## 3 Problem Statement
### Proposed Method  
**Step 1: Network Initialization:**  
Node Deployment: We deploy \(N\) nodes uniformly across an \(M\times M\) area. Each node \(i\) is initialized with non-replenishable initial energy \(E_{0}\), where residual energy at time t is calculated as:  
\[E_{residual,i}(t)=E_{0}-\sum_{j=1}^{t}P_{consumed,i}(j) \tag{10}\]  
where \(P_{consumed,i}(j)\) represents the power consumed in each discrete time interval \(j\) up to \(t\). This energy consumption model ensures that node \(i\) operates within its energy constraints, making energy optimization crucial for network longevity. Direct Communication with BS:  
The power \(P_{transmit}\) required by each node \(i\) to transmit data to the \(BS\) is calculated based on the distance \(d_{i,BS}\) as follows:  
\[P_{transmit}=P_{0}\bigg{(}\frac{d_{i,BS}}{d_{0}}\big{)}\gamma \tag{11}\]  
where \(P_{0}\) is the reference power at distance \(d_{0}\), and \(\gamma\) is the path-loss exponent (typically between 2 and 4). Nodes with minimal \(d_{i,BS}\) consume less energy per transmission, enhancing the overall efficiency. **Step 2: GAN-based Cluster formation:**  
Cluster Generation with GANs: Clusters are generated using a GAN, with the generator. G (z; \(\theta_{g}\)) producing synthetic cluster centroids from random noise \(z\), and the discriminator D (x; \(\theta_{d}\)) distinguishing real data from generated clusters. The GAN objective function is:  
Figure 4: Application architecture of proposed method. \[Min\;Max\mathbb{E}_{x}\sim P_{data}\left[\log D(x)\right]+\mathbb{E}_{z\sim p_{z}}[ \log(1-D(G(z)))] \tag{12}\]  
where \(P_{data}\) represents the real data distribution and \(p_{z}\) the noise distribution. Convergence is reached when the discriminator can no longer distinguish real from generated clusters, ensuring realistic clustering of nodes. CH Selection Using ACO: For each cluster, a CH is selected by evaluating three factors: residual energy \(E_{CH}\), distance to the BS \(d_{CH,BS}\), and central position within the cluster. The selection criterion \(S_{CH}\) for each candidate node \(i\) is:  
\[S_{CH}=\alpha.\frac{E_{residual,i}}{E_{max}}-\beta.\frac{d_{i,BS}}{d_{max}}- \gamma.\frac{||x_{i}-C_{k}||}{d_{avg}} \tag{13}\]  
where \(E_{max}\) is the maximum energy in the cluster, \(d_{max}\) is the maximum distance to the BS, \(C_{k}\) is the cluster centroid, \(d_{avg}\) is the average node distance to the centroid, and \(\alpha\), \(\beta\), \(\gamma\) are weight factors. The node with the highest \(S_{CH}\) is chosen as the CH. **Step 3: Initial Pheromone Setting for Paths:**  
Neighbor Discovery and Pheromone Initialization: Nodes broadcast \"Hello\" messages to identify their neighbors and initialize pheromone levels \(\tau_{ij}\) on each link (\(i,j\)). The initial pheromone value is set as:  
\[\tau_{ij}(0)=\frac{1}{d_{ij}+\epsilon} \tag{14}\]  
where \(d_{ij}\) is the Euclidean distance between nodes \(i\) and \(j\), and \(\epsilon\) is a small constant to avoid division by zero. This initialization favors shorter distances, encouraging efficient path selection. **Step 4: Route Selection Probability Calculation:**  
The probability \(P_{ijd}\) of choosing a path from node \(i\) to \(j\) for destination \(d\) is:  
\[P_{ijd}=\frac{\left(\tau_{ijd}\right)^{\alpha}\left(D_{ijd}\right)^{\beta} \left(\eta_{ijd}\right)^{\beta}\left(BP_{ijd}\right)^{\delta}\left(E_{ijd} \right)^{\mathrm{k}}}{\sum i\epsilon N_{i}\left(\tau_{ijd}\right)^{\alpha} \left(D_{ijd}\right)^{\beta}\left(\eta_{ijd}\right)^{\tilde{\gamma}}\left(BP _{ijd}\right)^{\delta}\left(E_{ijd}\right)^{\mathrm{k}}} \tag{15}\]  
In this context, the symbols \(\alpha\), \(\beta\), k, \(\tilde{\gamma}\), and \(\tau\) represent weight regulators associated with various network parameters. These regulators are linked to specific aspects of network performance and resource utilization:  
\(\alpha\) is related to energy, indicating its influence or weight in the overall calculation or decision-making process.