## 2 System Model and Problem Formulation
### Problem Formulation
#### 2.3.1 Trust Score Computation  
The trustworthiness of each vehicle \(v_{i}\) at time \(t\) is quantified using a novel trust model by integrating dynamic weighting mechanisms and multidimensional factors. In addition, trust evolution over time is considered through implicit decay mechanisms. These include cumulative behavioural anomalies, prolonged disconnection periods, and decreased model contribution relevance. This approach ensures that trust does not remain static but evolves based on both short-term and long-term indicators of vehicular reliability and consistency. Such temporal modelling supports the exclusion of clients exhibiting passive degradation even in the absence of acute security events. The overall trust score \(\tau_{i}(t)\) is calculated as  
\[\tau_{i}(t)=\tau_{i_{l}}(t)\cdot\Big{(}w_{N}\cdot\tau_{N_{l}}(t)+w_{VS}\cdot \tau_{VS_{i}}(t)+w_{E}\cdot\tau_{E_{i}}(t)+w_{C}\cdot\tau_{C_{i}}(t)+w_{P} \cdot\tau_{T_{l_{l}}}(t)+w_{U}\cdot\tau_{U_{i}}(t)\Big{)} \tag{1}\]  
In this model, the following is true:  
* \(\tau_{I_{l}}(t)\) is the IDS alert score, set to 1 if no alerts are generated and 0 otherwise. * \(\tau_{N_{i}}(t)\), \(\tau_{VS_{i}}(t)\), \(\tau_{E_{i}}(t)\), \(\tau_{C_{i}}(t)\), \(\tau_{D_{i}}(t)\), \(\tau_{U_{i}}(t)\) are scores for network status, vehicle status, energy efficiency, compliance with traffic regulations, FL participation quality, and user feedback, respectively. * \(w_{N},w_{VS},w_{E},w_{C},w,w_{U}\) are weights indicating the relative importance of each factor. Factor Weights  
The weights in the trust score calculation reflect two key considerations: First, a security perspective that prioritiises critical factors (IDS alerts) due to their importance in identifying threats and safeguarding local data. Second, a correlation approach that adjusts weights based on their statistical relevance. In particular, Pearson's correlation coefficients are periodically computed to assign higher weights to strongly correlated factors like network stability (\(\tau_{N_{i}}\)) while minimising the influence of less significant ones. This ensures that the weighting adapts effectively to evolving IoV conditions. Fed-DTB employs Equation (2), a dynamic mechanism for weight adjustment that adapts to evolving IoV conditions and operational feedback. Unlike static schemes, this approach recalculates the correlation between each factor's score (\(\tau_{k}\)) and the overall trust score (\(\tau_{i}\)) after periodic training rounds, adjusting weights based on their statistical relevance. For example, factors exhibiting stronger correlations, such as network stability (\(\tau_{N}\)), are assigned higher weights, while less significant factors receive lower weights. The weights are computed as  
\[w_{k}(t+1)=\frac{\text{Corr}(\tau_{k},\tau)}{\sum_{j=1}^{n}\text{Corr}(\tau_{ j},\tau)} \tag{2}\]  
where \(\text{Corr}(\tau_{k},\tau)\) is the correlation between factor \(k\)'s partial trust score \(\tau_{k}\) and the overall trust \(\tau\), \(\sum_{j=1}^{n}\text{Corr}(\tau_{j},\tau)\) sums these correlations for all \(n\) factors (indexed by \(j\)), and \((t+1)\) denotes the updated weight for the next federated learning round. The correlation term \(\text{Corr}(\tau_{k},\tau)\) in Equation (2) is computed explicitly using Pearson's correlation coefficient based on vehicle trust-score data collected from recent federated learning rounds. Specifically, the correlation is calculated across a sliding window of the latest \(L=5\) training rounds as follows:  
\[\text{Corr}(\tau_{k},\tau)=\frac{\sum_{i=1}^{n}(\tau_{k,i}-\tau_{k})(\tau_{i}- \tau)}{\sqrt{\sum_{i=1}^{n}(\tau_{k,i}-\tau_{k})^{2}}\sqrt{\sum_{i=1}^{m}( \tau_{i}-\tau)^{2}}} \tag{3}\]  
where \(\tau_{k,i}\) denotes the partial trust score for factor \(k\) of vehicle \(i\), \(\tau_{i}\) represents the overall trust score of vehicle \(i\), and \(\tau_{k}\) and \(\tau\) are their respective mean values computed over all participating vehicles (\(m\)) within the current training round.