Spiral of Silence in Recommender Systems

Transcription

1 Spiral of Silence in Recommender Systems Dugang Liu, Cen Lin Department of Computer Science, Xiamen University Xiamen, Cina ABSTRACT Yangua Xiao Scool of Computer Science, Fudan University Sangai, Cina Alibaba Group Hangzou, Cina It as been establised tat, ratings are missing not at random in recommender systems. However, little researc as been done to reveal ow te ratings are missing. In tis paper we present one possible explanation of te missing not at random penomenon. We verify tat, using a variety of different real-life datasets, tere is a spiral process for a silent minority in recommender systems were () people wose opinions fall into te minority are less likely to give ratings tan majority opinion olders; (2) as te majority opinion becomes more dominant, te rating possibility of a majority opinion older is intensifying but te rating possibility of a minority opinion older is srinking; (3) only ardcore users remain to rate for minority opinions wen te spiral acieves its steady state. Our empirical findings are beneficial for future recommendation models. To demonstrate te impact of our empirical findings, we present a probabilistic model tat mimics te generation process of spiral of silence. We experimentally sow tat, te presented model offers more accurate recommendations, compared wit state-ofte-art recommendation models. KEYWORDS Spiral of Silence, Recommender System, Missing not at Random, Hardcore ACM Reference Format: Dugang Liu, Cen Lin, Zilin Zang, Yangua Xiao, and Hangang Tong. 29. Spiral of Silence in Recommender Systems. In Te Twelft ACM International Conference on Web Searc and Data Mining (WSDM 9), February 5, 29, Melbourne, VIC, Australia. ACM, New York, NY, USA, 9 pages. ttps://doi.org/5/ INTRODUCTION Recommender Systems (RS) ave received extensive attentions from bot researc communities and industries. Te power of an RS is igly dependent on te assumption tat te collection of Permission to make digital or ard copies of all or part of tis work for personal or classroom use is granted witout fee provided tat copies are not made or distributed for profit or commercial advantage and tat copies bear tis notice and te full citation on te first page. Copyrigts for components of tis work owned by oters tan ACM must be onored. Abstracting wit credit is permitted. To copy oterwise, or republis, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from permissions@acm.org. WSDM 9, February 5, 29, Melbourne, VIC, Australia 29 Association for Computing Macinery. ACM ISBN /9/2... $5. ttps://doi.org/5/ Zilin Zang Scool of Computing Science, Simon Fraser University Burnaby, Canada Hangang Tong Scool of Computing, Informatics and Decision Systems Engineering, Arizona State University Tempe, U.S.A. Table : A toy example of 5 users ratings on 5 movies. Alice s ratings on Aliens and Eskiya are idden. Aliens Ben-Hur Casino Dangal Eskiya Alice (2) 3 3 (5) Bob Clare Diane Elle ratings correctly reflects te users opinions. Most recommender systems, owever, suffer from extremely sparse rating data. More callengingly, it is rare tat users tell te trut and te wole trut at all times. Wen a missing rating occurs due to te user s coice of non-response, te representativeness of te ratings is degraded and te inference of a recommendation model is distorted. Consider a conventional collaborative filtering RS running on a toy example illustrated in Table. Suppose, for some reason, Alice is not willing to give er ratings on te movie Aliens and Eskiya. Te RS will make a wrong judgement tat Alice s nearest neigbor is Bob, based on te two common items Alice and Bob ave, wile in fact Diane sares te most similar taste wit Alice. In te literature of RS, models [ 6] wic assume ratings are Missing Not At Random (MNAR models) are recognized to ave a superior ranking performance. Existing MNAR models mimic te generation of responses under different euristics, i.e. te possibility of a response is related to te exact value of te rating [2 4] or to an unknown feature of te item [5, 6]. Unfortunately, none of te euristics is empirically verified on real datasets, or supported by teoretical social studies. In real scenarios, tere could be various factors tat lead to missing responses. Our goal in tis paper is to provide a possible explanation for missing ratings and identify te key factors underlying users decision process of weter or not to rate an item in recommender systems. Towards tis goal, we first empirically examine te response patterns in recommender systems. We verify te existence of a spiral process in wic users are more and more likely to rate if tey perceive tat tey are supported by te opinion climate (i.e. te dominant opinion), wile te minority opinion olders are more and more reticent. Suc a spiral process can be explained by te Spiral of Silence Teory [7], wic as been acknowledged In te remaining of tis paper, te Spiral of Silence teory will be referred to as te teory.

2 as one of te most influential recent teories of public opinion formation [8]. We igligt te unique caracteristics of our empirical study. () We study te beavior of giving ratings, wile previous empirical studies focus on te biases in ratings [ 3, 33]. (2) We study te dynamic aspect wic as never been addressed by existing MNAR models [ 6]. For example, as time passes by, te domination of majority opinion grows, wic results in te increased willingness to rate for majority opinion olders and te decayed willingness for te minority opinion olders. Two callenges arise in te empirical study. Firstly, survey studies in a lab environment [ 4] is problematic, because te findings are based on ypotetical willingness instead of actual willingness [5]. A recent survey [6] sows tat te ypotetical willingness is a poor indicator of actual willingness. To address tis callenge, we form our empirical study on te basis of actual willingness. Two scenarios are included: () Scenario I considers tat a user s willingness to display or ide is/er ratings is offered (Sec. 3); (2) Scenario II considers tat a user s inclination to rate is not available (Sec. 4). Secondly, tere are counter cases according to te Spiral of Silence Teory, i.e. te minority opinion tat remains at te end of te spiral is called te ardcores. Mixing ardcore and non-ardcore users can urt te performance of a recommendation model as te two user groups beave differently. To tackle tis callenge, we present formal definition to distinguis ardcore users and study te caracteristics of ardcore users in Sec. 5. In order to demonstrate te impact of our empirical study, we present a straigtforward application of te two major findings, i.e. te existence of spiral and te existence of ardcore users. We develop a Missing Conditional on Persona (M) model wic mimics te generation process of ratings and responses. Te probability of giving response is related to te perceived opinion climate, individual rating, and te persona of te user (i.e. ardcore user or non-ardcore user). Experiments sow tat te M model outperforms state-of-te-art recommendation models, including models wit and witout MNAR assumptions. Our contributions are tree-fold. Practical contributions. We verify te existence of a spiral of silence by large-scale empirical studies on 8 real recommendation data sets. We validate te group of ardcore users and provide detailed insigts into te personal traits of ardcore users. Tese findings are particularly useful in nice marketing. Metodological contributions. We design solutions to conduct large-scale field researc of te spiral of silence teory on recommendation systems. For example, we give formal definitions to te core concepts including majority opinion olders and ardcore users. In ypotesis testing, we conduct a series of trend studies to capture te dynamic aspect of te teory wic as ardly been addressed by previous studies. Model contributions. We present a M model based on te major findings of our empirical study. We experimentally sow tat te M model outperforms state-of-te-art recommendation models. Te significant improved performance reveals te potential impact of our empirical findings. Te paper is organized as follows. Sec. 2 summarizes te related works. In Sec. 3 and Sec. 4, we testify te existence of a spiral of silence over different scenarios. Sec. 3 is devoted to empirical study in wic a user s willingness to rate is available. Sec. 4 corresponds to conducting empirical study on recommender systems in wic a user s willingness to rate is not available. In Sec. 5, we study te existence of ardcore users in te formation of te spiral of silence, and we reveal te caracteristics of ardcore users. Sec. 6 presents te M model wic is developed by embedding te empirical findings. Sec. 7 presents and analyzes te experimental results. Finally, Sec. 8 concludes our contributions and insigts into te future work. 2 RELATED WORK Biases in Recommender System is related to our researc. Some researcers observed a trend of increasing average ratings [7 9]; oters found tat later ratings are on average lower [2]. Hu et.al observed a J-saped distribution [2]. A recent work [22] sowed te existence and strengt of conformity. Tese works focused on te biases in ratings, wile we study te biases in responses, i.e. minorities are less likely to give ratings. In addition, te explanations in previous studies are not appropriate for all recommender systems, i.e. te coice-supportive bias [8] only applies to recommender systems wit reviews. Te ric gets ricer (Mattew effect) clice is anoter line of researc we want to distinguis our work wit. Toug te ric gets ricer assumption generates a similar penomena to te spiral process, it does not explain te formation of public opinion as te spiral teory does. Te Mattew effect suggests group dynamics. It is not suitable to derive a recommendation model because personalization is not retained. MNAR Models in RS are aware tat ratings are missing not at random. Probabilistic models were presented to relate a missing to various factors, e.g. te value of a idden rating [ 4] or to te item to be rated [, 5, 6]. As we mentioned above, tey were based on euristics tat are neiter empirically verified nor teoretically proven. Furtermore tey are unable to explain te evolution of ecology and several penomena in te recommender systems, e.g. a ig rated item gets more praises. Our work aims to reveal tese idden patterns from a social science perspective, and tus serves as a guiding ligt for future MNAR models. Empirical Study on Spiral of Silence Teory as a long istory. Tey adopted a train test type of experiments, i.e. te subjects are questioned about teir willingness to discuss wit a stranger on a train about any topic. Most works [ 4] observe a positive correlation between perceived opinion climate and willingness to rate, bot of wic are collected during te survey of train test. However te result is based on ypotetical willingness. We believe tat our work is te first to verify te spiral model in large scale real life recommender systems. Moreover, tey only proved te social conformity ypotesis" [9]. Empasis on time in te formation of te spiral as not been reflected on te metodologies. On te contrary, we acknowledge te dynamic nature of te spiral model. 3 EXISTENCE OF SPIRAL: SCENARIO I In tis section we testify te fundamental assumption of te teory in recommender systems: te spiraling process, in wic two key activities repeatedly occur. () A user is prompted to sow is

3 rating if e perceives a majority opinion similar to is own but is restrained to sow is rating wen e believes is opinion belongs to te minority. (2) Suc a response pattern leads to an even stronger majority opinion, wic in turn encourages more majority opinion olders to sow. We need to define te core concepts: te majority opinion of ratings denoted as M a, te minority opinion denoted as M i. We also need to quantify te possibility to sow a rating in majority opinion (denoted as p) and te possibility to sow a rating in minority opinion (denoted as q). We present two solutions to automatically identify M a, M i,p,q in Sec. 3., a tresold-based approac and a model-based approac. Te data set used in tis section is te extended epinions data set (Epinions) [23]. Te data set contains ratings wit teir timestamps in a 5-star range. Te number of users, items and ratings are sown in Table 2. Specially, tis data set includes te display status of eac rating, i.e. weter te user as cosen to sow or ide is rating. Tus tis data set is convenient for computing p,q. Table 2: Statistics of te data set in Sec 3 3. Metodology Dataset #Users #Items #Ratings Epinions 2, ,76 3,668,32 To start wit, we assume tat te majority opinion M a is a set of ratings wic are similar to te perceived opinion climate, and te minority opinion M i, a set consists of ratings wic are significantly divergent from te perceived opinion climate. We define te rating divergence d(r i, j,t ) of user i on item j at time t as: d(r i, j,t ) = r i, j,t r j,t ˆ, () were r i, j,t is te rating by user i on item j at timestamp t, r j,t ˆ is te perceived opinion climate at time t. Te value of r j,t ˆ is defined as te average rating on item j before time t. Tresold-based Approac. Most previous studies assume te majority opinion is a group of ratings wit smaller absolute values of rating divergence and te minority opinion is wit iger absolute values of divergence. We use a percentile based tresold. Since divergence is symmetric, we order all te rating divergences on eac item and select a range [Q s, Q e ] around d(r i, j,t ) =, were Q s is te starting percentile, and Q e is te ending percentile. We define te majority opinion M a = {r i, j,t Q s < d(r i, j,t ) < Q e } as a a set of ratings tat are positioned between te percentile range [Q s, Q e ]. Te rest of ratings are assigned to minority opinion M i. We experiment wit different ranges. For example, if we coose te 25t percentile for Q s and 75t percentile for Q e ten te value of eac rating in te majority opinion is iger tan te bottom 25% of all ratings and lower tan te top 25% of all ratings. Given M a, M i and V te set of ratings wic are displayed, we compute te willingness p, q as: p = ( M a V )/( M a ), (2) q = ( M i V )/( M i ). (3) Model Based Approac. Instead of determining directly te range of majority opinion, we also propose a Gaussian Mixture Model wic assumes te majority opinion follows a Gaussian distribution. Consider te Epinions data set as a collection of samples in te form of (d(r i, j,t ), z i, j,t ), were d(r i, j,t ) is rating divergence and z i, j,t is te display status, i.e. z i, j,t = indicates te user cooses to display te rating, z i, j,t = indicates te user cooses to ide te rating. We introduce a latent variable M i, j,t to indicate weter te rating belongs to te majority opinion. If te user considers imself as a majority opinion older, is willingness to sow te rating is p = P(z i, j,t = M i, j,t = ). Oterwise q = P(z i, j,t = M i, j,t = ). Inspired by [24], we assume tat te divergence for a majority opinion rating is normally distributed wit mean and standard deviation σ, P(d(r i, j,t ) M i, j,t = ) N(, σ 2 ). Te mean is set to be zero because intuitively a majority opinion older is most likely to be consistent wit te opinion climate. σ decides te range of majority. We assume tat te absolute value of rating divergence for a minority opinion older is distributed normally wit mean µ 2 and standard deviation σ 2, P(d(r i, j,t ) M i, j,t = ) N(µ 2, σ2 2). Using te EM algoritm, we can infer parameters p,q, σ, µ 2, σ 2 and derive te latent variable M i, j,t. Given te formal definition of M a, M i,p,q, we develop te following two ypoteses. Te first ypotesis focuses on a static penomenon (corresponds to key activity ). Te second ypotesis empasizes on te dynamic nature in te formation of te spiral (corresponds to key activity 2). Hypotesis (H). A majority opinion older as a larger possibility to sow te rating tan a minority opinion older, i.e. p > q. Hypotesis 2 (H2). As te majority opinion becomes more dominant, te tendency for a majority opinion older to sow te rating is on te rise, wile te tendency for a minority opinion older to sow te rating is on te decline, i.e. p increases over time wile q decreases. To testify ypotesis H, we identify M a, M i on te wole Epinions dataset and compare q, p. To testify ypotesis H2 we construct K snapsots. For eac item, we sort te ratings in cronological order and equally divide tem into K disjoint sets. We identify M a, M i on te K snapsots and conduct trend analysis on p,q. 3.2 Results and Analysis Table 3: Willingness to sow rating for majority opinion is significantly larger tan minority opinion p > q. Metodology p q Tresold-based Model-based Table 4: Increasing willingness to sow rating for majority opinion (positive p cange) and decreasing willingness to sow rating for minority opinion (negative q cange). Tresold-based approac K= snapsot P cange q cange Model-based approac K= snapsot P cange q cange

4 probability to rate majority opinion olders(p) minority opinion olders(q) Snapsots Figure : A case study: trend of p,q of item #49832 on Epinions, wit 5 snapsots, by model-based approac. As sown in Table 3, p is significantly larger tan q, wic means tat te majority opinion olders are muc more willing to sow rating tan minority opinion olders. p >.5 and q <.5 for bot approaces, suggesting tat tis conclusion is not only in a comparative sense but also in an assertive sense. A majority opinion older is likely to sow rating (p >.5) and a minority opinion older is prone to not sowing (q <.5). Terefore H is verified. For te tresold-based metods, we find tat reasonably expanding or narrowing te range does not cange te values of p,q muc. For ranges from (Q s = 25% Q e = 75%) to (Q s = 5% Q e = 85%) te values of p,q are in a narrow range p (.5757,.5773), q (.363,.373). For te model based approac, te solutions for parameters also give us some insigts. Firstly, te mean parameter for te minority opinion is µ 2 =.3524, µ 2 > µ =, suggesting tat te rating divergence for a minority opinion older is muc larger tan tat of a majority opinion older. Secondly, te variance parameters are σ =.744, σ 2 =.752. Note tat in model-based approac, we do not explicitly require σ 2 > σ, ence, te result is reasonable because a minority opinion will be more divergent. Fig. illustrates te values of p,q at different snapsots for a typical item on Epinions. We can see tat at te beginning te values of p,q are bot close to.5, because majority opinion as not yet been formed. At tis time bot majority and minority opinion olders are active in discussion. p keeps rising until it reaces above.9 in te final snapsot, indicating tat wit a strong domination, most majority opinion olders are likely to give ratings. On te contrary q is srinking to.35 in te tird snapsot. We also notice tat te fourt snapsot is a turning point. An interesting penomena occurs in te fort snapsot, in wic q sligtly increases to.5. Te minority opinion olders are arguing in defiance. We observe similar patterns in many oter cases tat te ardcores attempt to save te minority opinion before te turning point. Due to te limit of space, we do not give more case studies. After te turning point we observe steep slopes bot in p rising from to.95 and q declining from 5 to.5, wic indicates tat once majority opinion becomes powerful minority opinion olders are quickly pused back and soon no alternative opinions exists. For a detailed trend analysis, we report te difference between p,q in te last snapsot and p,q in te first snapsot in Table 4. We observe positive differences in p ( p is larger in te last snapsot) and negative canges of q (q is smaller in te last snapsot). It Table 5: Statistics of te data sets in Sec 4 to Sec 5 Dataset #Users #Items #Ratings Amazon-books 8,26,324 2,33,66 22,57,55 Amazon-clotes 3,7,268,36,4 5,748,92 Amazon-electronics 4,2, ,2 7,824,482 Amazon-movies 2,88,62 2,94 4,67,47 Epinions 22,66 296,277 92,44 Ciao , ,65 s Movielens2M 38,493 3,262 2,,263 Eacmovie 6, ,558,87 means te tendency for a majority opinion older to sow rating is rising, wile te tendency for a minority opinion older to sow rating is falling. H2 is verified. We manually ceck te values of p,q in eac snapsot and find tat p is monotonically increasing, wic is an even stronger indicator tat te spiral process exists. We do not observe monotone in q for all items, because of te revolt by ardcore people wo are trying to save te minority opinion. Furtermore, we see tat wit more snapsots, te number of ratings in eac snapsot is smaller, tus te trend is more significant, i.e. larger increase of p in te last snapsot. Summary and remarks. In scenario I were we know weter te user wants to ide or sow is/er ratings, we verify te existence of a spiral of silence by sowing () a majority opinion older as a larger possibility to sow rating tan a minority opinion older; (2) te possibility of a majority opinion older to sow rating is increasing as te majority opinion becomes more dominant. 4 EXISTENCE OF SPIRAL OF SILENCE: SCENARIO II In most common settings, we can only obtain te ratings of recommender systems, witout any oter indicators tat te user is willing to rate, i.e. weter te rating is idden or displayed. Direct computation of willingness to rate is infeasible. To deal wit recommender data sets witout explicit willingness, we testify te existence of a spiral process by observing te trend of majority opinion percentages. Te majority opinion percentage is monotonically increasing if and only if tere is a majority opinion. To see tis we consider te consequence wen a spiral of silence is triggered. If te majority opinion is strong, te majority opinion olders will be more active, resulting in an enanced percentage of majority opinion. A majority opinion is necessary to trigger te silent spiral. Wen tere is no clearly majority opinion, te opinion wic is supported by more users tan any oter opinion is called a plurality opinion. A plurality opinion will not induce a spiraling process, tus te opinion percentages will not increase. We use 8 real data sets, including four Amazon product ratings [25], product ratings datasets Epinions and Ciao [26] and movie rating datasets Movielens 2M [27] and Eacmovie [28]. All te ratings are timestamped. No oter information is provided. 4. Metodology Te first problem tat needs to be solved is to filter items wit a majority opinion. Intuitively, if an item as a majority opinion, its ratings will be concentrated in a small range to form a peak. On te contrary, if an item does not ave a majority opinion, te

5 distribution of ratings will be flatter. Kurtosis is usually adopted to measure te level of consensus in social attitudes [29]. We use kurtosis to capture tis information of eac item, defined by: k(j) = [E(x j µ) 4 ]/[σ 4 ] 3, (4) were random variable x j is te rating of item j, µ is te mean of x j, σ is te standard deviation of x j, and E( ) is te expectation of a random variable. A normal distribution as kurtosis of. If te kurtosis is positive, te item as a majority opinion. Oterwise if te kurtosis is negative, te item does not ave a clear majority opinion (but it as a plurality opinion). Given t te index of snapsots, M a (j) t is te fraction of majority opinion (or plurality opinion for items witout a majority opinion) at time t, and M e (j) t is te associated majority opinion expression for item j at time t. Next, we present two strategies to compute M a (j) t and M e (j) t. Numerical Approac. Te model based approac in Sec. 3 is not applicable in tis scenario, as witout te response variable it will be difficult to distinguis majority and minority. However it can be combined wit te tresold-based approac to form a numerical approac. We first compute te rating divergence to te current average rating on te item for eac rating as defined in Equ.(). Note tat in Sec. 3 we obtain σ 2 =.75, wic suggests tat te variance of te majority opinion is close to. Hence, to obtain te group of majority opinion olders, we define majority opinion on item j as a set of ratings M a (j) t = {i d(r i, j,t ) (, +)}. Plurality opinion is also calculated in te same way. Majority opinion expression is defined as a floor function of te average rating M e (j) t = r j,t ˆ. We use te floor function because fluctuations in te convergence of average rating r j,t ˆ is obviously limited in te range of (, +). For example, if r j,t ˆ rose from 2.4 to 2.5, we see tat te majority opinion does not cange because te value of majority opinion expression remains M e (j) = 2. Discrete Approac. Since te above metod is based on a numerical estimation of opinion, one may argue tat it is sometimes natural to represent an opinion by its polarity. Terefore in addition to te numerical representation of majority opinion, we present a discrete approac. We derive tree segments, SP = [3, 4, 5] are ratings for a positive opinion, SE = [2, 3, 4] are ratings for a neutral opinion and SN = [, 2, 3] for negative opinions 2. We allow te overlap of SP, SE, SN because suc segmentation is more tolerant to different standards, i.e. some users will give a rating 2 to poor quality items wile oters will consider 2 a neutral opinion. We calculate te fraction of population on eac segment and coose te igest segment as te majority opinion M a (j) t = {i r i, j,t S, S = arg max{ SN, SE, SP }}. For example, if te percentages of population to rate, 2, 3, 4, 5 are 3%, 2%, 2%, 8%, 2% respectively, ten te majority opinion is negative. We also calculate plurality opinion in te same way. Te majority opinion expression M e (j) t in discrete metods is expressed as negative, neutral or positive. It is only meaningful to compare te majority opinion fractions Ma(j) t for different timestamp t under te same majority opinion expressions. Hence, for a period of time s t e, M e (j) t is identical, ten te sequence < M a (j) s,, M a (j) e > is used in te following two ypoteses. 2 Anoter commonly used segmentation, i.e. SP = [4, 5], SE = [3], S N = [, 2], also gives similar results to Table 7 Hypotesis 3 (H3). For items wit majority opinion, te proportion of majority opinion olders in population is monotonically increasing overtime until it reaces a stable status. Matematically, j, k(j), < M a (j) s,, M a (j) e > is monotonically increasing. Hypotesis 4 (H4). If te item as no clearly majority opinion, te proportion of its plurality opinion is unlikely to monotonically increase overtime. Matematically, j, k(j) <,,< M a (j) s,, M a (j) e > is not monotonically increasing. Te non-parametric Mann-Kendall (MK) test is commonly employed to detect monotonic trends in time series data. Te MK test compares eac observation wit its preceding observation and computes te following MK statistic (S) by n S = n sдn(x i X k ), (5) k= i=k+ were sдn is sign function and X is time series sample, i.e.m a (j) n. Note tat we ave to conduct MK test for eac item. 4.2 Results and Analysis Table 6: Percentage of items (%) wit monotonically increasing < M a (j) s,, M a (j) e > by numerical approac. Significance level..5. Dataset k k < k k < k k < books clotes electronics movies Epinions Ciao Movielens2M Eacmovie Table 7: Percentage of items (%) wit monotonically increasing < M a (j) s,, M a (j) e > by discrete approac. Significance level..5. Dataset k k < k k < k k < books clotes electronics movies Epinions Ciao Movielens2M Eacmovie Te ratings in te datasets are transferred into a 5-star scale in te experiment. We remove te items wit less tan 5 ratings in eac dataset, because we need enoug ratings to fully reflect te formation of opinions. We coose to use ratings as a time window to segment time intervals. In Table 6, we report te percentages of MK positive series < M a (j) s,, M a (j) e > satisfying te different significance levels for items wit a majority opinion (k ) and witout a majority

6 opinion (k < ), determined by te numerical approac. We can see tat, no matter wat te significance level we coose, for most items wit a majority opinion (k ), te portion of majority opinion olders in population is increasing. On te contrary, for items witout a majority opinion, very few of tem (e.g. less tan 4% at significance level p.) sow a rising portion of plurality opinion. Tus te two ypoteses H3 and H4 are verified. In Table 7, we report te MK positive series percentages determined by te discrete approac. Te results are similar. Hypoteses H3 and H4 are again verified. Te percentages obtained by te discrete approac are larger tan tat by te numerical approac. Because by discrete approac we will ave a broader range for majority (or plurality) opinion, tus te possibility for observing a rising trend is bigger. For example, an item wit a rating distribution centered on 2, 3 and 4 and its mean is 2.9, te majority opinion by discrete metod is neutral (including 2, 3, 4), te majority opinion by numerical approac only includes ratings 2, 3. Summary and remarks. In scenario II were no indicators of willingness are available, we verify te existence of a spiral of silence in recommender systems by sowing () most items wit a majority opinion ave a monotonically increasing portion of majority opinion; (2) for items witout a majority opinion, it is very unlikely tat te proportion of its plurality opinion will monotonically increase. 5 FORMATION OF SPIRAL: HARDCORE Hardcore is a key factor in te spiral of silence. In tis section our objective is to understand te possible causes for users to act as ardcore. 5. Preliminaries In RS, a ardcore group is a bunc of users wo will give ratings no matter ow te ratings diverge from te majority opinion. To define a ardcore user, we compute a ardcore score, = n i /n i, (6) were n i is te number of ratings a user i gives to all items, n i is te number of ig divergent ratings of user i. Using te results obtained in Sec. 3, te ig divergent ratings are ratings r i, j were { r i, j,t r j,t ˆ > µ 2 =.4}. 5.2 Hardcore Users Our first question is weter ardcore is an inner caracter tat sapes a user s beavior. We use te recent Yaoo! data set. Te data set contains two sets of ratings: Yaoo!user and Yaoo!random. Yaoo!user set consists of ratings supplied by users during normal interactions, i.e. users pick and rate items as tey wis. Yaoo!user resembles a traditional recommender system, wic corresponds to a setting were users are free to ide teir responses.yaoo!random set consists of ratings collected during an online survey, wen te same group of users in Yaoo!user set were asked to provide ratings on exactly ten items. Yaoo!random is different because te items are randomly selected by te system instead of te users temselves. Yaoo!random corresponds to a setting were users are forced to respond, against is actual willing. Te dataset offers a unique opportunity to testify weter ardcore is a personality. Table 8: Statistics of te data sets used in Sec 5 Dataset #users #Items #Ratings Yaoo!user 5,4 3,74 Yaoo!random 54 54, If ardcore is an inner caracter ten te user will beave similarly under different settings. Terefore we present te following ypotesis. Hypotesis 5 (H5). Hardcore group in te user selected setting is similar to te ardcore group in te random setting. To testify H5, we first detect ardcore groups in bot yaoo data sets by Eq.(6), and ten compare te ardcore users in two subsets. For simplicity, we ignore te possibility tat users use pseudonyms. We assume tat eac user is unique and represents one user in RS. To see weter te two ardcore groups are identical, we conduct Mann-Witney U test to compare te overlap percentage between te two ardcore groups wit a baseline overlap percentage given tat users beave randomly (i.e. uniformly sample ardcore users from te two datasets). We find in Table 9 tat, te two ardcore groups (.5) in different settings are identical, i.e. te overlap percentage of ardcore users is significantly larger tan te baseline overlap. Furtermore, we discover tat non-ardcore users ( <.5) are different under te two settings. Terefore H5 is verified. If a user is ardcore under one setting, e tends to be also ardcore under anoter setting. Table 9: Percentage of ardcore group overlap. indicates te actual overlap is significantly larger tan baseline wit significance level p.5 based on Mann-Witney U test. Users Non-ardcore Hardcore Tresold < Actual Baseline Our next question is weter ardcore people are more likely to give extreme ratings. It is natural to relate ardcore wit attitude certainty, wile attitude certainty is represented by an extreme rating value. To study tis, we set.5 to detect ardcore users, and plot te ratio of extreme ratings (i.e. ratings wit values, 5) for ardcore and non-ardcore users in all data sets. We can see from Fig. 2 tat () in all data sets, ardcore users ave a iger median ratio of extreme ratings. Te IQR of for ardcore users is iger tan te IQR of for nonardcore users, wic suggests tat in recommender systems ardcore users are likely to give more extreme ratings. (2) Compare te two Yaoo!sets, we can see tat ardcore users in Yaoo!random ave a iger ratio of extreme ratings (i.e. smaller box and sorter tail). Tis observation is consistent to te spiral of silence teory, because wen users are forced to rate (te Yaoo!random set), tey can not ide extreme ratings (wic will be missing in Yaoo!users as tey are certainly different from te majority opinion). 5.3 Hardcore and Items It is mentioned in [32] tat ardcore is related to personal interest or importance. Some recommender systems encourage social tagging

7 non-ardcore (a) Epinions ardcore non-ardcore (b) Ciao ardcore non-ardcore ardcore (c) Movielens2M (a) Epinions (b) Ciao (c) Movielens2M.5.3. non-ardcore ardcore (d) Eacmovie non-ardcore ardcore (e) Yaoo!user non-ardcore ardcore (f) Yaoo!random (d) Eacmovie (e) Yaoo!user (f) Yaoo!random Figure 2: Ratio of, 5 ratings ( ) for ardcore and non-ardcore users in various datasets. Figure 3: Hardcore score under two moral situations, (criticize a positive item) and (praise a negative item). to describe contents of items. In recommender systems wit tags, personal interest is depicted by te number of ratings a user gives under a certain tag. Terefore we present te following ypotesis. Hypotesis 6 (H6). Under te most rated tag, an individual user as a iger ardcore score. To testify tis assumption, we use four data sets wit tags, i.e. Epinions, Ciao, Movielens and Eacmovie. We count te number of ratings per user for eac tag. For eac user, we select te most rated tag and least rated tag (wit at least ratings to avoid bias). We compute te ardcore score in Equ. 6 for eac user s most rated tag and least rated tag, were n i is te number of ratings a user i gives to all items associated wit te tag, n i is te number of ig divergent ratings i gives to items wit te certain tag. We report te median over all users in Table. We can see tat in all data sets, is significantly iger under a most rated tag tan a least rated tag. Tus H6 is verified. Tis is probably because for te most interesting items, users are more confident in teir own experience and are more courageous to give a deviant opinion. Anoter question is weter ardcore is related to moral basis. We define two moral situations in Recommender Systems, one is to praise a (wrongly) criticized item, te oter is to criticize an (improperly) appreciated item. Following te definition of ardcore score, we compute under two moral situations (), were n i is te number of ratings tat users give positive feedback (r i, j > ˆr +.4) to items wit average negative feedback (ˆr < 3), n i is te number of ig divergent ratings among n i. (2) : n i is te number of ratings tat users give negative feedback (r i, j < ˆr.4) to items wit average positive feedback (ˆr 3), n i is te number of ig divergent ratings among n i. As sown in Fig. 3, in most cases people feel more obligated to underrate a igly appreciated item tan to save a criticized item. Summary and remarks. In tis section we study te ardcore factor in te formation of spiral of silence. We verify tat () ardcore is a personality wit wic users are likely to give deviant ratings in different settings. (2) Hardcore is positively related to Table : Te median ardcore score for personally most rated tag and personally least rated tag. indicates significance level p. based on Mann-Witney U test. Dataset most rated tag least rated tag Epinions.7 Ciao.476 Movielens 2M.625 Eacmovie 5 personal interest. We visualize tat () ardcore users give more extreme ratings. (2) Users are more willing to criticize a (wrongly) appreciated item. 6 MODEL In tis section, we develop a Missing Conditional on Persona (M) model. We embed two major empirical findings in te M model. () Users are more likely to give ratings if teir ratings are consistent wit te perceived opinion climate. (2) Hardcore users are more likely to give ratings tat are not similar to te perceived opinion climate. 6. Preliminaries As wit most matrix factorization models, we assume tat tere are K idden aspects. Te user preference is denoted as a vector U i R K for user i, and te item feature is denoted as a vector V j R K for item j. Te rating given by user U i to item V j is denoted by X i, j. Te intuition of Probabilistic Matrix Factorization (PMF) [34] is tat, a user will give a ig rating if te item matces is/er preference. Terefore, te rating X i, j approaces to U i V j + BU i + BV j, wit a zero-mean Gaussian error, X i, j N(U i V j + BU i + BV j, σ 2 r ), were BU i and BV j are user specific and item specific bias. U i,v j, BU i, BV j are all zero-mean Gaussian random variables.

8 6.2 Modeling te Spiral of Silence Wit respect to te user rating process, we assume tat tere are tree distinctive stages: pre-rating stage, rating stage, and postrating stage. As sown in Fig. 4, te M model mimics te following generation process. Te pre-rating stage. In tis stage, te user preference U i, user specific bias BU i, item specific bias BV j and item features V j are generated from Gaussian distributions. BU i, BV j N(, σ 2 b ), U i N(, σ 2 u), V j N(, σ 2 v ). To model te split of users between ardcore and non-ardcore groups, we introduce a persona variable, denoted by π i R 2, an -of-2 coding for te persona indicator. Te persona variable π i Bern(β) is generated from a ardcore persona distribution. β (, ) is generated from a Beta distribution β Beta(ξ a, ξ b ) wit yper-parameters ξ a, ξ b. To model te beavior of ardcore and non-ardcore users, eac persona is associated wit a strengt parameter τ z N(, σ τ ), z {, }. Te rating stage. Similar to PMF, user i generates a rating X ij for item j based on is/er rating bias, te item s rating bias, is/er preferences and te item features: X ij N(U i V j + BU i + BV j, σ 2 x ) (7) Te ratings are semi-observed. Weter te rating X ij is observed is denoted by a binary response variable R i, j, were R i, j = indicates te rating is observed and oterwise te rating is missing. Te post-rating stage. In tis stage, te user decides weter or not to reveal is/er rating. Te user will first perceive te opinion climate E ij, wic in tis model is an observed variable. E ij is defined as te average rating on item j before i s rating if te rating X ij is observed. If X ij is not observed, we use te average rating on item j to approximate te opinion climate at te time of rating. As verified in our empirical studies, () te user is more likely to ide te rating if it is divergent to te perceived opinion climate; (2) te user is more likely to display te rating if e/se is a ardcore user π i, =. Tese two findings togeter give us te following generation process: z= P(R ij = X ij, E ij, π i,τ) = exp (τ z= z X ij E ij )] π. (8) i,z We apply a Generalized EM algoritm to infer te model parameters U,V, BU, BV, β,τ. In te E-step we estimate latent variable π i for eac user. In te M-step, we update te model parameters by gradient descent. Te inference can be found in supplementary material. 7 EXPERIMENT We describe a brief experimental analysis of te M model in tis section. More experimental results can be found in supplementary material. Te major evaluation metric is N DCG@L, wic is a standard measure for ranking systems. Te comparative N DCG is conducted on Yaoo!random dataset. Evaluating N DCG@L on a randomly missing data set, suc as Yaoo!random, as been used as te primary criteria in many MNAR researces [, 4]. Experiments on oter non-randomly missing σ v σ b σ x σ u V j E ij BV j σ τ X ij R ij τ N 2 BU i U i π i β ξ a Figure 4: Plate grap of te proposed M model. Variables σ ξ BV j BU i U i V j β π i X ij R ij E ij M Table : Notations for M model Explanations Hyper-parameters Variance for Gaussian distributions Hyper-parameters for Beta distributions Hidden-variables Bias for item j Bias for user i Preference vector for user i Feature vector for item j Hardcore persona probability Binary persona variable for user i Observations Rating on j by i Binary Response on j by i Perceived opinion climate before R ij datasets could be misleading as te ground trut does not accurately reflect user preferences. Terefore, N DCG on Yaoo!random dataset is te primary evaluation metric in tis work. We compare our model to a wide range of available models, including conventional memory-based and model-based collaborative filtering recommenders and MNAR models. Te comparative models include ()UKNN: te user based K-Nearest Neigbor collaborative filtering recommender; (2) IKNN: te item based K-Nearest Neigbor collaborative filtering recommender; (3) MF: te standard matrix factorization model [35]; (4)PMF: te probabilistic matrix factorization model [34]; (5)T-v and (6) Logit-vd: bot from te first MNAR models []; (7) MF-MNAR [4]: te recent probabilistic MNAR model wic masks te rating matrix by a response matrix; (8)RAPMF [2]: a recent MNAR model wic incorporates users response models into te probabilistic matrix factorization. Te parameters (including number of aspects K and variance σ) for te above models are tuned by cross validation. Te yper-parameters for M model is K = 5, ξ a = ξ b = 2, σ =.5 for all variances. Learning rate is initialized wit e 8 ξ b

9 and decayed every rounds. Convergence is determined after a maximal number of 3 rounds. Te reported results are averaged over 5-fold validation. We can see in Fig.5 tat M performs consistently best in all NDCGs. It boosts te performance for about 5% tan te best of MNAR models, i.e. MF-MNAR. Tis result demonstrates te competency of our model. Furtermore, it is wort-noting tat te persona specific strengt parameter learnt for M model τ = 2 for non-ardcore users and τ = for ardcore users. Te interpretation for tis value is tat, for te same rating tat falls in te minority opinion wit ig divergent X ij E ij, a ardcore user is more likely to display te rating tan a non-ardcore user. Tis result is consistent wit te empirical findings. NDCG PMF MF IKNN UKNN RAPMF TV logit_vd MF-MNAR MPC Figure 5: Comparable NDCG performance at top L items 8 CONCLUSION In tis paper we bring a social science perspective to te empirical study of missing not at random ratings in recommender systems. We verify te spiral of silence teory in large-scale real recommendation systems. We study te factors wic contribute to te formation of te spiral of silence, i.e. te existence of ardcore users and te caracteristics of a ardcore person. Our findings not only reveal tat ratings in recommender systems are not missing at random, but also capture te mecanism of missing ratings. To demonstrate te impact of our empirical findings, we use te findings to guide te developments of a MNAR recommendation model. We experimentally sow tat suc a model outperforms state-of-te-art models wit and witout MNAR assumptions. In te future, we will also use te findings to model te evolution of public opinions and peer groups. ACKNOWLEDGMENTS Cen Lin is supported by te Natural Science Foundation of Cina (no ). Yangua Xiao is supported by National Key R&D Program of Cina under No.27YFC837 and No.27YFC22, by te National NSFC (No.67324, No , No.U5923, No.U63627), by Sangai Municipal Science and Tecnology project (No.6522, No.6JC424), Sangai STCSMs R&D Program under Grant (No.6JC424). Cen Lin is te corresponding autor. REFERENCES [] B. M. Marlin and R. S. Zemel. Collaborative prediction and ranking wit nonrandom missing data. In RecSys 29, pages 5 2. L [2] H. Yang, G. Ling, Y. Su and MR. Lyu. Boosting response aware model-based collaborative filtering. IEEE Transactions on Konwledge and Data Engineering, 27(8): , 25. [3] YD. Kim and S. Coi. Bayesian binomial mixture model for collaborative prediction wit non-random missing data. In RecSys 24, pages [4] JM. Hernández-Lobato, N. Houlsby and Z. Garamani. Probabilistic matrix factorization wit non-random missing data. In ICML 24, pages [5] P. Gopalan, JM. Hofman and DM. Blei. Scalable Recommendation wit Hierarcical Poisson Factorization. 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