Strategic Timing of Content in Online Social Networks

Transcription

1 Strategic Timing of Content in Online Social Networks Sina Modaresi Department of Indstrial Engineering, University of Pittsbrgh, Pittsbrgh PA 56, Jan Pablo Vielma Sloan School of Management, Massachsetts Institte of Technology, Cambridge MA 39, Tahid Zaman Sloan School of Management, Massachsetts Institte of Technology, Cambridge MA 39, Today, having a strong social media presence is an important isse for large and small companies. A key social media challenge faced by these companies marketing teams is how to maximize the impressions or views of the content they post in a social network. Optimizing the posting time of content to increase impressions is an approach not considered before becase it was not clear how to systematically select the optimal posting time and what wold be the potential gain in impressions. In this work we show how to select posting times to maximize impressions and the potential gains of this strategic timing. We se data from several Twitter sers to bild a model for how sers view content in a social network. With this model we are able to provide a simple eqation that gives the impression probability for a piece of content as a fnction of its posting time. We show that for several real social media sers strategic timing can significantly increase impressions. Frthermore, this increase in impressions comes at no cost becase choosing the time to post is free. In addition, all calclations se pblicly available data, so this approach can be implemented very easily. Finally, we consider the sitation where strategic timing becomes widely adopted and posting times are schedled by an online application. This sitation leads to potentially intractable optimization problems and a natral trade-off between aggregate performance and fairness objectives. However, we show that srprisingly, increasing fairness actally seems to improve aggregate performance in this setting. In addition, we show that soltions that are nearly optimal for both objectives can be easily constrcted. Key words: marketing, advertising and media, social media, social networks, statistics, optimization. Introdction Having a strong social media presence is becoming more and more important for a wide range of companies (Pozin ). Modern social networks provide the opportnity for companies to easily reach a massive adience. The social media strategy of a company involves designing and posting content to grow their cstomer base and engage with existing cstomers. The marketing team of a company will typically be responsible for posting important information on social networks, sch as new prodct releases or promotional material, with the hope of reaching as large an adience as possible. Social networks have an incredibly large potential adience for these posts: the social network Twitter has over 3 million sers (Twitter 5) and the

2 Athor: Strategic Timing of Content in Online Social Networks image based social network Instagram has 3 million sers (Instagram 5). While this potentially large adience is attractive, the challenge is that today the nmber of competing posts from other social network sers is hge, with over 5 million tweets posted per day on Twitter (Twitter 5) and 7 million photos and videos posted per day on Instagram (Instagram 5). This large volme of posts means that it can be difficlt for a given post to reach the intended adience. Social network sers who post content wish to have their adience engage with their content. This is becase engagement is an active interaction with the post and represents a better reflection of sers who liked the post verss sers who saw the post and did not care for it. Engagement comes in different varieties depending on the specific social network. On Twitter engagement incldes favoriting a tweet or retweeting a tweet (forwarding a tweet to others). On Instagram engagement is done by liking a post. All these different forms of engagement involve clicking on the post and creating an easily measred signal. Therefore, engagement allows one to estimate how many people actally viewed the post and also deemed it interesting enogh to interact with it in some manner. Engagement can only occr if a ser actally sees the content, which is referred to as an impression. Therefore, increasing the nmber of impressions can potentially increase the overall engagement. Many factors determine the nmber of impressions content receives in social networks. One factor impacting the nmber of impressions is how engaged sers are on the social network. If no one ever checks for content on the social network, then there will be no impressions. If the overall qality of content improves on the social network, this can make sers check it more freqently. However, this factor cannot be affected by an individal ser trying to maximize his own impressions. Another factor impacting the nmber of impressions is follower cont. Having a larger nmber of followers means there is a larger potential adience for the posts. Therefore, increasing follower cont is one way to gain more impressions. There is another important, bt very often overlooked factor impacting the impression cont that has not received mch attention: the timing of the post. A very natral way to generate impressions is to post content when there are many sers checking the social network. If there is a large adience at a certain time, then that is a good time to post. However, in a social network, one s post mst compete with other sers posts for the attention of followers. Therefore, it is logical to also avoid times when there are many others posting. Combining these two effects, there is a clear trade-off between the nmber of sers checking the social network and the nmber of sers competing to attract their attention. Therefore one can see that the best time to post is when a large nmber of followers are checking the social network and when not many others are posting. In other words, if yo want someone to hear what yo have to say, say it when they are arond to listen and also when no one else is talking to them... Or Contribtion The above discssion sggests that strategic timing can be sed to increase the nmber of impressions received by a piece of content in a social network. In this work we show that this is indeed the case and

3 Athor: Strategic Timing of Content in Online Social Networks 3 that the gains from strategic timing can be qite large. We consider the sitation where a ser is trying to maximize the probability that a target follower sees his post. We begin by proposing a model for how the follower creates an impression for the ser s post in a social network. This model is fairly general and applies to several major social networks sch as Twitter and Instagram. The main components of the model are ) the arrival process of the follower to the social network, ) the posting process from all sers the follower follows, which we call the timeline process, and 3) how many posts the follower views each time he arrives to the social network, which we call the ctoff window. With this model we are able to calclate the optimal times for the ser to post content in order to maximize the probability that it is seen by the target follower. We also provide simple, tractable approximations that allow for easy calclation of the impression probability. In order to calclate the impression probability sing or model, we need to know the rates for the arrival and timeline processes. To do this, we develop a parametric Bayesian model for the rate fnctions. The rate fnctions captre the key temporal patterns seen in typical ser behavior on social networks, sch as daily and weekly oscillations in the arrival and timeline rates. Using data from actal sers on Twitter, we estimate these model parameters and obtain vales for the rate fnctions. We then se the estimated rate fnctions to calclate impression probabilities. We find that strategic timing of content can significantly increase the nmber of impressions received. This is a hge gain given that strategic timing has no additional cost and reqires virtally no extra effort on the part of the ser. In addition, we show both theoretically and empirically that or impression probability calclations are robst to ncertainty in model parameters. This is an important reslt since some model parameters, sch as the arrival rate and ctoff window, are difficlt to measre. Therefore, we show that strategic timing can be achieved sing pblicly available data which is easily accessible by anyone. We then stdy the impact of wide adoption of strategic timing of content. In particlar, we consider a hypothetical application that optimizes the posting times of several sers at the same time. This application wold natrally have to balance the sal trade-off between the aggregate performance and fairness objectiveness. Frthermore, both objectives can lead to a potentially intractable global optimization problem. However, we will show the somewhat srprising reslt that improving fairness tends to improve average performance and that soltions that are nearly optimal for both objectives can easily be constrcted. This work has immediate applications to firms or individal sers who tilize social media for promotional prposes or to gain inflence. Strategic timing is an easily implementable and cost effective tactic for increasing the impressions for content in social networks and can seamlessly be integrated with a broader social media campaign or strategy. It reqires no modification to the type or volme of content created. The method presented here can be directly applied by any social network ser becase the reqired data is pblicly available and selecting the time of a post does not incr any direct costs.

4 4 Athor: Strategic Timing of Content in Online Social Networks The rest of the paper is strctred as follows. We begin with a review of previos relevant work in Section.. In Section we present or model for creating impressions in social networks and or calclation of the impression probability. In Section 3 we present or Bayesian models for the arrival and timeline processes and their estimation sing data from sevaral Twitter sers. We calclate impression probabilities for these sers and show the robstness of the calclations to ncertainty in model parameters in Section 4. We show how real Twitter sers can modify their posting times to increase their impressions and the corresponding impression gains in Section 5. The optimization of posting times is presented in Section 6. We then smmarize the main insights regarding strategic timing in Section 7. All proofs are located in the Appendix... Previos Work Or work belongs to the body of research concerning the spread of information in social networks. In this area the qestions of interest are how to model and predict the diffsion of the information and determine what featres are condcive to rapid spreading of the information. We now review the major work in these areas. Several researchers have looked at how featres of individals or the content affect the spread of information. Aral and Walker () condct a randomized experiment with.3 million Facebook sers and find that featres sch as gender, age, and marital stats are predictive of inflence and ssceptibilty. Katona et al. () stdy adoption data for an online social network and find that featres of the local network strctre of an individal sch as their degree and edge density impact their adoption probability. A similar stdy done in Ugander et al. () on Facebook adoption data shows that the nmber of connected components of a ser s local network impacts the adoption probability. In Twitter, stdies of the information spreading problem have focsed on retweets. Several works have stdied what featres of the sers and content of the tweet case sers to retweet (Peng et al. ), (Sh et al. ) (Naveed et al. ) (Petrovic et al. ). Throgh varios prediction techniqes they show that ser featres sch as degree, and tweet featres sch as the presence of hashtags or URLs impact the likelihood of a retweet. Other work on the retweet problem aim not to predict the likelihood of an individal retweet, bt the total nmber of retweets received by a tweet. Hong et al. () and Bandari et al. () se a variety of algorithms to predict not the exact nmber of retweets, bt rather a coarse interval for the nmber of retweets of a tweet. In particlar, Hong et al. () investigates the factors inflencing information propagation in Twitter inclding message content, temporal information, and sers social graph. Bandari et al. () se regression and classification algorithms to show that it is possible to predict ranges of poplarity of tweets with reasonable accracy. Zaman et al. (4) predict the final retweet cont of a tweet sing the time-series path of its retweets. They se a Bayesian approach to develop a probabilistic model for the evoltion of the retweets sing the retweet times and the social network strctre. Their approach predicts the exact nmber of retweets of a tweet within mintes of its posting time with a very low error.

5 Athor: Strategic Timing of Content in Online Social Networks 5 The problem of maximizing the spread of information in a social network was framed as an optimization problem in Kempe et al. (3). Here the goal is to select the best set of seed sers to initialize with the information in order to reach the maximm nmber of sers throgh natral diffsion on the social network. A greedy algorithm was proposed to solve this combinatorial optimization problem. Variants of this problem and algorithmic soltions were sbseqently stdied in Kempe et al. (5), Chen et al. (9), Chen et al. (). Or work is also related to the research on competition for attention in social networks. Several athors have looked at the problem of competing for attention in a crowded social network. Van Zandt (4) first proposed the idea that receiver attention will become a bottleneck as it becomes easier to send messages or share information. Anderson and De Palma () and Anderson and De Palma (3) stdy a similar model and show that higher costs for posting content or messages can increase the average tility of a message and increase the overall viewing of messages. Iyer and Katona (5) propose a model for the incentives for entering a social network and posting content. They investigate the choice of sers to post or jst view content and also how the network strctre affects entry to the network and competition for attention. They show that this model predicts the so called participation ineqality phenomenon where as the commnication span of the social network increases, a smaller fraction of the sers actally post content, bt do so at higher freqency. Or work differs from the approach of this extant work becase we do not explicitly consider costs or incentives in theoretical models of social networks. Rather, we se data from real social networks to model the dynamics of ser behavior and show how this model can be sed to select the timing of a post in order to maximize the chance of it gaining someone s attention. Since or work is based pon real ser data, it can be directly applied to many poplar social networks sch as Twitter or Instagram.. Impression Probability Model We now present a model of how a ser views content, or generates an impression, in a social network. Or model assmes that content is displayed to the ser in chronological order, with the most recently posted content being seen first. This method of displaying content is sed by social networks sch as Twitter and Instagram. We assme that sers access the social network throgh an application on their mobile phones, which is the predominant way to access many social networks. For instance, over 8% of Twitter sers access the social network throgh a mobile application (Twitter 5). The main implication of this assmption is that becase of the size of a mobile phone screen, additional ser action is reqired in order to see older content which is not immediately displayed on the screen. As we will show, the assmptions on the chronological ordering of content and the ser action reqired to view older content, pls the temporal dynamics in the behavior of sers of the social network (both content creators and consmers) lead to significant differences in the impression probability of content as a fnction of the time it is posted. We

6 6 Athor: Strategic Timing of Content in Online Social Networks next discss the elements of or model in detail and show how they can be sed to calclate the impression probability... Model Components Or model is for a specific ser seeing a piece of content (creating an impression). We refer to this ser viewing the content as the follower. We want to calclate the probability that a piece of content posted at time t by another ser is seen by the follower. To do this, we mst model three components of the follower: his arrival process to the social network, his timeline process, and the ser interface for the social network application. The arrival process models the times when the follower checks the social network application for new content. Throghot the paper we refer to the content as a post, and se both terms interchangeably. In a social network a follower can choose to follow other people. We will se the Twitter terminology and refer to these sers as the follower s friends. When the follower checks the social network application, he will look at his timeline, which contains posts from all of his friends. The timeline process models the arrivals of posts to the follower s timeline. The ser interface characterizes how the follower views the content. The timeline posts are arranged in chronological order, with the most recent post located at the top of the timeline. As time passes, older posts are pshed down on the timeline. Typically, a follower only looks at a certain nmber of new posts when he checks his timeline. We refer to this nmber of posts as the follower s ctoff window. Or main assmption regarding the arrival and timeline processes is that they can be modeled as nonhomogeneos Poisson processes with time varying rates. For any times t and t + s (t, s > ) and a follower, we denote the nmber of arrivals in the arrival and timeline processes in the time interval [t, t + s) as M (t, t + s) and N (t, t + s), respectively. In other words, M (t, t + s) represents the nmber of times the follower checks the social network for new content in the time interval [t, t + s) and N (t, t + s) represents the nmber of new posts on the follower s timeline in the time interval [t, t + s). The rate of the arrival process is (t) and the rate of the timeline process is of arrivals in a time window [t, t + s) for s for the arrival process and E [M (t, t + s)] =, which is given by (t). Another sefl definition is the mean nmber (t, t + s)= Z t+s Z t+s E [N (t, t + s)] = (t, t + s)= t t ( )d () ( )d () for the timeline process. We will specify the parametric form of these rates and their temporal dependence in Section 3 when we look at ser data from Twitter. The ctoff window of a follower is denoted by C. For convenience, we list the model parameters in Table.. Under these modeling assmptions, we will next show how to calclate the probability of an impression being created by the follower for a piece of content as a fnction of the time it was posted.

7 Athor: Strategic Timing of Content in Online Social Networks 7 Table Parameter Definition Follower index C Ctoff window M (t, t + s) Arrival process cont N (t, t + s) Timeline process cont (t) Arrival process rate (t) Timeline process rate (t) Average arrival process cont (t) Average timeline process cont Description of parameters for components of the impression probability model... Impression Probability In order for a follower to see a piece of content, or create an impression, the following events mst occr. First, the content is posted at a time that is chosen by the content prodcer. The follower mst check his timeline at a time after the content is posted. Second, the time when the follower checks his timeline is determined by the arrival process. In the time interval between the posting of the content and the follower arrival, there will be a nmber of new posts on the follower s timeline, which is determined by the timeline process. If this nmber of posts is less than the follower s ctoff window, the follower will see the content and an impression is created, otherwise the content is not seen. We define the probability that a follower sees a piece of content posted to his timeline at time t as q(t). This will be referred to as the impression probability. We have the following reslt for the vale of the impression probability in this model. THEOREM.. For a piece of content posted at time t>, let q(t) be the impression probability of a follower with arrival rate (t), timeline rate (t), and ctoff window C. For s>, let the mean nmber of arrivals in the arrival and timeline processes be given by eqations () and (), respectively. Then, q(t)= CX k= Z k! (t + s)( (t, t + s)) k e ( (t,t+s)+ (t,t+s)) ds. (3) For any follower we are able to obtain a closed form expression for their impression probability as a fnction of posting time given their arrival rate, timeline rate, and ctoff window. Eqation (3) appears complicated pon first glance, bt we will see that in practice, it can be greatly simplified withot sacrificing very mch in terms of operational performance..3. Proportional Timeline and Arrival Rates To gain insight into the impression probability fnction, we consider the sitation where the timeline and arrival rates of a follower are proportional to each other. Assming non-homogeneos Poisson processes for both the timeline and arrival process, we obtain a simple expression for the impression probability given by the following lemma.

8 8 Athor: Strategic Timing of Content in Online Social Networks LEMMA. For a piece of content posted at time t>, let q(t) be the impression probability of a follower with timeline rate (t), arrival rate (t)=a (t) for some a>, and ctoff window C. Then, q(t)= (a + ) (C+). (4) According to this expression, when the arrival and timeline rates are proportional, then the impression probability is constant in time and there is no vale for strategic timing for a content prodcer. Also, this expression clearly shows the impact of the ctoff window on the impression probability. Not srprisingly, we see that q(t) approaches one as C increases. This means that if a follower has a large ctoff window and checks a large nmber of posts, then it is more likely he will see any post. In addition, for proportional arrival and timeline rates, the convergence to one is exponentially fast in C. We next look to examine the meaning and impact of the constant a in the above expression. To do this, we assme that all friends of the follower have the same arrival rate and that each time they arrive, they post a new piece of content. For a follower with F friends, this means that the timeline rate will be or eqivalently, (t)=a = F (t), and a = F. If we sbstitte this expression for a in eqation 4, we find that q(t)= C+ F. F + This shows that the impression probability is a decreasing fnction of the nmber of friends F. This is expected becase having a larger nmber of friends will reslt in a larger timeline rate, which means a post is visible for a shorter amont of time. Therefore, it will be harder for a post to be seen and we expect q(t) to be smaller. Thogh this analysis made strong assmptions abot follower behavior, it does align with or intition abot what increases the impression probability. Or reslt shows that followers with a higher impression probability do not have a large amont of competing posts on their timelines (small friend cont) and check many older posts on their timelines (large ctoff window)..4. Approximating the Impression Probability While simple expressions can be obtained for the impression probability when we assme proportional timeline and arrival rates, they are difficlt to obtain in a more general setting. However, in certain parameter regimes which occr in practice, simple approximations which provide sefl insight can be obtained. We now provide and analyze sch approximations. In eqation (3) we have assmed that the timeline process is a non-homogeneos Poisson process. To simplify the expression for q(t), we now assme the timeline process is deterministic with rate (t), so that the nmber of arrivals in an interval [t, t + s) is given by N(t, t + s)= (t, t + s). Define the residal time for an arrival given that there has not been an arrival at time t as S t. For a non-homogeneos Poisson process with rate ( ), it can be shown that the random variable S t has a density given by f St (s) =

9 Athor: Strategic Timing of Content in Online Social Networks 9 (t + s)e (t,t+s), where we have defined (t, t + s)= R s (t + )d (Cox and Isham 98). With this assmption, an impression is created if (t, t + S t ) apple C. The probability of this is given by q (t)=e [ ( (t, t + s) apple C)] = Z = Z = e ( (t, t + s) apple C) f St (s)ds (t + s) e (t,t+s) ds (t,t+ ), (5) where we define =sp{s : (t, t + s) apple C}. (6) The time is the amont of time the content posted at time t has before it is pshed ot of the ctoff window and as a reslt is not seen by the ser. We will refer to this as the lifetime of the content. We obtain the following bond on the error of approximation q (t). THEOREM.. Let the arrival rate (t) and the timeline rate (t) of a follower be sch that F apple (t)/ (t) for some constants <F. Let q(t) be the exact impression probability given by eqation (3) and let q (t) be the approximation of the impression probability given by eqation (5). Then the error of the approximation is bonded by q(t) q (t) apple p C F. The approximation q (t) will be accrate for followers with small ctoff windows and a large timeline rate relative to their arrival rate. This approximation is simpler than the exact expression, bt still rather complex becase it contains the content lifetime and the mean arrival process. To provide a more intitive expression, we make the following additional approximations. First we expand the exponential in eqation (5) to first order in (t, t + )to obtain e (t,t+ ) (t, t + ). Next, we se a first order Taylor approximation abot t for the arrival process mean to obtain (t, t + ) (t). Finally, we approximate the timeline process mean sing a first order Taylor expansion abot t to obtain (t, t + s) (t)s for s apple. Using this approximation for (t, t + s) along with eqation (6) we obtain (t, t + ) C (t), or more simply C/ (t). These Taylor approximations will be accrate as long as is small compared to the time scale over which the timeline and arrival rates vary. This will typically be tre becase is on the order of mintes, whereas the arrival and timeline rate variations are on the order of hors or days (more on this in Section 3). Combining these approximations we obtain the following simple eqation for the impression probability

10 Athor: Strategic Timing of Content in Online Social Networks q (t)=c (t) (t). (7) The above approximation is a simple, intitive expression for the impression probability which captres the main aspects of or model. We can easily see the following important facts regarding the impression probability from eqation (7). First, if a follower checks many posts pon each visit to the social network (large ctoff window C), then the impression probability is higher. Second, the times when the follower is likely to be on the social network correspond to times when the arrival rate (t) is large, and this is also when the impression probability is high. Finally, the times when there are a large nmber of posts on the follower s timeline correspond to times when the timeline rate (t) is large, and it is at these times when the impression probability will be lower. Frthermore, for followers whose arrival rate is mch smaller than their timeline rate, eqation (7) is a very accrate approximation for the impression probability. This is made precise in the following theorem. THEOREM.3. Let the arrival rate (t) and the timeline rate (t) of a follower be sch that F apple (t)/ (t) apple F for some constants <F apple F. Let q(t) be the exact impression probability given by eqation (3) and let q (t) be the approximation of the impression probability given by eqation (7). Then the error of the approximation is bonded by q(t) q (t) apple p C + C + C(F F ). F F The approximation error for q (t) is slightly larger than for q (t) de to the extra approximations made. However, the regimes where both approximations are good are similar: large timeline rate and small ctoff window. We will see in Section 4 that many real Twitter sers fall in this regime and therefore both approximations are highly accrate..5. Impact of Arrival Rate verss Posting Rate To calclate the impression probability in or model the follower s arrival rate is reqired. In practice this may be difficlt to measre. However, measring the posts of a follower can be done rather easily. For instance, this data can be obtained for a Twitter ser from the Twitter API. The posting rate will be less than the arrival rate becase in order to post, a follower mst first arrive to the social network. We model the posting process as a random sample of the of the arrival process, with sampling probability (, ]. This means that each time the follower arrives, he posts with probability. The posting process is then a non-homogeneos Poisson process with rate (t). We show here that having ncertain knowledge of the vale of does not impact key predictions of or model, namely the optimal time to post.

11 Athor: Strategic Timing of Content in Online Social Networks We establish the following reslt for the impact of sing the posting rate instead of the arrival rate to find the optimal time to post. From eqation (7) we define the following approximation for the impression probability when one ses the posting rate (t) instead of the arrival rate (t) as bq (t)=c and we define the time which maximizes this expression as We also define the time that maximizes the tre impression probability as (t) (t), (8) bt = arg max bq (t). (9) t t = arg max q(t) () t We are concerned with the difference between the impression probability if one posts at bt verss t.we have the following reslt, which can be seen as a simple application of Theorem. COROLLARY. Let the arrival rate (t) and the timeline rate (t) of a follower be sch that F apple (t)/ (t) apple F for some constants apple F apple F. Let q(t) be the exact impression probability given by eqation (3). Let t be given by eqation () and for have that q(t ) q(bt ) apple (, ], let bt be given by eqation (9). Then we p C + C + C(F F ). F F This reslt shows that even if an approximation for the impression probability is sed with the posting rate and not the arrival rate, the actal impression probability will not be changed by a very large amont for sers with a small ctoff window relative to the ratio of their timeline rate to their arrival rate. Therefore, the posting rate can be sed in place of the arrival rate and the approximation q (t) can be sed instead of q(t) to determine the optimal time to post content. This is important becase it is difficlt to measre the arrival rate for a ser, while the posting rate is readily available..6. Impact of Ctoff Window The ctoff window is another model parameter that may be difficlt to measre. For companies rnning social networking applications this parameter can be measred by recording what content is displayed on a follower s screen. However, this information is not readily available to those not within the company. De to this restriction, we wold like to nderstand how sensitive the impression probability is to this parameter. We define q (t, C ) as the vale of the approximation to the impression probability given eqation (7) for a ctoff window C. For an arbitrary vale of the ctoff window C > we define the time which maximizes q (t, C ) as bt C = arg max q (t, C ). () t

12 Athor: Strategic Timing of Content in Online Social Networks The time that maximizes the tre impression probability sing the correct ctoff window C, arrival rate (t) and timeline rate simply an application of Theorem. (t) is t as defined in eqation (). We have the following reslt, which again is COROLLARY. Let the arrival rate (t) and the timeline rate (t) of a follower be sch that F apple (t)/ (t) apple F for some constants apple F apple F. Let q(t) be the exact impression probability given by eqation (3) sing the tre ctoff window C. Let t be given by eqation () and for C >, let bt C given by eqation (). Then we have that q(t ) q(bt C ) apple p C + C + C(F F ). F F Here we see that as long as the arrival rate is mch smaller than the timeline rate, then the impression probability is robst to errors in the ctoff window. This is an important reslt becase of the difficlty in measring this parameter. 3. Bayesian Model For Timeline and Posting Processes We have seen how to calclate the impression probability once the follower arrival rate and timeline rate are both known. We also saw that we can get a good approximation to the impression probability if we se the posting rate instead of the arrival rate. In this section we present Bayesian models for the timeline and posting processes for sers in Twitter. We focs on the posting process instead of the arrival process becase we can obtain data on the posting process throgh the Twitter API. We take a Bayesian approach becase it provides a very natral way to obtain credibility intervals for all model parameters and the model estimation procedre is fairly direct. This then allows for simple calclation of credibility intervals for the impression probabilities, which we present in Section Exploratory Data Analysis To begin or model development, we perform an exploratory analysis of the posting and timeline processes for 498 Twitter sers. These sers are random samples of the followers of prominent Twitter sers sch as Barack Obama (@BarackObama), Taylor Swift (@taylorswift), and several others. To obtain the timeline of each ser in or dataset we collected the posts of a maximm of their friends. This allowed s to reconstrct a sampled version of the timeline process for each ser. We only collected the posts of friends per ser becase of rate limitations by the Twitter API. All post times are in GMT, which is the defalt for the Twitter API. We begin by stdying the temporal variation in the timeline processes. We define the smoothed timeline rate as the posts per nit time in a sliding six hor window. We chose this window size to prodce a smooth crve which wold show the main qalitative featres of the timeline rate. We plot the smoothed timeline rate for different Twitter sers in Figre. As can be seen, there are clear oscillations over a one day period. Also, the peak time of these oscillations are different for each ser. be

13 Athor: Strategic Timing of Content in Online Social Networks Timeline Rate [posts/day] Timeline Rate [posts/day] Sat Sn Mon Te Wed Th Fri Sat Sn Mon Te Wed Th Fri Sat Sn Mon Te Wed Th Fri Sat Sn Mon Te Wed Th Fri Time Time 6 6 Timeline Rate [posts/day] Timeline Rate [posts/day] Sat Sn Mon Te Wed Th Fri Sat Sn Mon Te Wed Th Fri Time Figre 5 Fri Sat Sn Mon Te Wed Th Fri Sat Sn Mon Te Wed Th Time The smoothed timeline rate for Twitter sers with screen names (top (top (bottom (bottom Each timeline was constrcted from friends of the ser. The timeline rate was smoothed sing a sliding six hor window. We next look more closely at the daily variations in both the timeline and posting proceses. We plot a histogram of the hor of posts in the timeline and posting process for each of the sers from Figre in Figre. As can be seen, for these sers there is considerable variation in the nmber of posts verss hor for both their timeline and posting processes. Also, the horly distribtion for the two processes can be very different. The one day period of oscillation for the timeline rate is clearly visible in the data. We next examined variations over a one week period. We show a histogram of the day of the posts in the timeline and posting processes for the same sers in Figre 3. As with the hor data, we see variation across these sers in the day distribtion for both processes. Also, the timeline and posting processes can be very different in their day distribtions for a given ser. To frther show the difference in the peak times for the posting and timeline processes, we plot in Figre 4 the peak hor and peak day for the timeline and posting process for each ser in or dataset. The points in the plot are jittered with Gassian noise with standard deviation.5 in order to make them more clear. As can be seen, there is very little correlation between the timeline and posting processes in terms of peak day and peak hor. For the two processes, the correlation of the peak hor is.3 (p-vale < 6 ) and the correlation of the peak day is.6 (p-vale =.4). There is more correlation in the hor than in the day, bt both vales are mch smaller than one.

14 4 Athor: Strategic Timing of Content in Online Social Networks Freqency Timeline Posting Freqency.6 Timeline Posting Hor Timeline Posting 5 5 Hor.4 Timeline Posting... Freqency.8.6 Freqency Hor 5 5 Hor Figre Histograms of the hors of posts from the posting and timeline process for Twitter sers with screen names (top (top (bottom (bottom This exploratory analysis provides two important conclsions. The first is that the posting and timeline processes of sers show variations with a one day and one week period. This is not a srprising reslt given the manner in which sers typically se social networks. Second, the peak hor and day of the posting and timeline processes are not aligned for many sers. We saw in Section.3 that when the timeline and arrival rates were eqal, then the impression probability was constant in time and there was no gain from strategic timing. However, when the rates are not proportional, strategic timing can prodce sbstantial gains. Becase of the observed misalignment of the peak hor and day for the two processes for several sers, we expect strategic timing of content to be beneficial in practice. 3.. Parametric Form of Rate Fnctions We now propose a parametric form for the timeline rate and posting rate fnctions. Becase we assme that the posting process is a random sample of the arrival process, this parametric model for the rate of the arrival process will be a scaled version of the rate of the posting process. Timeline posts are an aggregation of ser posts, so it is reasonable to assme that the posting and timeline processes have a similar parametric form (bt with different parameter vales). Based pon the analysis in Section 3. we assme that all rates have oscillations with a one day and one week period.

15 Athor: Strategic Timing of Content in Online Social Networks 5. Timeline Posting. Timeline Posting Freqency.5. Freqency Sn Mon Te Wed Th Fri Sat Day Sn Mon Te Wed Th Fri Sat Day.5 Timeline Posting. Timeline Posting..5 Freqency.5. Freqency..5.5 Sn Mon Te Wed Th Fri Sat Day Sn Mon Te Wed Th Fri Sat Day Figre 3 Histograms of the days of posts from the posting and timeline process for Twitter sers with screen names (top (top (bottom (bottom 7 Peak posting process hor 5 5 Peak posting process day Peak timeline process hor Sn Mon Te Wed Th Fri Sat Peak timeline process day Figre 4 (left) Plot of the peak hor of the timeline process verss the posting process for several Twitter sers. (right) Plot of the peak day of the timeline process verss the posting process for several Twitter sers. All data points are jittered with Gassian noise with a standard deviation of.5.

16 6 Athor: Strategic Timing of Content in Online Social Networks In or model the timeline process of a ser consists of the following parameters: {,b h,b d,t h,t d}. The timeline rate is then given by (t)= (+b h cos (! h (t t h))) (+b d cos (! d (t t d))). () The parameters t h and t d indicate the peak hor and day of the timeline rate. The strength of the horly and daily variations are modeled by the terms b h and b d. The average timeline rate is captred by. If we measre the time t in days, then the freqencies above become! h = (one day period) and! d = /7 (seven day period). We model the posting process rate sing a similar parametric form as in () keeping! h and! d the same, bt sing different parameters {, h, d, h, d }. (t)= (+ h cos (! h (t h ))) (+ d cos (! d (t d ))). (3) The parametric form of the rates allow s to express the mean vale of the processes in closed form. The mean timeline process is given by (t, t + s)= s + sin (! h (r t! h)) h b h and the mean posting process is given by t+s + b d r=t b hb d (! d +! h ) sin ((! d +! h )(r (t d + t h))) b hb d (! d! h ) sin ((! d! h )(r (t d t h)))! d sin (! d (r t d)) t+s r=t t+s r=t + t+s r=t + (4) (t, t + s)= s + h h! h sin (! h (r h )) d t+s + r=t (! d +! h ) sin ((! d +! h )(r ( d + h ))) h 3.3. Bayesian Model Specification d (! d! h ) sin ((! d! h )(r ( d h ))) b d! d sin (! d (r d )) t+s r=t t+s r=t + t+s r=t +. (5) To learn the model parameters we will se a Bayesian approach. This will provide s with posterior estimates and credibility intervals for all model parameters in a very natral way. The Bayesian framework also allows for ease of model fitting when there is not sfficient data, as is the case for many sers posting processes. Becase of the difference in amont of data for the timeline and posting processes, we se different model strctres for each process. For the timeline process we have sfficient data for each ser and therefore learn each timeline rate independently. For the posting process, becase of sparse data for many

17 Athor: Strategic Timing of Content in Online Social Networks 7 sers, we se a hierarchical model which allows for information sharing between sers. We now specify these models in detail. The timeline process of a ser provides s with observations of the n times of the timeline posts t = {t,t,...t n }. The timeline model parameters for ser are = {,b h,b d,t h,t d}. Becase we model the timeline process as an non-homogeneos Poisson process with rate observations is given by with Y P (t ) = n i= (t i ) e (t i,t i ) Yn = e (t,t n ) i= (t), the likelihood of the (t i ), (6) (t) and (s, t) given by eqations () and (4). We se ninformative hyperpriors for the timeline model parameters. For we se a gamma distribtion with shape and scale one and,. For the remaining parameters we se normal priors with zero mean and standard deviation. For the posting process, many times we will have sers who do not provide a large amont of data. This is in contrast to the timeline process which typically has a large amont of data for each ser. Therefore, it may not be possible to accrately learn the posting rate parameters of each ser individally. However, if we assme that similar behavior is shared between different sers, then we may be able to better learn the parameter vales. For this reason, we se a hierarchical Bayesian model for the posting process. We first define for a ser a set of posting rate parameters = {, h, d, h, d }. For each ser we observe the corresponding m posting times s = {s,s,...s m}. The likelihood of the observations conditioned on the ser parameters is similar to that for the timeline observations and is given by P (s ) =e (s Yn,s n ) i= (s i ), (7) with (t) and (s, t) given by eqations (3) and (5). To cople the ser parameters, we make them each independent conditional on a set of global paramters. We define this set of global parameters as =, h, d, h, d, h, d, h, d. The parameters characterize the typical vale of the ser parameters across all sers. The parameter characterizes the typical posting rate. The strength of the horly and daily variations and the same goes for the and terms characterize the typical terms model how variable individal sers are. The terms and the corresponding peak hors and days. The distribtions of the ser parameters conditioned on the global parameters are Exp (8) h h, N h h, d d, N d d, h d (9) ()

18 8 Athor: Strategic Timing of Content in Online Social Networks h h h h h h d d d s i Posting time i d d d User Figre 5 Graphical model of the hierarchcial Bayesian model for the posting process. The plates denote replication over posting times S i and sers. Hyperpriors are omitted for simplicity. h h, h N h, d d, d N d, h () d, () where Exp (x) denotes an exponential distribtion with mean x and N (µ, ) denotes a normal distribtion with mean µ and standard deviation. Hyperpriors are chosen to be ninformative and conjgate when possible. The hyperprior on is inverse gamma with parameters one and one. For all means of the normal conditional distribtions we se normal priors with zero mean and standard deviation. All the variances of the normal conditional distribtions have inverse gamma hyperpriors with parameters one and one. For convenience we inclde a visal depiction of the strctre of the hierarchical model for the posting process in Figre 5. For N sers, let the observed posting times be S = {s, s,...,s N } and let the observed timeline times be T = {t, t,...,t N ser }. We define the set of all ser posting rate parameters as = {,,... N } and the set of all ser timeline rate parameters as ser = {,,... N }. The posterior distribtion of the timeline process model parameters given T is given by NY P ( ser T) / P (t ) P ( ). The posterior distribtion of the posting process model parameters given S is given by NY P (, ser S) /P ( ) P (s ) P ( ). = = We sample from these posterior distribtions sing a Markov Chain Monte Carlo (MCMC) sampler. For the timeline rates the model decoples and we can sample the parameters for each ser timeline individally. For the posting process becase all sers are copled throgh the global parameters, we mst jointly sample all ser parameters. Details of or MCMC sampler are provided in the Appendix.

19 Athor: Strategic Timing of Content in Online Social Networks 9 Freqency Freqency Freqency λ [posts/day] b h.5 t h [hor] Freqency b d Freqency Sn Mon Te Wed Th Fri Sat t d Figre 6 Posterior histograms of the timeline rate parameters of for the timeline process model Model Estimation Reslts We estimate or model on a sbset of 94 sers from or dataset. These sers are selected at random from or dataset with the reqirement that they have at least posts in their timeline and posting processes. Each timeline model was learned separately for each ser. For the posting processes, we estimated the parameters for all sers jointly sing the hierarchical model. For each model estimation (timelines and posting process), we generated posterior samples sing three independent MCMC chains with dispersed starting points rn for 6, iterations and discarding a brn-in period of, iterations. Convergence of the MCMC sampler was assessed sing the Gelman-Rbin statistic (Gelman and Rbin 99). We plot the posterior histograms of timeline parameters for one ser s timeline in Figre 6. To visally demonstrate the qality of the in-sample fit, in Figre 8 we show the posterior in-sample mean timeline process verss the tre timeline process for this ser, where we define this as b (,t)=e [ (,t )]. We also plot the empirical and posterior in-sample mean timeline rates in Figre 8. The empirical timeline rate is calclated sing a six hor sliding window. As can be seen, the in-sample fit is qite good for the timeline and captres mch of the daily and weekly variation, which is most clearly visible in the timeline rate plots. We plot posterior histograms for the posting process for the same ser in Figre 7. In Figre 8 we show

20 Athor: Strategic Timing of Content in Online Social Networks Freqency Freqency Freqency ψ [posts/day] β h.5 τ h [hor] Freqency.4.3. Freqency β d Sn Mon Te Wed Th Fri Sat τ d Figre 7 Posterior histograms of the posting rate parameters of for the timeline process model. this ser s posterior in-sample mean vale posting process b (,t)=e [ (,t )] verss the tre posting process. As can be seen, the in-sample fits are qite good for the posting process as well and captres mch of the daily and weekly variation. We do not plot the posting rates becase the sparse nmber of posts (relative to the timeline process) prodces a very noisy empirical rate crve. The posterior histograms of the global parameters is shown in Figre 9. Here we see that the median posting rate for these sers is 7 posts per day. The median peak hor is 3 (in GMT) and the median peak day is between Wednesday and Thrsday. The median standard deviation of the ser peak hor is 4 hors. This indicates that there is a wide spread in the peak hor of posting across sers. However, the median standard deviation for the peak day is.3 days, indicating less variation in the day of most ser activity Model Comparisons To assess the qality of fit of or model, which we refer to as Model 3, we compare it to two other benchmark models. The first model, which we refer to as Model, assmes no temporal dependence in the timeline and posting rates. The second model, Model, only assmes a one day periodicity in the rates, bt ignores h

21 Athor: Strategic Timing of Content in Online Social Networks Timeline Posts 4 x Data Model Fit Timeline Rate [posts/day] 5 Data Model Fit Fri Sat Sn Mon Te Wed Th Fri Sat Sn Mon Te Wed Th Fri Sat 9 Data Model Fit Time Sat Sn Mon Te Wed Th Fri Sat Sn Mon Te Wed Th Fri Time User Post Cont Fri Sat Sn Mon Te Wed Th Fri Sat Sn Mon Te Wed Th Time Figre 8 Posterior fits of the timeline process (top left), timeline rate (top right) and posting process (bottom left) for The ble crves are the 9% posterior credibility intervals for the model fit crves. the weekly periodicity. Both models assme a hierarchy on the posting rate parameters, similar to Model 3 proposed in Section 3.3. We now provide detailed Bayesian specifications of these two models. The timeline and arrival rates of Model are assmed to be constant. The fll model is given by (t)= The hyperprior for (t)= (, ) is a gamma distribtions with shape and scale parameters one and, respectively. For, the hyperprior is inverse gamma with shape and scale parameters both eqal to one. The timeline and arrival rates of Model are given by (t)= (+a (+b h cos (! h (t t h ))))