Which is more influential,“Who” or “When” for a user to rate in online review site?

Abstract

At its heart the act of reviewing is very subjective, but in reality many factors would influence user’s decision. This can be called social influence bias. We pick two factors, “Who” and “When” and discuss which factor is more influential when a user posts his/her own rate in an online review system. We consider two kinds of users: real and virtual. In the former each user has its own metric, but in the latter the metric is assigned to the order of review posting actions (rating). We propose a weighted multinomial generative model that can learn the factor metric quite efficiently from a vast amount of data already available in many online review systems. If the model can explain the data well enough, this implies that such a social bias does exist. We evaluate the proposed method and confirm its effectiveness by five review datasets, and empirically clarify that there is no universal solution, but the social bias does exist. In reality the influential factor depends on each dataset, the majority of users is normal (average), and there are two small groups of users, each with high metric value and low metric value.

Keywords

Social media influence review score machine learning

1. Introduction

The emergence of Social Media has provided us with the opportunity to collect a large number of user reviews for various items, e.g., products and movies. The act of reviewing at its heart is necessarily subjective on the basis of the reviewer’s personal experience and preference, but in reality and in particular in case of online review system, the reviewer (user)1

¹
A user means a reviewer in this paper.

can read other reviewers’ comments and ratings which may influence the reviewer’s decision about his/her actions, e.g., buy the product, see the movie, and eventually the rating.

Namely, our basic assumtion is that in such online review systems, a user would read other users’ reviews/ratings prior to completing their own, and thus, their reviews/rating must have been influenced by others’. We want to validate this hypothesis by a standard hypothesis testing approach employed in many scientific fields, that is, by building a model that takes these influences into account, learning these influences by a part of the observed data (used as a training dataset), and testing its generalization capability by the remaining observed data (used as a test dataset).

It is thus important to be able to assess how the review posted by a user exerts influence over other users’ reviews. If such an influence is assessed to exist, it can be called social influence bias.2

Later in Section 2 we cite [2] in which they define Social Influence Bias (SBI), which is more specific. We use this notion as a general concept.

Our interest here is to find what kind of factors play an important role for a user to make a decision in online review system. There must be many factors, but we can easily think of two major factors: “Who” (user influence) and “When” (order influence). The former implies that each user is not the same in exerting his/her influence to others. Some user is more influential than others. We want to identify influential users. The latter implies that the timing of when to post is more importat than who posts. Some user who reviewed and posted early before other users do may be more influential than those other users who reviewed later. We want to know whether this is the case or not.

Analyzing a review in depth needs natural language processing. Fortunately many social media sites offer numeric review scores. We use this score instead of review itself. A score is defined as a rating given by a user and their values vary across users and items, say, tens of thousands of users and items. If we want to identify high-quality items in a given category in an efficient way from these scores, the simplest way would be to rely on naive ranking. Naive ranking simply ranks items according to the number of reviews or the average review score. There the emphasis is more on the statistical reliability without taking any account for the influence of the review of each user over others. If we are able to incorporate the notion of influence, the assessment of high-quality items could be more plausible.

There are a series of studies [11, 12, 7, 13] that incorporated the trust relationship which is represented by trust strength (i.e., the local trust-value information) into low-rank matrix factorization techniques [18, 9] to improve the performance in rating prediction. Tang et al. [21] proposed a method of incorporating the trustworthiness of users (i.e., the global trust-value information) into this framework, which further improved the performance in rating prediction. In their work, the trustworthiness of a user was estimated by applying the PageRank algorithm [1] to the trust network.

As we see, the above approach assumes a separate source of information, often represented by a trust network, which is not easy to obtain due to its intrinsic difficulty in quantification and time varying nature. We aim to avoid collecting such information and try to quantify the influence solely from the observed data, i.e., the time series review score data, that is, the data consisting of 4-tuple $(u,i,t,s)$ , meaning user $u$ rated item $i$ at time $t$ and gave score $s$ . Thus, another purpose in this paper is to investigate whether it is possible to automatically learn user influence metric only from observable data sets of review rating results. In our experiments, we will present a number of empirical evidences suggesting that user influence metric can be learned.3

Our early work showed that this could be possible [17].

We assume that each user rates an item only once and the score is approximated by a multinomial distribution.4

⁴

In [17] we called the model a modified Voter model (for Voter model, see [19, 3]), but in this paper we name it a weighted multinomial model to directly refer to the working mechanism of the model.

A score of a user at a certain time point is approximated by multinomial distribution that can be inferred from the integrated score distribution to that time point and the weight of each user.

We mentioned that each user is assigned a weight. To be more precise we introduce two different notions of user, real and virtual, to cope with the two different factors, “Who” and “When”, and formulate a generalized user influence model, i.e., weighted multinomial model, which learns and assesses the influence of each user, be it actual or virtual, from the observed record of review scores. In case of actual user each user has its own metric value. In case of virtual user, the metric is assigned to the order of reviewing actions (rating). Users who rated first, second, third, …, are grouped respectively and each order has its own metric.

One of the advantage of our model is that the goodness of the metrics, i.e., whether the considered metric is meaningful or not, can be quantitatively evaluated by the generalization performance (predictive power of future rating). Evidently, we can expect that the model with high generalization performance can learn the intrinsic properties in the dataset, implying that our model with influence reflects a reality. Users with higher influence metric must be notably representative and influential reviewers. Thus, their rating scores are considered to be trustable.

The number of users is huge, as many as tens of thousands. Thus, leaning influence metric, i.e., weight, for each user is not trivial. We show that it is indeed learnable by maximizing the log likelihood, i.e., by making the inferred users’ review score distribution best match the observed score distribution. The learning uses iterative scheme but is very efficient. Data is divided into two parts of about equal size and the former half is used for learning the parameters of the model, i.e., influence metrics, and the generalization capability of the learned model is evaluated by the estimated likelihood from the unseen latter half data to determine the optimal value of the regularization factor which is introduced to avoid overfitting. Once the regularization factor is fixed, all the data are used to relearn the model.

We tested the proposed method by applying it to five review systems. Anime review dataset, Cosmetics review dataset, Kakaku review dataset, Tabelog review dataset and MovieLense dataset.5

⁵

For detail, see Section 5.1.

The influence metric distributions of both user and order depend on the characteristics of dataset used. Generalization capability of each metric differs from one dataset to another. Thus, there are no single conclusion. Basically there are three groups of datasets. “Who” group, “When” group and other group. Anime dataset clearly indicates that user influence is more dominant than order influence while Kakaku dataset clearly indicates order influence is more dominant than user influence. However, MovieLense dataset is totally different. None of the two metrics seems to work, which implies that the review is mostly subjective. We have to conclude that humans behaviour is different if the target category, e.g., cosmetics, various products, restaurants, etc. of online review system is different, but we do conclude that the two types of influence factors indeed exist. We tried to characterize them in terms of other basic phenomena, but we are only partially successful. For datasets that belong to “When” group, the user influence metric is hard to interpret, but for datasets that belong to “Who” group the user influence metric plays an important role. It has a strong positive correlations with the following three more basic metrics: 1) the number of reviews a user made, 2) the number of the user’s followers who rate the same item, 3) the number of the user’s followers who gave the similar rate. Influential users are those who have large values for these three basic metrics. Those influential users are not necessarily early raters. However, the majority of the people have the average influence metric and only a small fraction of people have high or low metrics. Thus, the method can identify two interesting groups of people that is worth paying attention to. High score users are influential to drive the expected consensus score, i.e., lead global behaviors. Users with low influence metrics deviate from the average behavior, i.e. somewhat eccentric users.

The rest of the paper is organized as follows: We describe related work in Section 2. We then propose a weighted multinomial model in Section 3 and the weight learning algorithm in Section 4. We report the results of our experiments and their analysis in Section 5, and conclude the paper by summarizing the main results and needed future work in Section 6.

2. Related work

A large amount of data for peoples’ trust relationships is now available due to the emergence of Social Media such as Epinions, and there is a growing interest in measuring how much an online user can trust another user. For such trust metrics, [5] proposed TidalTrust, and [14] proposed MoleTrust. Also, [20] proposed mTrust and [4] proposed PushTrust. mTrust and PushTrust exploit, in addition to the trust network, the user-item rating matrix for each category in a product review site, and successfully applied it to rating prediction. Especially, PushTrust considers trusted, distrusted, and neutral friends in a network and successfully ranks them based on the similarity to the latent features of a target user so that the trusted friends come to the top portion of the list, while the distrusted ones are pushed to its bottom. The neutral friends appear in the middle. Their method enables us to quantify the strength of a trust link for each category. It is based on a latent factor model, and is a kind of collaborative filtering by use of a low-rank matrix factorization. These trust metrics are regarded as local trust metrics since they provide a personalized trust-value for a given user $u$ of an evaluating user $v$ .

Low-rank matrix factorization techniques were extensively applied to rating prediction in online review systems [18, 9]. Several researchers [11, 12, 7, 13] incorporated the information of trust relationships represented by trust strengths (i.e., the local trust-value information) into low-rank matrix factorization techniques, and improved the performance in rating prediction. [21] proposed a method of incorporating the information of trust worthiness of users (i.e., the global trust-value information) into this framework, and further improved the performance in rating prediction. They measured the trust worthiness of a user by simply applying the PageRank algorithm [1] to the trust network.

Unlike a local trust metric, a global trust metric measures how much a given user $u$ is collectively trusted by all users, and reveals the reputation of the user $u$ in the whole network. What we propose in this paper is the same in the spirit in that the influence metric of each user is a global metric. We do not call it trustworthiness (trust means more than influence). It is closer to conformity. The proposed method does not need to use the information of trust relationships, which is crucially different from the work [21]. It is based on time series data of review scores for items given by users. In that sense, the work [16] by Pan et al. is closer to our method. In the framework of social update recommendation that aims at recommending to users articles posted to social media, they embedded some features derived from information diffusion processes in a latent-feature-based collaborative filtering model without using any trust relationship. In their work, “when” is considered by the temporal popularity of topics, while “who” is considered by the celebrity score of a user that is a simple ratio of the number of followers of the user over the number of his/her followees. But, they only consider whether an article is diffused, or not, i.e., a binary decision, but not multiple rating scores.

Social Influence Bias (SIB) [2] is also closely related to our work because it means the tendency to conform to the perceived norm in a community. In the context of recommender systems, it means that a review score is affected by the mean or median value of scores that have already been given by others until then. Krishnan et al. proposed a framework to learn, analyze, and mitigate SIB in recommender systems [10]. Their method needs three kinds of ratings i.e., the initial rating given before seeing the median rating, the median rating itself, and the final rating given after seeing the median rating to learn and analyze SIB. But, in general, it is difficult to collect such triplets through ordinary review sites. SIB itself is an effect that can be actually observed. But, within this concept, it is impossible to analyze factors that affect users’ ratings more in depth as we do in this work.

3. Model

We denote the sets of users and items by $V=\{u,v,w,\cdots\}$ and $I=\{i,\cdots\}$ , respectively. When a user $v\in V$ reviewed an item $i\in I$ , we denote its timestamp and score by $t_{v,i}$ and $s_{v,i}$ , respectively, where each score $s_{v,i}$ is denoted by a positive integer in $S=\{1,\cdots,|S|\}$ , and $|S|$ stands for the number of elements in $S$ . Then, we can express our observed data set as $D=\{\cdots,(v,i,t_{v,i},s_{v,i}),\cdots\}$ . Hereafter, let $V(i)=\{v∼{}|∼{}(v,i,t_{v,i},s_{v,i})\in D\}$ be a set of users who reviewed an item $i$ . For users in $V(i)$ , let $U(i,t)=\{u\in V(i)∼{}|∼{}t_{u,i}<t\}$ be the set of users who reviewed item $i$ before $t$ , and, among them, $U(i,t,s)=\{u\in U(i,t)∼{}|∼{}s_{u,i}=s\}$ the set of users who gave score $s$ .

As mentioned earlier, reviews are thought to be fundamentally subjective, but users may decide their review scores of each item on the basis of not only their own evaluations and preference, but also past majority scores or those submitted by trustable/influential users. In order to stochastically cope with the opinion decision problem that is affected by majority scores, we can employ the basic multinomial model, and define the probability that a user $v$ gives a score $s$ to an item $i$ as follows.

$\displaystyle P_{bm}(s|i,t)=\frac{1+|U(i,t,s)|}{|S|+|U(i,t)|},$ (1)

where we employed a Bayesian prior known as the Laplace smoothing. Here we note that the Laplace smoothing of Eq. (1) corresponds to the assumption that each user initially holds one of the $|S|$ scores with equal probability. Note also that the Laplace smoothing corresponds to a special case of Dirichlet distribution that is very often used as a prior distribution in Bayesian statistics, and in fact the Dirichlet distribution is the conjugate prior of the categorical distribution and the multinomial distribution.

Thus far, we assumed that all the past user scores are of equal importance, i.e., equally weighted. However, it is naturally conceivable that trustable users should be more influential, i.e, have larger weights. In order to reflect this kind of effects into the model, we consider introducing a positive influence metric $\exp(\theta_{u})$ to each user $u$ , where $\theta_{u}$ is a parameter which controls the influence metric. Hereafter, we denote the vector consisting of these parameters by $\mbox{\boldmath$\theta$}=(\cdots,\theta_{u},\cdots)$ . Then, by extending the basic multinomial model defined in Eq. (1), we can build our weighted multinomial model, referred to as user influence model, and define the probability that a user $v$ gives a score $s$ to an item $i$ at time $t$ as follows:

$\displaystyle P_{um}(s|i,t;\mbox{\boldmath$\theta$})=\frac{1+\sum_{u\in U(i,t,% s)}\exp(\theta_{u})}{|S|+\sum_{u\in U(i,t)}\exp(\theta_{u})}.$ (2)

Here, we should mention that our weighted multinomial model defined in Eq. (2) can also be viewed as a model that relates some users’ prior ratings to a new user’s rating. Good predictive performance would support the hypothesis that some users’ ratings are influenced by other users’ prior ratings.

In this paper, in order to estimate $\theta$ from the observed data set $D$ , we consider maximizing the following logarithmic likelihood function based on Eq. (2):

$\displaystyle L_{um}(D;\mbox{\boldmath$\theta$})=\sum_{i\in I}\sum_{v\in V(i)}% \log P_{um}(s_{v,i}|i,t_{v,i};\mbox{\boldmath$\theta$}).$ (3)

Here note that each pair of review score $s$ and $t$ is replaced by the observed pair of $s_{v,i}$ and $t_{v,i}$ in Eq. (3).

Next, we consider another model that assumes that the weights are attributed to the order when the rating is made. Namely, we consider introducing a positive influence metric $\exp(\phi_{r})$ to each review order $r\in\{1,2,\cdots\}$ , where $\phi_{r}$ is a parameter which controls the influence metric of review order $r$ , which does not depend on any individual user $u$ . Here, we also denote the vector consisting of these parameters by $\mbox{\boldmath$\phi$}=(\cdots,\phi_{r},\cdots)$ . Then, by extending the basic multinomial model defined in Eq. (1), we can build another weighted multinomial model, referred to as order influence model, and define the probability that a user $v$ gives a score $s$ to an item $i$ at time $t$ as follows:

$\displaystyle P_{om}(s|i,t;\mbox{\boldmath$\phi$})=\frac{1+\sum_{u\in U(i,t,s)% }\exp(\phi_{|U(i,t_{u,i})|+1})}{|S|+\sum_{u\in U(i,t)}\exp(\phi_{|U(i,t_{u,i})% |+1})}.$ (4)

Again, in order to estimate $\phi$ from the observed dataset $D$ , we consider maximizing the following logarithmic likelihood function based on Eq. (4):

$\displaystyle L_{om}(D;\mbox{\boldmath$\phi$})=\sum_{i\in I}\sum_{v\in V(i)}% \log P_{om}(s_{v,i}|i,t_{v,i};\mbox{\boldmath$\phi$}).$ (5)

Now, we consider a set of virtual users denoted by $\tilde{V}=\{\tilde{v}_{1},\cdots,\tilde{v}_{r},\cdots\}$ , where the review order of user $\tilde{v}_{r}$ is always $r$ for any item, i.e., $\tilde{v}_{r}$ in general contains multiple users who all rate a respective item for the $r$ -th time. Similarly to $U(i,t)$ and $U(i,t,s)$ , we can also define $\tilde{U}(i,t)$ and $\tilde{U}(i,t,s)$ from $\tilde{V}$ . Moreover, we can rewrite the probability of the order influence model defined in Eq. (4) as follows:

$\displaystyle P_{om}(s|i,t;\mbox{\boldmath$\phi$})=\frac{1+\sum_{u_{r}\in% \tilde{U}(i,t,s)}\exp(\phi_{u_{r}})}{|S|+\sum_{u_{r}\in\tilde{U}(i,t)}\exp(% \phi_{u_{r}})}.$ (6)

Therefore, since Eq. (6) is reduced to Eq. (2) just by replacing $\tilde{U}$ to $U$ , we can regard the order influence model as the virtual user model with $\tilde{V}$ . Hereafter, under this interpretation, we abbreviate $P_{um}(s|i,t;\mbox{\boldmath$\theta$})$ and $P_{om}(s|i,t;\mbox{\boldmath$\phi$})$ as $P(s|i,t;\mbox{\boldmath$\theta$})$ , and $L_{um}(D;\mbox{\boldmath$\theta$})$ and $L_{om}(D;\mbox{\boldmath$\phi$})$ as $L(D;\mbox{\boldmath$\theta$})$ unless there is a risk of confusion, and refer to this integrated model as our generalized user model. Here note that we can derive a wide variety of models by assuming the other virtual users.

4. Learning algorithm

The number of users (both real and virtual), which corresponds to the dimensionality of $\theta$ , easily becomes quite large, say tens of thousands. Thus, in order to avoid an overfitting problem, we consider minimizing the following objective function with a standard regularization term:

$\displaystyle J(\mbox{\boldmath$\theta$})=-L(D;\mbox{\boldmath$\theta$})+\frac% {\eta}{2}\|\mbox{\boldmath$\theta$}\|^{2},$ (7)

where $\|\mbox{\boldmath$\theta$}\|^{2}=\sum_{u\in V}\theta_{u}^{2}$ , and $\eta$ stands for a regularization factor. In order to minimize $J(\mbox{\boldmath$\theta$})$ with respect to $\theta$ , we propose a learning algorithm based on the gradient descend method equipped with the second-order optimal step-length calculation for an auxiliary objective function of $J(\mbox{\boldmath$\theta$})$ .

Now, let $I(v)=\{i|(v,i,t_{v,i},s_{v,i})\in D\}$ be a set of items which were reviewed by a user $v$ . For users $V(i)$ , let $W(i,t)=\{w\in V(i)|t_{w,i}>t\}$ be the set of users who reviewed item $i$ after $t$ and, among them, $W(i,t,s)=\{w\in W(i,t)|s_{w,i}=s\}$ be the set of users who gave score $s$ . Further, we define the following functions just like posterior probabilities typically used in the EM algorithm.

$\displaystyle q_{v,i,t}(\mbox{\boldmath$\theta$})=\frac{\exp(\theta_{v})}{|S|+% \sum_{u\in U(i,t)}\exp(\theta_{u})},$ (8) $\displaystyle q_{v,i,t,s}(\mbox{\boldmath$\theta$})=\frac{\exp(\theta_{v})}{1+% \sum_{u\in U(i,t,s)}\exp(\theta_{u})}.$ (9)

Then, by setting the search direction $\mbox{\boldmath$\delta$}=(\cdots,\delta_{v},\cdots)$ to the negative gradient, i.e., $\delta_{v}=-\partial J(\mbox{\boldmath$\theta$})/\partial\theta_{v}$ , we can calculate $\delta_{v}$ as follows:

$\displaystyle\delta_{v}=\sum_{i\in I(v)}\sum_{w\in W(i,t_{v,i},s_{v,i})}q_{v,i% ,t_{w,i},s_{v,i}}(\mbox{\boldmath$\theta$})-\sum_{i\in I(v)}\sum_{w\in W(i,t_{% v,i})}q_{v,i,t_{w,i}}(\mbox{\boldmath$\theta$})+\eta\theta_{v}.$ (10)

From the first term of the right-hand-side in Eq. (10), we can easily see that $\theta_{v}$ of user $v$ tends to increase if $|W(i,t_{v,i},s_{v,i})|$ is relatively large for many items $i$ .

Next, by using the following terms,

$\displaystyle N(v,i;\mbox{\boldmath$\theta$})=\log(1+\sum_{u\in U(i,t_{v,i},s_% {v,i})}\exp(\theta_{u})),$ $\displaystyle D(v,i;\mbox{\boldmath$\theta$})=\log(|S|+\sum_{u\in U(i,t_{v,i})% }\exp(\theta_{u})),$

we note that our objective $J(\mbox{\boldmath$\theta$})$ can be rewritten as follows:

$\displaystyle J(\mbox{\boldmath$\theta$})=-\sum_{i\in I}\sum_{v\in V(i)}N(v,i;% \mbox{\boldmath$\theta$})+\sum_{i\in I}\sum_{v\in V(i)}D(v,i;\mbox{\boldmath$% \theta$})+\frac{\eta}{2}\sum_{v\in V}\theta_{v}^{2}.$

Now, by using the following term

$\displaystyle M(v,i;\mbox{\boldmath$\theta$}|\bar{\mbox{\boldmath$\theta$}})=% \sum_{u\in U(i,t_{v,i},s_{v,i})}\theta_{u}q_{u,i,t_{v,i},s_{v,i}}(\bar{\mbox{% \boldmath$\theta$}}),$

we consider the following auxiliary objective function $Q(\mbox{\boldmath$\theta$}|\bar{\mbox{\boldmath$\theta$}})$ to describe our method for stably calculating the step-length $\lambda$ with respect to $\delta$ .

$\displaystyle Q(\mbox{\boldmath$\theta$}|\bar{\mbox{\boldmath$\theta$}})=-\sum% _{i\in I}\sum_{v\in V(i)}M(v,i;\mbox{\boldmath$\theta$}|\bar{\mbox{\boldmath$% \theta$}})+\sum_{i\in I}\sum_{v\in V(i)}D(v,i;\mbox{\boldmath$\theta$})+\frac{% \eta}{2}\sum_{v\in V}\theta_{v}^{2},$

where $\bar{\mbox{\boldmath$\theta$}}$ stands for the current estimate of $\theta$ . Here, by introducing a dummy user $z\not\in V$ whose parameter value is fixed at zero, i.e., $\theta_{z}=0$ , an augmented set defined by $UA(i,s,t)=U(i,s,t)\cup\{z\}$ for each pair of item $i$ and score $s$ , and the following cross-entropy term,

$\displaystyle H(v,i;\mbox{\boldmath$\theta$}|\bar{\mbox{\boldmath$\theta$}})=-% \sum_{u\in UA(i,t_{v,i},s_{v,i})}q_{v,i,t_{u,i},s_{u,i}}(\bar{\mbox{\boldmath$% \theta$}})\log q_{v,i,t_{u,i},s_{u,i}}(\mbox{\boldmath$\theta$}),$

we can obtain the following equality:

$\displaystyle N(v,i;\mbox{\boldmath$\theta$})=M(v,i;\mbox{\boldmath$\theta$}|% \bar{\mbox{\boldmath$\theta$}})+H(v,i;\mbox{\boldmath$\theta$}|\bar{\mbox{% \boldmath$\theta$}}).$

Thus, we can see that the objective function $J(\mbox{\boldmath$\theta$})$ is expressed as follows:

$\displaystyle J(\mbox{\boldmath$\theta$})=Q(\mbox{\boldmath$\theta$}|\bar{% \mbox{\boldmath$\theta$}})-\sum_{i\in I}\sum_{v\in V(i)}H(v,i;\mbox{\boldmath$% \theta$}|\bar{\mbox{\boldmath$\theta$}}).$ (11)

Therefore, since the second term of right-hand-side in Eq. (11) is minimized at $\mbox{\boldmath$\theta$}=\bar{\mbox{\boldmath$\theta$}}$ , the objective function $J(\mbox{\boldmath$\theta$})$ is optimized by minimizing the auxiliary objective function $Q(\mbox{\boldmath$\theta$}|\bar{\mbox{\boldmath$\theta$}})$ with respect to $\theta$ .

Now, by considering a univariate objective function defined as $F(\lambda)=Q(\bar{\mbox{\boldmath$\theta$}}+\lambda\mbox{\boldmath$\delta$}|% \bar{\mbox{\boldmath$\theta$}})$ , we can calculate the second-order optimal step-length $\lambda$ as $\lambda=-F^{\prime}(0)/F^{\prime\prime}(0)$ where $F^{\prime}(0)$ and $F^{\prime\prime}(0)$ mean the first- and second-order derivatives of $F(\lambda)$ with respect to $\lambda$ at $\lambda=0$ . Here note that

$\displaystyle F^{\prime}(0)=-\|\mbox{\boldmath$\delta$}\|^{2},$ (12)

and we can efficiently calculate $F^{\prime\prime}(0)$ as follows:

$\displaystyle F^{\prime\prime}(0)=\sum_{i\in I}\sum_{v\in V(i)}\sum_{u\in U(i,% t_{v,i})}\delta_{u}^{2}q_{u,i,t_{v,i}}(\bar{\mbox{\boldmath$\theta$}})-\sum_{i% \in I}\sum_{v\in V(i)}\left(\sum_{u\in U(i,t_{v,i})}\delta_{u}q_{u,i,t_{v,i}}(% \bar{\mbox{\boldmath$\theta$}})\right)^{2}+\eta\|\mbox{\boldmath$\delta$}\|^{2}.$ (13)

We can easily see that $F^{\prime\prime}(0)>0$ , $\lambda>0$ and $F(\lambda)$ has a unique global optimal solution due to $F^{\prime\prime}(0)>0$ . Namely, we can stably calculate the step-length without using a technique like safeguarded Newton’s method. Note that straightforward univariate objective function defined by $G(\lambda)=J(\bar{\mbox{\boldmath$\theta$}}+\lambda\mbox{\boldmath$\delta$})$ does not hold this kind of nice properties.

We show our learning algorithm in Algorithm 4, which outputs learned parameter vector $\theta$ from observed data set $D$ . The parameter vector is first initialized (Line 3), then it is iteratively updated until it converges (Lines 5–10). Line 5 computes the search direction $\delta$ by Eq. (10). $\epsilon$ in Line 6 is a small constant, say $\epsilon=10^{-2}$ , which controls the convergence criterion. The search step-length $\lambda$ is computed by Eqs (12) and (13) at Line 9.

[h] Learning algorithm[1] Input: observed data set $D$ Output: parameter vector $\theta$ Initialize each parameter value by $\theta_{v}=0$ for $v\in V$ not converged Calculate the search direction vector $\delta$ as $\delta_{v}\leftarrow-\partial J(\mbox{\boldmath$\theta$})/\partial\theta_{v}$ $\|\mbox{\boldmath$\delta$}\|<\epsilon$ Terminate the iteration Calculate the search step-length $\lambda$ as $\lambda\leftarrow-F^{\prime}(0)/F^{\prime\prime}(0)$ Update the parameter vector by $\mbox{\boldmath$\theta$}\leftarrow\mbox{\boldmath$\theta$}+\lambda\mbox{% \boldmath$\delta$}$ return $\theta$

5. Experiments

We applied the proposed method to five real world review datasets and compared the results with those which were obtained by other methods (explained later in Subsection 5.4) to empirically evaluate the performance of the proposed method and investigate characteristics of resulting influence metrics.

5.1 Datasets

We collected review score records from four famous review sites in Japan and constructed each dataset for our experiment. More specifically, we utilized the review scores for animes extracted from “anikore”,6

⁶
http://www.anikore.jp/.

cosmetics from “@cosme”,7

⁷

http://www.cosme.net/.

variety of products from “kakaku.com”,8

⁸

http://kakaku.com/.

and restaurants from “tabelog”.9

⁹

http://tabelog.com/.

Below these datasets are referred to as Anime, Cosme, Kakaku, and Tabelog in this order. The periods of the review records were from 2010/8/01 to 2012/8/08 for Anime, from 2008/12/07 to 2009/12/09 for Cosme, from 2005/12/07 to 2012/06/22 for Kakaku, and from 2005/03/29 to 2012/02/01 for Tabelog. In addition, we also employed MovieLens 10M/100k dataset,10

¹⁰

http://files.grouplens.org/datasets/movielens/ml-10m-README.html.

to experimentally evaluate the results of our influence models. A detailed description of this dataset is given in [6]. This dataset is the result of users interacting with MovieLens,11

¹¹

https://movielens.org/.

a steadily evolving recommender system, and users who rated at least 20 movies from 1995/1 to 2009/1 were selected at random for inclusion. In MovieLens, a user in general views items in a list of recommendations and rates those items, where the recommendations are performed on the basis of user-personalized rating values that its underlying collaborative filtering (CF) algorithm predicts. The employed recommendation algorithm has changed over time, and the mechanisms for searching and filtering have also evolved. Note that a user cannot see how other users rated a movie, and it is pointed out that the timestamps associated with each rating do not represent the date of consumption, i.e., watching movie, for example, ratings in MovieLens can happen many years after watching movies. Therefore, it is considered that this dataset represents users’ subjective behaviors rather than behaviors associated with their communications among other users. Recall that our hypothesis is that some users’ ratings are influenced by other users’ prior ratings in case of on-line review system. Thus, it is worth examining the generalization performance (predictive power of future rating) of our model for MovieLens dataset as an example of off-line review system. Table 1 summarizes the basic statistics of these five datasets consisting of the numbers of reviews (#reviews), users (#users), and items (#items), the score range (s. r.), and the average score (a. s.). In this paper we intend to evaluate the basic performance of our proposed method. Thus we deleted data with missing or noisy illegal values and used only clean and complete review results. Examining the effects of noise and/or missing values is left for our future study.

Table 1

Basic statistics of review datasets

	#reviews	#users	#items	s. r.	a. s.
Anime	300,327	13,112	1,790	1 to 5	3.9
Cosme	297,453	10,403	46,398	1 to 7	4.4
Kakaku	475,327	170,645	75,519	1 to 5	4.3
Tabelog	3,114,507	291,052	443,573	1 to 9	6.1
MovieLense	10,000,054	71,567	10,681	1 to 10	7.0

5.2 Learning results

First of all, we compared the proposed user and order influence models in terms of three aspects: how their learning performance is affected by the regularization factor $\eta$ , how efficiently they can be learned, and how different they are in the generalization capability. To this end, we divided each dataset into two parts of about equal size and learned the influence metric of each (real/virtual) user from the former half (training data) varying the value of $\eta$ , and then evaluated the generalization capability of the learned model by using the latter half (test data) in terms of generalization likelihood defined as $L(D_{\textit{test}};\hat{\mbox{\boldmath$\theta$}})/|D_{\textit{test}}|$ where $D_{\textit{test}}$ means the test data, $\hat{\mbox{\boldmath$\theta$}}$ a parameter vector learned from the training data, and $|D_{\textit{test}}|$ the number of elements in $D_{\textit{test}}$ .

First, we evaluate both models in terms of how efficiently their models can be learned. Figure 1 depicts how the norm of the gradient vector $||\mbox{\boldmath$\delta$}||$ changes in the learning phase as a function of the number of iterations by the proposed method until $||\mbox{\boldmath$\delta$}||$ converges to less than $10^{-2}$ for different regularization factors $\eta=10,20$ , and $30$ . Regularization parameter $\eta$ can be optimized for each dataset by providing validation dataset additionally, but as we see below its effects on convergence is mild. We, thus, chose its value based on the generalization performance so that it gives good performance, i.e., so as not to overly under- or over-estimate the likelihood in the final series of experiments.

Figure 1.

Norm of gradient vectors by the proposed algorithm as a function of the number of iterations for different values of $\eta$ .

From Fig. 1a to e it is seen that the number of iterations needed by the proposed method to learn these models is not affected by the value of $\eta$ so much and that the user influence model can be learned more efficiently than the order influence model. With exception to MovieLense dataset shown in Fig. 1e, about ten to several tens iterations are enough for learning the user influence model, while one hundred or more iterations are required for learning the order influence model. Interestingly MovieLense dataset needed one thousand or more iterations to learn both the user and order influence models.

Second, we compare the two models in terms of the generalization capability. Figure 2 shows how the generalization likelihood changes in the learning phase as a function of the number of iterations by the proposed method until $||\mbox{\boldmath$\delta$}||$ converges to less than $10^{-2}$ for the regularization factors $\eta=10,20$ , and $30$ . Here, note that the larger the value of generalization likelihood is, the better a learned model is in terms of the generalization capability.

Figure 2.

Generalization likelihood of the user and order models as a function of the number of iterations.

Figure 3.

Distributions of the resulting user influence metrics derived by the user influence model.

Figure 4.

Distributions of the resulting order influence metrics derived by the order influence model.

The user influence model is better than the order influence model in the generalization capability for Anime and Cosme datasets (Fig. 2a and b). The difference between them is smaller for Cosme dataset than for Anime dataset. Especially for Anime dataset the learning effect is small for the order influence model. This implies that for these datasets, the scores of users with high user influence value are better references to estimate the future average score than the scores given at an early stage. On the other hand, the trend is completely reversed for Kakaku and Tabelog datasets, for which the order influence model has higher generalization capability than the user influence model (Fig. 2c and d). Especially for Kakaku dataset the learning effect is very small for the user influence model. Thus, we can say that there is a possibility that early scores are better references for these datasets, although we have yet to confirm that the order metric monotonically decreases with the order (discussed later in reference to Fig. 4). The result that the generalization performance substantially degraded in MovieLense dataset as the learning proceeds both for the user and order influence models (Fig. 2e) indicates that the basic assumptions of our models do not hold for this dataset, i.e., ratings are based on users’ subjective behaviors.

Third, in terms of the effect of the regularization factor $\eta$ , we see that it does affect much the generalization likelihood except for MovieLense dataset for which we see the likelihood monotonically decreases as the learning proceeds. In case of the user influence model $\eta=10$ can achieve the best likelihood among the three values we tested for every dataset except Cosme dataset where the generalization likelihood for $\eta=10$ has a peak and eventually becomes lower than the value for $\eta=20$ . In case of the other datasets, there is no difference in the learning behavior for different values of $\eta$ .

5.3 Estimated influence metrics

We further investigated the distribution of the resulting user influence metrics and plotted them for each value of $\eta$ in Fig. 3. The horizontal axis is the rank of users in the descendant order of the user influence. It is clear that the trends observed from these figures (Fig. 3a to e) are quite similar for every dataset. Namely, regardless of values of $\eta$ , the majority of the users have the average value, i.e., $1.0$ and only a small fraction of them have high or low influence metric.12

¹²
We did not place a constraint that the average of the influence metric is 1.0, but the results are indeed very close to 1.0 for each $\eta$ .

Especially, these characteristic users are worthy of attention for Anime and Cosme datasets where the user influence model works reasonably well. The users with high influence metrics are the main focus in this paper and we try to characterize them in Subsection 5.4. Users with low influence metrics tend to rate an item in a different way from the majorities. Thus, their scores may be useful for a particular user with similar interests, i.e. eccentric users. Further, knowing these low score users helps preventing other users from being confused or deceived by such unusual ratings. Moreover, we should emphasize that in case of Cosme dataset in which each user page can have fan (or reader) links, the numbers of links were generally large for users with high influence metrics, but small for those with low metrics. Note that we cannot characterize users based on the order influence model that never assigns any value to real users.

We also investigated the distribution of the resulting order influence metrics and plotted them for each value of $\eta$ in Fig. 4a to e, where the horizontal axis is the order of reviews (note that the horizontal axis is logarithmic, and its length corresponds to the number of ranks for the item for which the number of reviews is the largest).

Here, we can categorize the datasets into two groups according to their curve types: 1) Anime and Kakaku datasets (Fig. 4a and c), 2) Cosme, Tabelog and MovieLense datasets (Fig. 4b, d and e). The first group clearly indicates that the earlier reviews have larger influence metrics although these two datasets have completely opposite characteristics in terms of the generalization capability as is shown in Fig. 2a and c. We can assume that the influential users are also early reviewers in Anime datasets, and this assumption is examined in Subsection 5.4. This also confirms the conjecture for Kakaku dataset in Subsection 5.2 that early scores are better references although the metric does not decrease strictly monotonically.

The second group indicates that earlier reviews do not necessarily have larger influence metrics. Especially for Cosme dataset the metrics of earlier reviews are less than the normal, i.e., $1.0$ , which indicates that these reviews are not so trustable. In summary, the social influence bias does exist but the way it appears depends on the characteristic of the review datasets.

5.4 Evaluation of user influence metrics

In what follows, we investigate how the users with high user influence shown in Fig. 3 can be characterized by other naive measurements. We exclude from analysis MovieLense dataset for which both the user and order influence models work poorly. For this purpose, we focus on the following three basic metrics: 1) the number of reviews a user made (number of item), 2) the number of the user’s followers who rate the same item (successors ratio), and 3) the number of the user’s followers who gave the similar rate (rating similarity). It is natural to consider that the necessary conditions for a user to be influential are that they have high values for all of these three metrics.

The number of reviewed items and the successors ratio of user $v$ are respectively given by

$\displaystyle\mbox{\#Items}(v)=|I(v)|,$ (14) $\displaystyle\mbox{S-ratio}(v)=\frac{1}{|I(v)|}\sum_{i\in I(v)}\frac{|W(i,t_{v% ,i})|}{|V(i)|},$ (15)

where $W(i,t)=\{w\in V(i)|t_{w,i}>t\}$ , i.e., the set of users who reviewed item $i$ after $t$ . The rating similarity between the score of user $v$ and ones given by its followers is defined as follows:

$\displaystyle\mbox{R-sim}(v)=1-\frac{1}{|I(v)|}\sum_{i\in I(v)}\frac{|s_{v,i}-% \mu_{v,i}|}{|S|-1},$ (16)

where $\mu_{v,i}$ means the average score of successors calculated by

$\displaystyle\mu_{v,i}=\frac{1}{|W(i,t_{v,i})|}\sum_{u\in W(i,t_{v,i})}s_{u,i}.$ (17)

Figure 5.

Scores of the basic three measures (the number of items reviewed, the average ratio of successors, and the rating similarity) for users ranked in terms of the influence metrics based on the user model.

Figure 5 plots the top- $k$ average values of these basic metrics of each user, defined by $g(k)=\sum_{h=1}^{k}f(r(h))/k$ , where $f()$ means one of #Items(), S-ratio() or R-sim(), and $r(h)$ stands for the user $v$ with the $h$ -th largest value with respect to the user influence degree. Here, we adopt $\eta=10$ for every dataset following the observations explained in Subsection 5.2. All the plots share the same vertical axis as their values are normalized into the range of $[0.0,1.0]$ . The horizontal axis is the rank of users in the descendant order of the user influence, similarly to Fig. 3a to d. Here, we focus on the users with high user influence metrics. Such users are limited to a top few percents of the whole users in each dataset as shown in Fig. 3a to d. We categorize the four datasets into two groups as before according to the generalization capability of the user influence model: 1) Anime and Cosme datasets and 2) Tabelog and Kakaku datasets. In terms of the generalization capability shown in Fig. 2a to d, the user influence model outperforms the order model in the first group, while this relation is reversed in the second group. What is common to the two groups is that the number of items decreases with the user rank except for the top 10 users. This is because for a user to have a high user influence metric, s/he has to rate many items in the first place.

For Anime dataset in the first group, we can see that overall all the three basic metrics: #Items, S-ratio and R-sim monotonically decrease with the user’s ranking (Fig. 5a). S-ratio decreases more slowly for the top 1,000 users than the other two. Similar tendency can be observed for Cosme dataset belonging to the same group, except for S-ratio that does not decrease for the top 1,000 users (Fig. 5b). Comparing the difference of behavior in Fig. 3a and b, there seem to be more influential users in Anime dataset who rated at an early stage than in Cosme dataset. Overall, we can say that the scores of the users with high user influence satisfy the necessary conditions above and can be a good reference to estimate the future average score of items.

For Tabelog and Kakaku datasets in the second group shown in Fig. 5c and d the generalization capability of user influence model is smaller than the order model. We can clearly see that the users with relatively high influence metric have smaller ratios of successors. Kakaku dataset exhibits stronger tendency to this. This means that no users with high influential metrics exhibit the above three characteristics.

6. Conclusion

In this paper, we addressed the problem of quantitatively assessing the influence of a user in the context of rating items, which is generally considered as the social influence bias problem, and proposed an efficient algorithm that learns the influence metric of each user from observed review scores. Our hypothesis is that some user may rate an item taking into account not only his/her own opinion but also scores already given to the item by other users, and the reliability of scores depends on who rated them when. We considered two social influence bias: user influence (who part) and order influence (when part). To deal with the two different kinds of bias, we introduced a notion of virtual users for the latter. We modeled this rating process as a stochastic decision making process and used a weighted multinomial generative model. The two influential metrics: user and order were quantified as the solution of the maximum likelihood learning problem. Both influence metrics can be efficiently learned by an iterative algorithm within a few tens of iterations. The model’s generalization capability is insensitive to the value of the regularization factor.

Empirical evaluation on the five real world review datasets uncovered some interesting findings about the two different influence metrics learned by the proposed algorithm. These datasets are Anime review dataset, Cosmetics review dataset, Kakaku review dataset, Tabelog review dataset and MovieLense dataset.13

¹³

For detail, see Section 5.1.

Distributions of influence metrics (both user and order) depend on the characteristics of dataset used. Generalization capability of each metric differs from one dataset to another. Thus, no single conclusion can be drawn. Datasets are categorized in three groups: “Who” group, “When” group and other group. Anime and Cosme datasets belong to “Who” group. Kakaku and Tabelog datasets belong to “When” group. In particular Anima and Kakaku datasets are representative of each group. MovieLense dataset is totally different. None of the two metrics worked, implying that the review is mostly subjective.

We have to conclude that humans behaviour is different if the target category, e.g., cosmetics, various products, restaurants, etc. of online review system is different, but we do conclude that the two types of influence factors, i.e., social influence bias, indeed exist. Here recall that high influence users identified by our model have strong correlations between their ratings and their successive users’ ratings, which turns out to bring good generalization performance (predictive power of future rating) of the model. We tried to characterize them in terms of other basic phenomena. We are only successful for user influence but not for order influence. For datasets that fall in “When” group, the user influence metric is hard to interpret. For datasets that fall in “Who” group the user influence metric plays an important role. It has a strong positive correlations with the following three more basic metrics: 1) the number of reviews a user made, 2) the number of the user’s followers who rated the same item, and 3) the number of the user’s followers who gave the similar rate. Influential users are those who have large values for these three basic metrics. Those influential users are not necessarily early raters, meaning that early adoptors are not necessarily influential. Further, there are only a small fraction of people who have high influence metrics. The majority of the people have the average influence metrics. Interestingly there are also people of a small fraction who have low influence metrics. Thus, the method can identify two interesting groups of people that is worth paying attention to. High score users are influential to drive the expected consensus score, i.e., lead global behaviors. Users with low influence metrics deviate from the average behavior, i.e. somewhat eccentric users.

Our immediate future work includes investigating the characteristics of users with high influence metric more in depth and discuss how they compare with trustworthy users. Another immediate future work is to investigate the effects of noise in data on the results. In addition, we plan to refine our method so that it can incorporate a decay factor to discount the effect of rating behaviors from the distant past.

Footnotes

Acknowledgments

This work was partly supported by Asian Office of Aerospace Research and Development, Air Force Office of Scientific Research under Grant No. AOARD-13-4042, and JSPS Grant-in-Aid for Scientific Research (C) (No. 17K00314).

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Which is more influential,“Who” or “When” for a user to rate in online review site?

Abstract

Keywords

1. Introduction

1 A user means a reviewer in this paper.

3. Model

5.1 Datasets

6 http://www.anikore.jp/.

12 We did not place a constraint that the average of the influence metric is 1.0, but the results are indeed very close to 1.0 for each η .

Footnotes

Acknowledgments

References

¹
A user means a reviewer in this paper.

⁶
http://www.anikore.jp/.

¹²
We did not place a constraint that the average of the influence metric is 1.0, but the results are indeed very close to 1.0 for each $\eta$ .