## 3 Trust-Optimized 6G-IoT Framework
### Trust Management System
#### 3.2.2 Maintaining Resilience and Robustness

Resilience \(\mathbf{R}(t)\) and robustness are mathematically characterized by a coupled system of stochastic partial differential equations (SPDEs) reflecting the dynamic interactions among devices (Appendix A). The evolution of the resilience state \(\mathbf{R}(t)\) is given by

\[d\mathbf{R}(t) =\Bigg{(}\mathbf{A}_{r}(t)\mathbf{R}(t)+\sum_{i=1}^{N}\mathbf{B} _{i}(t)\tau_{i}(t)\Bigg{)}dt\] \[\quad+(\mathbf{C}_{r}(t)+\mathbf{D}_{r}(t)\mathbf{W}_{r}(t))d \mathbf{W}_{r}(t), \tag{17}\]

where \(\mathbf{A}_{r}(t)\) captures deterministic dynamics, \(\mathbf{B}_{i}(t)\) represents the influence of individual trust scores on resilience, and \(\mathbf{W}_{r}(t)\) denotes a multi-dimensional Wiener process. The predictive failure model \(\mathbf{F}_{i}(t)\) is extended using a time-dependent hazard function:

\[\mathbf{F}_{i}(t)=\lambda_{i}(t)\exp(\gamma\mathbf{X}_{i}(t)), \tag{18}\]

where \(\lambda_{i}(t)\) is the baseline hazard rate, and \(\gamma\) is a vector of regression coefficients. #### 3.2.3 Security Protocols

The security level \(\mathbf{S}(t)\) is enhanced by integrating cryptographic verification with dynamic compliance checks. The integrity score \(\mathbf{D}_{i}(t)\) is computed as

\[\mathbf{D}_{i}(t) =-\log(1-\mathbb{P}(\text{Integrity breach}))\] \[\quad+\alpha_{2}\mathbf{D}_{i}(t)+\alpha_{3}\mathbf{S}_{i}(t)\]0

where \(\mathbf{Q}_{i}(t)\) is a compliance matrix, and \(\mathbf{G}_{i}(t)\) captures device-specific security attributes. Compliance \(\mathbf{S}_{i}(t)\) is modeled using a dynamic risk assessment function:

\[\quad+\alpha_{2}\mathbf{D}_{i}(t)+\alpha_{3}\mathbf{S}_{i}(t)\]1

where \(\mathbb{I}(\cdot)\) is an indicator function, \(\mathbf{H}_{i}(\tau)\) represents the device's risk profile, and \(\theta\) is a threshold level. The overall security level \(\mathbf{S}(t)\) is a weighted sum:

\[\quad+\alpha_{2}\mathbf{D}_{i}(t)+\alpha_{3}\mathbf{S}_{i}(t)\]2

where \(\alpha_{7}\), \(\alpha_{8}\), and \(\alpha_{9}\) are weighting factors, and \(\mathbf{L}_{i}(t)\) accounts for ledger updates in the blockchain-based security protocol. ### Economic Optimization

In the TEO-IoT framework, economic optimization is important to ensure that resource management will be carried out effectively. This section discusses an advanced approach of dynamic pricing and cost-efficiency algorithms together with incentive mechanisms integrated within a mathematically sound framework that illustrates the workflow necessary for achieving optimum economic performance in 6G-enabled IoT networks. Figure 2 depicts how various strategies and workflows can be aptly integrated in this process of economic optimization of resource management in 6G-enabled IoT networks. \[\quad+\alpha_{2}\mathbf{D}_{i}(t)+\alpha_{3}\mathbf{S}_{i}(t)\]3 Architectural diagram illustrating the economic optimization process in the TEO-IoT. #### 3.3.1 Dynamic Pricing Strategies

We model the dynamic pricing as an optimization problem, whose objective is to update the price at every instant \(\pi(t)\) based on (i) the real-time demand \(D(t)\), (ii) trust scores \(\tau_{i}(t)\), and (iii) network conditions \(\eta(t)\). \[\quad+\alpha_{2}\mathbf{D}_{i}(t)+\alpha_{3}\mathbf{S}_{i}(t)\]4

where \(\pi_{0}\) is the base price and \(\lambda_{1}\), \(\lambda_{2}\), and \(\lambda_{3}\) are scaling factors. The function of revenue \(R(t)\) is

\[\quad+\alpha_{2}\mathbf{D}_{i}(t)+\alpha_{3}\mathbf{S}_{i}(t)\]5

where \(S(t)\) is the resource usage, \(\gamma\) is the penalty parameter, and \(\pi_{\text{opt}}\) is the optimal price when maximum efficiency is reached. To maximize the cumulative revenue over a time horizon \(T\), we define the total revenue \(R_{\text{total}}\) as

\[\quad+\alpha_{2}\mathbf{D}_{i}(t)+\alpha_{3}\mathbf{S}_{i}(t)\]6

The optimization problem is to find the price \(\pi(t)\) that maximizes \(R_{\text{total}}\).