## 3 Proposed Model
### Measuring Similarity Based on Probability Distribution of User Interactions  
We measure the nodes' similarity based on their probability distributions in comparison with other nodes. It is difficult to find similarity between nodes in heterogeneous networks, as indirect links might exist between different nodes. The similarity is measured to identify the neighboring nodes. Figure 3: Frequency table of user–item clicks. Table 1 presents the notation we used to define different interaction paths. We denote the set of users as U and a single user as \(u_{m}\). Similarly, the set of all the items is denoted by I, and a single item is represented by \(i_{n}\). Now, as we are finding the probability distribution of user clicks for items, we define the click probability of a user \(u_{m}\) for a single item \(i_{n}\) as \(P_{u_{m}}(i_{n})\), and their click probability for all items I is defined as \(P_{u_{m}}(\text{I})\). After this, we find the similarity between two users by measuring the distance between their click distributions for items. Hence, we define it as \(d(P_{u_{1}}(\text{I}),\ P_{u_{2}}(\text{I}))\). \(Xu_{ij}\) is a collection of all the clicked items in an individual user's set. It can be defined according to Equation (1). \[Xu_{ij}=\text{ Qu}_{i}\ \cup\ \text{Q}u_{j}\,\ (u_{i}\,\ \ u_{j}\ \in U) \tag{1}\]  
where \(\text{Q}u_{i}\ and\ \text{Q}u_{j}\) are sets of clicked items of any two users of \(U\). To formulate this similarity, different distance metrics on the probability distribution can be used. We use both Jensen-Shannon divergence (JS divergence) and Kullback-Leibler divergence (KL divergence), as both have their pros and cons [45]. For example, JS divergence is symmetrical, while KL divergence is sensitive towards small changes in data. It is symmetrical. [PERSON] et al. [46] used Kullback-Leibler (KL) divergence, also called KL distance, from the perspective of using probability distributions to evaluate the distance (similarity) between a pair of items. As JS divergence is derived from KL divergence, we first define and calculate the KL divergence. For the distance between two users \(u_{i}\) and \(u_{j}\) in terms of item clicks, the preference can be defined according to Equation (2). \[D_{KL}\Big{(}P_{u_{i}}(\text{I}),\ P_{u_{j}}(\text{I})\Big{)}=\sum_{i_{m}\in \text{ }x_{u_{ij}}}\ P_{u_{i}}\ (i_{m})\ \ln\frac{P_{u_{i}}\ (i_{m})}{P_{u_{j}}\ (i_{m})} \tag{2}\]  
Before formulating the JS divergence, we need to define an additional factor, i.e., \(P_{M}\) according to Equation (3). \[P_{M}=\frac{1}{2}P_{u_{i}}(\text{I})+\ P_{u_{j}}(\text{I}) \tag{3}\]  
For the distance between two users \(u_{i}\) and \(u_{j}\) in terms of item clicks, the preference can be defined according to Equation (4). \[D_{JS}\Big{(}P_{u_{i}}(\text{I}),\ P_{u_{j}}(\text{I})\Big{)}=\frac{1}{2}\ D(P_{u_{i}},\ P_{M})+\ \frac{1}{2}\ D\Big{(}P_{u_{j}},\ P_{M}\Big{)} \tag{4}\]  
As shown in Figure 4, once the distributions of the clicks' probability have been estimated, the similarity between two users is calculated based on their distribution distances. \begin{table}
\begin{tabular}{c c} \hline \hline
**Notations** & **Definitions** \\ \hline U & Set of all users \\ \hline I & Set of all products \\ \hline \(P_{u_{m}}(i_{n})\) & Click probability of user m for item \(i\) \\ \hline \(P_{u_{m}}(\text{I})\) & Click probability of user m for all items I \\ \hline \(d(P_{u_{1}}(\text{I}),\ P_{u_{2}}(\text{I}))\) & The similarity between two users \(u_{1}\) and \(u_{2}\) \\ \hline Qu\(u_{i}\) & Set of user \(u_{i}\)’s clicked items \\ \hline \(Xu_{i}\) & Union set of all the items in any of two users \\  & from U consisting of clicked items \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Different notation used with its respective definitions. According to Equation (5), we can say that users' similarity increases if the distance among their probability distributions decreases.