# Graph Layer Security: Encrypting Information via Common Networked Physics
## 4 Simulations and Results
### Experimental Setting  
The IEEE 39-Bus dynamics are configured in the following. Figure 2 illustrates the network structure with \(N=39\) nodes, in which \(i=30,\ldots,39\) are generator buses, and \(i=1,\ldots,29\) are load buses.  
The physical dynamic model over the network is expressed by following differential and algebraic equations (DAEs) [37; 40]:  
\[\frac{d\theta_{i}(t)}{dt}=\Delta\omega_{i}(t) i\in\{1,\ldots,39\} \tag{26}\] \[\frac{d\Delta\omega_{i}(t)}{dt}=\frac{\omega_{s}}{2H_{i}}\cdot(T_ {i}(t)-P_{i}(t)-D\cdot\Delta\omega_{i}(t)) i\in\{30,\ldots,39\}\] (27) \[T_{i}(t)=-\bigg{(}K_{P}\cdot\Delta\omega_{i}(t)+K_{I}\int_{0}^{ t}\Delta\omega_{i}(\tau)d\tau\bigg{)} i\in\{30,\ldots,39\}\] (28) \[P_{i}(t)=\sum_{j=1}^{39}G_{i,j}cos(\theta_{i}(t)-\theta_{j}(t))+ B_{i,j}sin(\theta_{i}(t)-\theta_{j}(t)) i\in\{30,\ldots,39\}\] (29) \[PL_{i}=\sum_{j=1}^{39}G_{i,j}cos(\theta_{i}(t)-\theta_{j}(t))+B_ {i,j}sin(\theta_{i}(t)-\theta_{j}(t)) i\in\{1,\ldots,29\}. \tag{30}\]  
In Equations (26)-(30), \(\theta_{i}(t)\) is the phasor's angle at \(i\) bus, and \(\Delta\omega_{i}(t)\) is the phasor's speed deviation (i.e., the difference between actual speed and the standard synchronous speed \(\omega_{s}=2\pi\cdot 60\) rad/s). For generator node \(i\) (i.e., \(i=30,\ldots,39\)), \(H_{i}\) is the inertia constant, \(T_{i}(t)\) is the mechanical torque controlling the generator's speed deviation (with constants \(K_{I}\) and \(K_{P}\)), and \(P_{i}(t)\) is the electric air-tap torque affected by the neighbour nodes' angles. \(G_{i,j}\) and \(B_{i,j}\) are, respectively, the real and imaginary parts of the \((i,j)\)th element of admittance matrix, which characterises the admittance of power lines between buses. The aforementioned parameters are assigned according to the work in [37]. In Equation (30), \(PL_{i}\) represents the load under bus \(i\), following a Gaussian distribution with empirical expectation and variance, which accounts for the dynamical input of the system that affects the networked dynamics (i.e., the angles and the speed deviation). We also emphasise here that the aforementioned DAEs are only used for physical dynamic generation, and are unknown for the encryption and communication processes (e.g., process is data-driven and model-agnostic).  
With the help of the dynamic model, we select the speed deviation as the physical networked signal for GLS encryption. To be specific, each bus (node) \(i\) is equipped with an IoT sensor (e.g., the phasor measurement unit PMU), aiming at measuring the speed deviation from time \(t=0\) to \(t=5\), by sampling rate \(T_{s}=10^{-3}\) s [41] (i.e., discrete-time  
Figure 2: Network topology of IEEE 39-Bus system, with \(N=39\) buses (nodes) including 10 generators (buses 30–39) and 29 load buses (1–29).  
\(K=5000\)). As such, the \((i,k)\)th element of matrix \(\mathbf{X}\) in Equation (1) is assigned as the measured \(\Delta\Omega_{i}(t=k\cdot T_{s})\), with a physical measurement noise (not to be confused with communication SNR), which is used to encrypt (for Tx), process (for relays), and decrypt (for Rx) the transmitted information. In this simulation, any pair of nodes are tested as a transmitter (Tx) and a receiver (Rx). The information is assigned as the discrete on-off keying (OOK) with length \(K=5000\). The channels for such communications are wireless and are independent with the physical networked dynamics.  
For attackers, we test the aforementioned passive and active types, respectively. In the passive mode, we assume the eavesdropper can access part of nodes (e.g., 5%, 10%, and 25%) and their time-varying electricity phases. Then, the eavesdropper will try to recover the whole networked dynamics using the steps in Section 3.3.1. In the active mode, we assume the attacker can inject the jamming perturbations into the nodes in order to interfere with the dynamic pattern (as is discussed in Section 3.3.2).