## 3 Sentiment Classification
### Probability Model  
In order to find the best configuration of user and content labels, the classification model uses a probability model that estimates the likelihood of a given distribution of user and content labels. This probability model was based on the [PERSON] assumption that the sentiment of user \(u_{i}\) (\(l_{i}^{u}\)) is influenced only by the sentiment of every piece of content \(c_{i}\) (\(l_{j}^{c}\)) authored by the user (\(P_{i}\)) and the sentiment labels of its neighbors in the network (\(N_{i}\)). Likewise, the sentiment of a piece of content \(c_{i}\) (\(l_{i}^{c}\)) is influenced by the sentiment label of its author. The label of a node (i.e., user or piece of content) may or may not be known in advance. If a label for a node is known, that node is said to be labeled. Labeled users (\(B^{u}\)) and content (\(B^{c}\)) are assigned a higher weight or influence on global probability. The model is defined as follows. Let \(l_{i}^{u}\) be the label for user \(u_{i}\), and let \(L^{u}\) be the vector of labels for all users. Let \(l_{i}^{c}\) be the label for content \(u_{c}\) and \(L^{c}\) be the vector of labels for all content. To simplify our notation, we will also use \(P_{i}\) as the subset of content that has been authored by user \(u_{i}\) and \(N_{i}\) as the subset of users who are connected to user \(u_{i}\) in the social context graph. Two users are connected when there is an edge between them, which can be chosen from the different types of relations and interactions available in the context, i.e., \(\{u_{i},u_{j}\}\in E,E\in\{R,I\}\). The probability of a configuration of labels (\(L^{u},L^{c}\)) is given by Equation (9):  
\[ \begin{split} log(P(L^{u},L^{c}))=\sum_{u_{i}\in U}\sum_{c_{j}\in P _{i}}\mu(l_{i}^{u},l_{j}^{c})\frac{\rho_{u}(u_{i})\cdot\rho_{c}(c_{j})}{|P_{i }|}\\ +\sum_{u_{j}\in N_{i}}\lambda(l_{i}^{u},l_{j}^{u})\frac{\rho_{ neigh}\cdot e_{ij}}{\sum_{u_{k}\in N_{i}}e_{i,k}}\\ -log(Z)\end{split}  \tag{9}\]  
where \(\rho_{neigh}\) is a constant that controls the weight of the effect of neighboring users, \(\rho_{u}\) and \(\rho_{c}\) determine the weight of each piece of content and each user, respectively, and \(e_{i,j}\) is the weight of the edge between neighboring users \(u_{i}\) and \(u_{j}\). The value of \(\mu(\alpha,\beta)\) and \(lambda(\alpha,\beta)\) models how a node labeled \(\beta\) affects  
Figure 2: Problem definition. The task is to predict the missing labels. a node labeled \(\alpha\) (\(\alpha,\beta\in Polarities\)). For the typical case, where \(\mathit{Polarities}=\{\mathit{positive},\mathit{negative}\}\), \(\mu\) and \(\lambda\) can be thought of as an array with four values, one per combination of the two polarities. For instance, the value of \(\mu_{\mathit{positive},\mathit{positive}}\) is the weight given to positive content by positive users. The weight of a specific user is controlled through \(\rho_{u}\) (Equation (10)), and \(\rho_{c}\) (Equation (11)) controls the weight of each piece of content. The values of both functions depend on whether the label for the specific user and or content is known a priori. For users with a known sentiment, the weight is \(\rho_{labeled}\), and for unknown values, it is \(\rho_{unlabeled}\). Based on previous works [2,32], we use the following values: \(\rho_{u,labeled}=\rho_{c,labeled}=1\), \(\rho_{u,unlabeled}=\rho_{c,unlabeled}=0.2\)Once again, \(e_{i,j}\) is the weight of the edge between users \(u_{i}\) and \(u_{j}\). Intuitively, this allows for some specific edges to represent stronger bonds and, hence, have a bigger impact on the result. The influence of neighboring agents \(\rho_{neigh}\) is a parameter that can be adjusted. \[\rho_{u}(u)=\begin{Bmatrix}\rho_{u,labeled}&:  
SampleRank can be used to determine the value of \(\phi\), which is divided into \(\mu_{a,\beta}\) and \(\lambda_{a,\beta}\). Ideally, the value of \(Z\) could be obtained through regularization, but in practice, this can be costly.