, even those with many friends, add few friends after the first year.

The result that users with many friends are more attractive as potential friends stands in contrast to the negative estimate found by Christakis et al. (2010) for the number of friends a potential friend has. We suspect that this result is due to the unique context of the online social network environment in which users come from many different countries and do not know one another in real life. In such an environment, popular users are more likely to be “known” by other users (i.e., visible to other users), which is a prerequisite for tie formation. In comparison, students attending the same school usually know each other, so popular students do not have an advantage in this regard. In addition, initiating and maintaining many connections may be less costly in the virtual world than in real life.

Next, the effect of the proportion of friends that user i and user j have in common is significant and positive (marginal effect of .0008 ), implying that users are more likely to connect with friends of friends. Two average users who have 50% of friends in common are 44% more likely to become friends than two average users who have no friends in common. This finding is in line with results in the literature (e.g., Aral, Muchnik, and Sundararajan 2009; Bhattacharya et al. 2019; Shalizi and Thomas 2011).

In terms of similarity, the proportion of common animes has a significant positive effect with a marginal effect of .0008 , suggesting that common interests increase the probability of forming a friendship. Further, similarity in the three demographic traits (i.e., age, gender, and country) also increases the likelihood of friendship formation, consistent with the finding in Christakis et al. (2010). Comparing the magnitudes of the effects of the three demographics (marginal effects in parentheses), we find that having similar age ( .0019 ) is the most important trait, followed by coming from the same country ( .0014 ) and then having the same gender ( .0009 ). Put together, two users who share one demographic trait are, on average, three times more likely to become friends as two users who do not share any demographic traits. Lastly, the coefficient for the standard deviation of the pair-specific random effect capturing latent homophily between individual i and individual j is positive but insignificant. This finding suggests that our flexible modeling approach, together with the observed variables included in our model, captures the similarity between two individuals well.

User j ’s number of written posts represents this user’s domain expertise and has a significant positive effect on friendship formation, albeit a small one (marginal effect of .0000 ). Thus, UGC publishing is conducive to the formation of a social relationship. ³⁴

Finally, we find significant effects for all our control variables. The coefficient for the dummy variable indicating whether user j showed any activity during the previous week is positive and significant. One likely reason is that user j ’s activities increased this user’s visibility and awareness among other platform users. User j ’s (cumulative) number of watched animes has a significant negative effect on this user’s formation of a friendship tie with user i , although the effect is very small. ³⁵ Further, the weekend dummy has a significant negative coefficient, suggesting that users are less likely to form friendships on weekends.

Online Activities

Next, we discuss our results related to a user's anime watching decisions. The number of animes watched by friends has a significant positive effect on a user's anime watching behavior. For example, if a focal user's friends watch two animes the previous day, the probability that the focal user watches animes the next day increases by 15%. However, the number of friends a user has does not have a significant influence. These results indicate that having more active friends increases a user's activity level because of the presence of direct peer effects. We do not find a spillover effect of friends’ posting behavior on a user's anime watching: a user is not more likely to watch an anime if a friend made a post the previous day. Further, our results reveal positive state dependence in anime watching: a user is more likely to watch an anime if the user did so the previous day. Lastly, the coefficient for the weekend dummy is positive and significant, implying that users are more likely to watch animes on weekends.

We now describe our results related to content generation. Similar to our findings for anime watching, we observe evidence of a direct peer effect on a user's content generation: the number of posts published by friends during the previous day has a significant and positive effect on a user's content generation decision on the following day. We also note a positive spillover effect: the number of animes watched by friends increases a user's UGC production. In addition, the coefficient for the number of friends a user has is significant and positive, suggesting that having more friends makes a user more active in publishing content. Further, there is evidence of positive state dependence in content generation; that is, we observe a significant positive effect of a user's posting on the user’s posting behavior the following day. And, lastly, the weekend dummy has a significant positive effect.

Comparing the magnitudes of the direct peer effect versus the spillover effect using marginal effects, we observe the direct peer effect to have a larger impact than the spillover effect in increasing activity levels for both anime watching and UGC publishing. ³⁶ For example, if a focal user's friends write two more posts the previous day, the probability that the focal user also writes a post the next day increases by 26%. In contrast, if a focal user's friends watch two more animes, the probability that the focal user writes a post the next day increases by only 17%.

Model Fit

We evaluate the fit of our models using two criteria: the Akaike information criterion and the Bayesian information criterion. Our main model outperforms the independent model and the homogeneous model on both criteria (see bottom of Table 3). We also compare the (in-sample) predictive performance of the three models using hit rates for three holdout periods: the last 30 days, the last 60 days, and the last 90 days of our sample period. The average hit rates across these three holdout periods for each of the three activities (i.e., friendship formation, anime watching, and UGC production) are presented in Table 4. The average hit rates are very similar across the three models with the main model performing slightly better than the other two models in predicting UGC posts.

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Table 4.

Hit Rates.

	Independent Model	Homogeneous Model	Main Model
Friending actions	99%	99%	99%
Adoption actions	72%	72%	72%
UGC actions	94%	95%	96%

Platform-Wide Stimulation

We compare the effects of different platform-wide stimulations on overall activity levels, that is, the sum of activities in all three areas. The overall activity levels serve as a proxy for the total site traffic or total time spent on the site. The predictive exercises are implemented as follows: in each scenario, we increase one type of activity (friendship tie formation, product adoption, or UGC generation) among all egos and all alters by the amount of that activity on a typical day for an average user. To put it differently, our stimulation doubles the amount of activity of a particular type on a given day for each ego and each alter. We do so on Days 30, 90, and 150 and predict users’ behavior going forward until Day 184. ³⁷ When presenting our findings, we focus on findings for the egos.

The results are presented in Table 5. The first column shows the changes in the average number of active egos per day, that is, the mean number of egos who perform at least one activity in a day, and the second column depicts the changes in the number of total activities performed by egos. Out of the three types of stimulations the platform can implement (i.e., to recommend friends, animes, or forum discussion topics), stimulating users to watch more animes is the most effective intervention, resulting in the highest overall increase in the number of active users and in the level of in-site activities. This is because anime watching is the most frequent activity of the three in-site activities and because the peer effect of friends’ anime watching is a stronger driver than the spillover effect of users’ anime watching. The effects of stimulating users to make more friends or to post more UGC are very similar in terms of magnitude.

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Table 5.

Prediction from a Platform-Wide Stimulation.

Stimulation Activity	Stimulation Day	Percentage Change in Active Users	Percentage Change in Activities
Friendship formation
	30	2.76	1.97
	90	3.23	.43
	150	12.66	2.63
	Average	6.22	1.68
Anime watching
	30	1.80	12.66
	90	4.03	2.36
	150	17.14	6.47
	Average	7.66	7.17
UGC production
	30	1.74	1.69
	90	.43	.44
	150	13.90	3.21
	Average	6.20	1.78

Friendship Recommendations

Friendship recommendation systems are a popular tool used by online social networks such as Facebook or LinkedIn to increase network connectedness (Sun and Taylor 2020). Here, we ask whom to recommend as a potential friend in an evolving social network, that is, what type of individual a platform should suggest for friendship.

We evaluate the effects of forming a friendship with five types of potential friends: (1) popular user, that is, the user with the largest number of friends; (2) friend of a friend, that is, the user who has the most common friends with the focal user; (3) active anime watcher, that is, the user who watches the most animes; (4) active UGC creator, that is, the user who publishes the most posts; and (5) user with similar interest, that is, the user who watches the largest number of common animes with the focal user. Furthermore, this predictive exercise relies on the assumption that a friend recommendation strategy is successful in convincing all egos and alters to accept one recommended user as a new friend on Day 150. ³⁸ We then simulate users’ behavior going forward until Day 184, and compare the increases in each of the three activities as well as the overall activity level among all egos.

Among the five friendship recommendation strategies, recommending a friend of a friend is the most effective strategy in stimulating subsequent friendship formation within the evolving network. ³⁹ This finding stands in contrast to the common result for static networks, for which recommending a popular user has been found to be the most effective strategy. ⁴⁰ A potential reason for our different result is that a popular user does not necessarily share interests with or is not necessarily similar to the focal user. Thus, friends of the popular user are also less likely to be potential candidates for new friendships, thereby limiting the cascading effect. However, a friend of a friend is more likely to share interests with or be similar to the focal user. Thus, friends of a friend of a friend are more likely to be potential candidates for new friendships, thereby enhancing the cascading effect. Our data and estimation results support this explanation: friends of a friend have, on average, 1.5 common friends with the focal user, whereas friends of a popular user have, on average, .3 common friends with the focal user. Friends of a friend also watch a larger number of the same animes than friends of a popular user (9.3 vs. 8.4). And, lastly, we find significant and positive effects of similarity variables in the estimation (see the “Results” section).

Seeding

Previous studies find that users often have varying degrees of activity in different areas (e.g., Iyengar, Van den Bulte, and Valente 2011; Manchanda, Xie, and Youn 2008). Our data confirm this pattern (see Figure 5). For example, a user might make many friends or publish many posts but watch only a few animes. Consequently, carefully choosing whom to target and which type of activity to stimulate are crucial decisions for platform owners in achieving a desired outcome, such as an increase in UGC production.

In this set of prediction exercises, we examine the effectiveness of different seeding strategies in increasing tie formations, anime watching, and UGC production in an evolving social network. For these prediction exercises, we select the 15% most/least active egos among all egos on the basis of their activity levels in each of the three activities (“selection activity”) as our seeding targets. ⁴¹ As a benchmark, we also randomly select 15% of egos as the initial seeds. Next, we increase the activity level of these selected egos in one of the three areas of activity (“seeding activity”) by the amount of that activity on a typical day for an average user. This is the same stimulation as in the “Platform-Wide Stimulation” section. We do so on Days 30, 90, and 150 (“stimulation date”) and simulate users’ behavior going forward until Day 184. To recap, we perform a total of 63 prediction exercises, including 3 × 3 × 3 = 27 prediction exercises for the 15% most active egos and 3 × 3 × 3 = 27 prediction exercises for the 15% least active egos, as well as 3 × 3 = 9 prediction exercises for our benchmark case of randomly selected users. When presenting our findings, we again focus on the results for egos.

We start by investigating the question of whether platform owners should seed to the most or least active users. We find that seeding to the 15% most active users is a more effective strategy than seeding to the 15% least active users when the goal is an increase in anime watching or in UGC production. However, for friendship formation, seeding to the 15% least active users is a more effective strategy than seeding to the 15% most active users. Seeding to 15% randomly selected users performs the worst for all three outcomes. ⁴²

Next, inspired by results for static networks, we evaluate whether seeding to the most-connected users, that is, users with the most friends, is also the most effective seeding strategy in terms of the choice of selection activity in evolving networks. In other words, we examine whether the most-connected users are indeed better candidates for seeding than users who are most active in anime watching or UGC creation in an evolving social network. In answering this question, we focus on the 27 prediction exercises that use the most active users as initial seeds. Our results indicate that, on average, seeding to users who post the most UGC is the most effective seeding strategy, followed by seeding to the most-connected users, and lastly by seeding to users watching the most animes. ⁴³ A similar pattern also holds for the individual prediction exercises: in 25 out of the 27 prediction exercises, seeding to the most-connected users is less effective than either seeding to users posting the most UGC or seeding to users watching the most animes. We conclude that, in contrast to the common finding for static networks, seeding to the most-connected users is not the most effective strategy in evolving networks.

Whereas the discussion in the previous paragraph focused on the choice of the best selection activity with the goal of increasing overall activity levels, here we investigate the most effective seeding strategy for each desired outcome in terms of choosing both a selection and a seeding activity. For all three potential desired outcomes of increasing the number of friendships, increasing anime watching, or increasing UGC posting, the most effective seeding activity is the desired activity itself. However, the selection activity does not follow the same pattern. If the goal is to increase the number of friendships, then choosing the users who watch the fewest animes and encouraging them to make more friends is the most effective strategy. To increase anime watching, selecting the most-connected users and encouraging them to watch even more animes is the best strategy. And, lastly, to increase UGC posting, the best strategy is to select users who create the most content and encourage them to post more content.

Accounting for Network Evolution

First, we examine by how much the effectiveness of seeding strategies is underestimated when the endogenous network formation is shut down. To do so, we rerun 54 of the 63 prediction exercises discussed in the “Seeding” section but do not allow users to form new friendships. ⁴⁴ We find that not allowing for endogenous network formation leads, on average, to an underestimation of seeding effectiveness by 41%. This result underscores the practical importance of modeling the coevolution of individual users’ friendship tie formations and their concurrent in-site activities to correctly measure seeding effectiveness.

Second, we investigate whether we can replicate the common finding in the literature that seeding to the most-connected users is the most effective strategy to increase online activity levels (anime watching and UGC posting in our case) in static networks. To do so, we focus on the subset of 27 prediction exercises in which users are not allowed to form new friendships and in which the 15% most active users (in terms of friends, anime watching, and UGC posting) are used as initial seeds; that is, we center on the selection strategy. Consistent with the literature (e.g., Aral, Muchnik, and Sundararajan 2013; Hinz et al. 2011; Trusov, Bodapati, and Bucklin 2010), we find that, in static networks, seeding to the most-connected users is, on average and in all but two individual prediction exercises, the most effective strategy to increase online activity levels.

Lastly, we investigate whether our results for the case when both the selection and the seeding strategy can be chosen continue to hold when the endogenous network formation is shut down. For the two desired outcomes of increasing anime watching or increasing UGC posting, we indeed find the same two strategies to be most effective: to increase anime watching, selecting the most-connected users and encouraging them to watch even more animes is the best strategy. To increase UGC posting, the best strategy is to select users who create the most content and encourage them to post more content.