# Recommendation-Based Trust Evaluation Model for the Internet of Underwater Things
## 5 Recommendation Evaluation Process
### Definition of Similarity
#### 5.2.1 Confidence Level as a Measure of Similarity  
This method leverages the level of confidence in the trust values assigned to different nodes based on the communication period with time. High confidence in a node typically suggests reliability, assuming that the source node consistently provides successful interactions. To explore the meaning behind this measurement in evaluating recommendations, we begin by further analysing the confidence in two known trust models. In the literature, two robust models have been extensively utilized to establish trust: the beta-based reputation model [36] and the subjective logic model [37]. Both models are recognized as foundational for assessing trust and reputation, allowing for the quantification of trust with a degree of uncertainty, hence supporting the development of confidence measures. Figure 4 illustrates the representation of each model measurement to construct the trust. The **beta-based model** employs the beta probability distribution to analyse the behaviour of entities within a system through observed actions, such as successful or failed packet deliveries. This model relies on the beta distribution, a type of continuous probability distribution that ranges between 0 and 1, to predict the likelihood of favourable outcomes based on historical data. The parameters \(\alpha\) and \(\beta\) define this distribution, representing the counts of positive and negative outcomes, respectively. Figure 4a shows the beta distribution for various \(p\) with a fixed good and bad experience factor. To address the problem of evaluating dishonest recommendations, several studies in the literature attempt to estimate the confidence score for recommending nodes [23; 38; 39; 40; 27]. Within a beta distribution framework, confidence is deduced from the variance in historical interactions. For instance, if node \(i\) engages in both positive and negative interactions with node \(k\) at time \(t\), the confidence is calculated by utilizing the deviation from the mean, emphasizing the stability and predictability of their interactions over time such as [40]:  
\[c_{ik}=1-\sqrt{\frac{12\alpha_{ik}\beta_{ik}}{(\alpha_{ik}+\beta_{ik})^{2}( \alpha_{ik}+\beta_{ik}+1)}}, \tag{3}\]  
where \(\alpha_{ik}\) represents the aggregated positive observations when a node forwards packets, and \(\beta_{ik}\) represents the aggregated negative observations when a node drops packets. One notable work, presented in [27], focuses on evaluating the honesty of recommending nodes based on three key factors: the confidence derived from interactions, compatibility of information (assessed through deviation tests), and closeness between nodes. In their model, the confidence value, denoted as \(V^{conf}_{ik}\), is derived as: \(V^{conf}_{ik}=1-\sqrt{12\alpha_{ik}}\), where \(\sigma_{ik}\) is the beta distribution variance between \(i\) and \(k\). This formula ensures that the confidence value lies within the interval [0, 1], where 0 indicates no previous interactions (hence, no confidence), and 1 signifies complete confidence based on substantial positive and negative  
Figure 4: Existing trust models. interactions. Similar to the above, [PERSON] et al. [40] introduces a trust model that includes several metrics derived from social trust to evaluate recommenders, which are honesty (the mean of beta distribution), frequency-based social trust (which represents the confidence similar to the previous work), and intimacy. The **subjective logic model** functions based on subjective perceptions of the world and employs opinions to signify these perceptions. The fundamental idea is to represent trust through belief, disbelief, and uncertainty [12]. The trust value \(T\) is given by: \(T=(b,d,u)\), in which \(b\) stands for belief, \(d\) is disbelief, and \(u\) accounts for the uncertainty related to the trustworthiness of a node, and \(b+d+u=1\). In the case of absence of belief or disbelief, a base rate (r) is defined as the prior probability.