## 3 Problem Formulation
### Problem Statement and Design Goals  
In this study, a federated learning system, called EdgeGuard, is developed using blockchain technology to enable safe and effective cooperation in the Internet of Medical Things (IoMT) networks with \(N\)-distributed edge devices. The challenge involves effectively carrying out federated learning tasks across many IoMT devices while reducing threats (\(\mu_{d}\), \(\mu_{p}\), and \(\mu_{i}\)). This is conducted to enhance the learning process's overall efficiency and provide strong security in a cooperative healthcare setting. During the federated learning process over periods \(\{t_{1},t_{2},\ldots,T\}\), the objective is to maximize data utility (\(DU\)) and model accuracy (\(MA\)) while minimizing communication overhead (\(CO\)), potential threats (\(\mu\)), and model divergence (\(MD\)) in the presence of unknown malicious clients, such that we have the following:  
\[\{\omega\ |\ \omega:\prod_{i=1}^{N}\prod_{j=1}^{M}\prod_{k=1}^{K}\omega_{ijk}\} \tag{11}\]  
where \(\omega_{ijk}=1\) if edge device \(e_{i}\) participates in learning round \(j\) at local edge devices. Equation (12) formulates the threat model by incorporating the considered potential threats (\(\mu_{d}\): data poisoning, \(\mu_{p}\): privacy breaches, and \(\mu_{i}\): integrity attacks), aiming to minimize these risks and enhance the overall security of the collaborative environment:  
\[\mu_{k}=\min(\mu_{d},\mu_{p},\mu_{i}) \tag{12}\]  
Equation (13) is formulated in a way that guarantees the effective distribution of edge devices to learning rounds across geographically dispersed data centers:  
\[\omega:\mathcal{E}\times\mathbb{T}\times\mathcal{S}\times\mu_{k}\to\{0,1\} \tag{13}\]  
The critical constraints that must be satisfied during the federated learning process within the IoMT network are stated in Equations (15)-(18):  
\[\text{C1}:\sum_{k=1}^{K}\sum_{i=1}^{N}\sum_{j=1}^{M}\omega_{ijk}=1 \tag{14}\] \[\text{C2}:\sum_{k=1}^{K}\sum_{i=1}^{N}\sum_{j=1}^{M}E_{CPU}^{i} \times\omega_{ijk}\leq\xi_{CPU}^{k}\] (15) \[\text{C3}:\sum_{k=1}^{K}\sum_{i=1}^{N}\sum_{j=1}^{M}E_{RAM}^{i} \times\omega_{ijk}\leq\xi_{RAM}^{k}\] (16) \[\text{C4}:\sum_{k=1}^{K}\sum_{i=1}^{N}\sum_{j=1}^{M}E_{BW}^{i} \times\omega_{ijk}\leq\xi_{BW}^{k}\] (17) \[\text{C5}:\sum_{k=1}^{K}\sum_{i=1}^{N}\sum_{j=1}^{M}\omega_{ijk} \leq\textit{AvailED} \tag{18}\]  
The constraint \(\text{C1}\) specifies that each edge device \(e_{i}\) must participate in exactly one learning round; the constraints \(\{\text{C2-C4}\}\) state that the available resource capacity (CPU, RAM, bandwidth) of the edge network must be greater than or equal to the total requesting resource capacity of participating edge devices; constraint \(\text{C5}\) specifies the geographical constraints, indicating the availability of edge devices within the IoMT network. The design goals of EdgeGuard are, thus, formulated as a multi-objective optimization problem:  
\[\min_{w_{g},\omega}\{CO,\mu,MD\}\quad\text{and}\quad\max_{w_{g},\omega}\{DU,MA\} \tag{19}\]  
subject to constraints C1-C5, where \(w_{g}\) denotes the global model parameters. To achieve these goals, EdgeGuard employs the following:  
* An adaptive aggregation function \(f\) to balance data utility and privacy: \[w_{g}^{t+1}=f(\{w_{i}^{t+1}\}_{i\in\mathcal{E}_{t}},\{q_{i}^{t}\}_{i\in \mathcal{E}_{t}},\{r_{i}^{t}\}_{i\in\mathcal{E}_{t}})\] (20)
* A blockchain security layer \(\mathcal{B}=(B,T,\sigma,V)\) to ensure integrity and traceability.
* A quality assessment module \(Q:\mathcal{D}\rightarrow[0,1]\) to evaluate data quality.
* A reliability evaluation function \(R:\mathcal{E}\times\mathbb{T}\rightarrow[0,1]\) to assess device trustworthiness.  
These components work in concert to create a secure, efficient, and privacy-preserving federated learning system for IoMT healthcare networks.