# Graph Layer Security: Encrypting Information via Common Networked Physics
## 4 Simulations and Results
### GLS Performance with Active Attackers  
We next evaluate the GLS performance against active attackers. As is described in Section 3.3.2), the active attackers here aim to degrade the dynamic dependency among network nodes by adding artificial jamming inputs into the network. In the context of the IEEE 39-Bus system, the jamming inputs are the variations of the power usages at each bus, following the Gaussian distribution with means equalling the reference power usages at each bus, and variance as 5 per unit (larger than that of the dynamic perturbation), which will then affect the underlying dynamics (i.e., the rotor speed deviations) for information  
Figure 4: Statistical distribution of all (Tx,Rx) pairs with respect to different levels of BERs with passive Eves.  
encryption and decryption. Four active attackers are considered and assigned with different levels of jamming rates, i.e., 0 s\({}^{-1}\), 10 s\({}^{-1}\), 100 s\({}^{-1}\), 200 s\({}^{-1}\), and 500 s\({}^{-1}\).  
In Figure 5, we provide the average BERs of the legitimate (Tx,Rx) pairs versus the measurement noise variance (i.e., \(\sigma^{2}\)), under different levels of jamming attacks. It is firstly seen that the BERs for all levels of jamming rates become lower, as the decrease of the measurement noise variance \(\sigma^{2}\). This indicates the GLS reliability that relies on the measuring accuracy of the physical networked dynamics for information encryption and decryption. Then, it is observed that with the increase of the jamming rates, the communication performance becomes worse. For example, at the point \(\sigma^{2}=3.16\times 10^{-5}\), the BER increases from an order of \(10^{-5}\) to \(10^{-2}\) when jamming rate increases from 0 s\({}^{-1}\) to 500 s\({}^{-1}\). This is because (i) the burst jamming injection degrades the linear-dependant property of the dynamics among the (Tx, relays, Rx) nodes, and (ii) with such increases of the jamming inputs, deteriorates the successful decryption rate at Rx.  
In Figure 6, we provide the statistical distribution of (Tx,Rx) pairs with respect to different BER regions, i.e., <\(10^{-3}\), between \(10^{-3}\) and \(10^{-1}\), and >\(10^{-1}\)), under different levels of the jamming rate from the active attackers. It is seen that, given the fixed physical measurement noise variance, with the increases of the attacker's jamming rate, the ratio of (Tx,Rx) pairs belonging to the low BER regions decreases. For instance, when measurement noise variance has an order such as \(\sigma^{2}=10^{-5}\), the ratio of (Tx,Rx) with no jamming belonging to BER \(<10^{-3}\) approaches 1, larger than those affected by jamming input (i.e., with jamming rate 10 s\({}^{-1}\), 100 s\({}^{-1}\), 200 s\({}^{-1}\) and 500 s\({}^{-1}\)). Nevertheless, with medium jamming rates and sensitive physical measurement noise regions, as is depicted in Figure 6 (i.e., jamming rate <=100 s\({}^{-1}\) and \(\sigma^{2}<=10^{-4}\)), the proposed GLS can still approach the 7 overhead hard-decision FEC limit (i.e., BER \(\approx 4.5\times 10^{-3}\)) [42], indicating an error-free BER performance with proper FEC codes. This statistical result, combined with the previous average result in Figure 5, demonstrates the encryption robustness and reliability of the proposed GLS to secure wireless communications against potential active attackers. It is also noteworthy that the results, although from the networked dynamics of the IEEE 39-Bus system, are able to show the general applicability of our GLS scheme for securing other IoT systems with correlated underlying networked dynamics, as the encryption steps/algorithm are irrelevant with the specific IEEE 39-Bus dynamic model, i.e., Equations (26)-(30).  
Figure 5: GLS performance with active attackers: Average BER versus physical measurement noise with different levels of jamming rate.