# Graph Layer Security: Encrypting Information via Common Networked Physics
## 3 GLS Encryption Using Physical Networked Dynamic
### Relay Selection & Weight Computation
#### 3.2.1 GFT Operator-Based Surrogate  
Suppose that the physical networked dynamic matrix can be decomposed as \(\mathbf{X}=\mathbf{\Gamma}\cdot\tilde{\mathbf{X}}\) with \(rank(\mathbf{\Gamma})=rank(\mathbf{X})\). Then, if \(\mathbf{\Gamma}\) is only related with the dynamic model, and is therefore invariant with different initialisation cases, such a \(\mathbf{\Gamma}\) can be used as the surrogate matrix for analysing the row (node) dependency of \(\mathbf{X}\) in Equation (9). To implement this idea, we adopt the graph spectrum analysis and the concept of graph-bandlimitedness from [30; 31; 32; 33; 34; 35]. Before we start, we give a brief introduction of the graph Fourier transform (GFT) and the graph-bandlimitedness. Given a GFT operator as \(\mathbf{U}^{-1}\) (typically an orthogonal matrix of size \(N\times N\), i.e., \(\mathbf{U}^{-1}=\mathbf{U}^{T}\)), the processes of GFT and inverse GFT are specified as follows [30; 31; 32; 33; 34; 35; 36]:  
\[\tilde{\mathbf{x}} =\mathbf{U}^{-1}\cdot\mathbf{x}, \tag{10}\] \[\mathbf{x} =\mathbf{U}\cdot\tilde{\mathbf{x}}, \tag{11}\]  
where \(\mathbf{x}\) of size \(N\times 1\) is the graph signal, and \(\tilde{\mathbf{x}}\) is its graph frequency response. We call \(\mathbf{x}\) graph \(\mathcal{R}\)-bandlimited to the GFT operator \(\mathbf{U}^{-1}\), if only the elements of \(\tilde{\mathbf{x}}\) with rows in the set \(\mathcal{R}\subset\{1,\cdots,N\}\) are non-zeros. According to the work in [29], this property holds for a vast variety of the time-varying networked dynamics in real-world systems, i.e., either the original \(\mathbf{X}_{:,k}\) or the time difference \(\mathbf{X}_{:,k}-\mathbf{X}_{:,k-1}\) are (or can be approximately treated as) graph \(\mathcal{R}\)-bandlimited to \(\mathbf{U}^{-1}\) for all time indices \(k\). Here, we denote \(\mathbf{Y}\) as the original graph signal or the corresponding time difference, i.e.,  
\[\mathbf{Y}=\left\{ \begin{aligned} \mathbf{X}& \mathbf{X}_{:,k}\text{are graph-bandlimited}\\ &[\mathbf{X}_{:,k}-\mathbf{X}_{:,k-1}]\text{ with all }k\\ &\mathbf{X}_{:,k}-\mathbf{X}_{:,k-1}\text{are graph-bandlimited} \end{aligned} \right. \tag{12}\]  
where the selection is dependent on difference scenarios. As such, by denoting \(\mathbf{Y}=\mathbf{X}\) or \([\mathbf{X}_{:,k}-\mathbf{X}_{:,k-1}]\) with all \(k\), a surrogate of \(\mathbf{Y}\) can be assigned as \(\mathbf{\Gamma}=\mathbf{U}_{:,\mathcal{R}}\), if \(\mathbf{Y}_{:,k}\) for all time indices \(k\) are graph \(\mathcal{R}\)-bandlimited to \(\mathbf{U}^{-1}\), i.e.,  
\[\mathbf{Y}_{:,k}=\mathbf{U}_{:,\mathcal{R}}\cdot\tilde{\mathbf{Y}}_{\mathcal{ R},k}. \tag{13}\]  
Here, for \(\mathbf{Y}=[\mathbf{X}_{:,k}-\mathbf{X}_{:,k-1}]\) with all \(k\), we need to replace the original physical graph signal \(\mathbf{X}\) with the corresponding time difference \(\mathbf{Y}\) in all GLS encryption and further processes, i.e., Equations (3)-(9). In Equations (10)-(13), the GFT operator \(\mathbf{U}^{-1}\) is typically assigned as the eigenvector matrix of the graph adjacent matrix denoted as \(\mathbf{W}\)[30; 32], or the graph Laplacian matrix, computed as \(\mathbf{L}=diag(\mathbf{W}\cdot\mathbf{1})-\mathbf{W}\)[33; 34; 35]. Accordingly, the graph-bandlimited set \(\mathcal{R}\) is truncated from \(\{1,\cdots,N\}\) to concentrate on the low-graph-bandlimited area (e.g., to minimise \(\mathbf{x}^{T}\cdot\mathbf{L}\cdot\mathbf{x}\) for Laplacian matrix). The problem here lies in the difficulty in measuring the actual elements in adjacent matrix \(\mathbf{W}\) (although with the knowledge of the existence status of each link). To address this, we exploit a data-driven method, as we notice that the columns of \(\mathbf{U}_{:,R}\) in Equation (13) are the orthogonal vectors derived from the columns in \(\mathbf{Y}\).