## 2 Method Analysis
### Average Metropolis Weight Local Average Consensus Algorithm  
The metropolis weight local average consensus algorithm [18] is an iterative algorithm to solve the average consensus problems in a distributed way, only the basic process of which is elaborated in this section. Its relationship with MVDR and how each node obtains the same MVDR estimated result with this method are provided in Section 2.3, with more details.  
The relations between different nodes in WASNs can be represented by a topology diagram [18; 19]. In order to describe the process of the consensus algorithm, some notations are needed. For the WASN with \(N_{d}\) nodes, we introduced an undirected graph \(G=(V,\varepsilon)\), where the vertices in \(V=\{1,2,\cdots,\ N_{d}\}\) correspond to the nodes in the network, and an edge \((i,j)\in\varepsilon\) corresponds to the communication links between node \(i\) and \(j\). The adjacency matrix \(\mathrm{A}=\big{[}a_{ij}\big{]}\) indicates whether the \(i_{\text{th}}\) node and the \(j_{\text{th}}\) node are connected. If \((i,j)\in\varepsilon\), \(a_{ij}=1\), which means the nodes \(i\) and \(j\) are interconnected and communication can be executed between them, otherwise \(a_{ij}=0\). The set of neighboring nodes of node \(i\) is \(C_{i}=\{j\in V\big{|}(i,j)\in\varepsilon\}\). The degree \(d_{in}(i)\) of node \(i\) is determined by the following equation:  
\[d_{in}(i)=\sum_{j=1}^{N}a_{ij}, \tag{15}\]  
Given the initial value \(g_{i}(0)\) at node \(i\) (in Section 2.3, the \(g_{i}(0)\) stands for \(\tilde{\mathbf{Y}_{i}}(0)\) and \(\tilde{\mathbf{N}_{i}}(0)\)), the metropolis weight local average consensus algorithm aims to find the average value \(g_{\text{arve}}=\frac{1}{N_{d}}\sum_{j=1}^{N_{d}}g_{i}(0)\) at all nodes by using an iterative scheme with only local information and local processing [14]. According to [15; 16; 18], the linear iterative algorithm can be used to compute the weighted mean of the error between the current value and average value, then the updated local estimate of node \(i\) is presented as follows:  
\[\begin{array}{l}g_{i}(t+1)=g_{i}(t)+\sum\limits_{j\in\mathbb{C}_{i}(t)}U_{ij }(t)\Big{(}g_{j}(t)-g_{i}(t)\Big{)}\\ =(1-\sum\limits_{j\in\mathbb{C}_{i}(t)}U_{ij}(t))g_{j}(t)+\sum\limits_{j\in \mathbb{C}_{i}(t)}U_{ij}(t)g_{i}(t)\\ =U_{ii}(t)g_{i}(t)+\sum\limits_{j\in\mathbb{C}_{i}(t)}U_{ij}(t)g_{j}(t),\end{array} \tag{16}\]  
where \(t\) is the iteration number, the suffix \((t)\) means the relevant value at iteration \(t\). \(g_{i}(t)\) is the updated value of node \(i\). \(U_{ij}(t)\) is the updated weight between node \(i\) and \(j\). The updated self-weight of node \(i\), \(U_{ii}(t)\) is defined as follows:  
\[U_{ii}(t)=1-\sum\limits_{j\in\mathbb{C}_{i}(t)}U_{ij}(t), \tag{17}\]Let the vector \(\mathbf{g}(t)=\left[g_{1}(t),g_{2}(t),\cdots,g_{N_{d}}(t)\right]^{\text{T}}\) denote the vector of values at iteration t. \(\mathbf{U}(t)=\left\{U_{ij}(t)\right\}(1\leq i,j\leq N_{d})\) is the \(N_{d}\times N_{d}\) dimensional consistency weight matrix. Equation (16) can be explained in matrix form as follows:  
\[\mathbf{g}(t+1)=\mathbf{U}(t)\mathbf{g}(t), \tag{18}\]  
The purpose of the local average consensus algorithm is to find a suitable consistency weight matrix \(U(t)\), then each node will reach the average value through communicating with neighboring nodes. The average metropolis weight method can be used to fulfill the above purpose by computing the updated weight using Equation (19), which can achieve a faster convergence rate and easier implementation than the metropolis weight method [16]:  
\[U_{ij}(t)=\left\{\begin{array}{c}\frac{2\theta}{d_{in}(i)+d_{in}(j)}\ (i,j)\in\varepsilon\\ 1-\sum_{k\in C_{i}(t)}U_{ik}(t)\ i=j\\ 0\ \text{others}\end{array}\right., \tag{19}\]  
where \(\theta\) (\(0<\theta<1\)) is the trade-off factor.