# Graph Layer Security: Encrypting Information via Common Networked Physics
## 2 Physical Dynamic Model for Data Encryption  
In this section, we introduce the physical networked dynamic model for further encryption of wireless communications. The networked dynamics are the time-varying networked signals in an IoT system (e.g., the electricity flows of the smart grid system, or a time-varying compound concentrations in the water distribution network). Such signals are irrelevant with the wireless communication aspects (i.e., the dynamic patterns in a smart grid system or water distribution network do not depend on or affect the communication channels and environment). The networked dynamic is described by the underlying network topology and the dynamic signal flow over it. The network topology is configured by a static graph, denoted by \(\mathcal{G}(\mathcal{N},\mathbf{W})\). Here, \(\mathcal{N}=\{1,\cdots,N\}\) representsa set of total node indices. **W** of size \(N\times N\) is the adjacent matrix, in which the \((m,n)\)th element \(w_{m,n}\in\{0,1\}\) reflects the existence of directed link from node \(n\) to \(m\).  
Given the topology of the network, the dynamic signal over each node evolves in accordance with its self-dynamic and the coupling interactions from its neighbouring (connected) nodes. We denote the signals for all \(k=1,\cdots,K\) (\(K\in\mathbb{N}^{+}\)) discrete time as a matrix **X** of size \(N\times K\), and the evolution function from discrete time \(k\) to \(k+1\) as \(\textbf{F}:\mathbb{R}^{N}\rightarrow\mathbb{R}^{N}\). Then, the networked dynamic model is expressed as  
\[\textbf{X}_{,k+1}=\textbf{F}(\textbf{X}_{,k})+\textbf{b}_{k}, \tag{1}\]  
where \(\textbf{X}_{,k}\) represents the \(k\)th column of **X**.  
In Equation (1), \(\textbf{b}_{k}\) is the perturbations that account for the randomness of the networked dynamics. For example, in a smart grid system (in our case-study), \(\textbf{b}_{k}\in\mathbb{R}^{N}\) represents the electricity usages on \(N\) nodes at \(k\) time-index. Such usages by the users are random, and thereby will make the whole networked dynamics in Equation (1) random. Here, we specify \(\textbf{b}_{k}\) by the compositions of the unknown injection amplitude \(\textbf{b}_{k_{i}}\) of size \(N\times 1\) with respect to a random injection time (i.e., \(k_{i}\)), governed by the Dirac delta function, i.e.,  
\[\textbf{b}_{k}=\sum_{k_{i}\in\mathbb{N}^{+}}\textbf{b}_{k_{i}}\cdot\delta(k- k_{i}). \tag{2}\]  
In Equation (2), both the injection amplitudes and time are unknown, and may have different distributions for different dynamic systems. For this work, we do not rely on the exact distribution of the perturbations, but assume its sparse appearance (e.g., the contaminant injection into a water distribution network, or the steep variations of the power usages in an electrical bus system are sparse). As such, the combination of the networked evolution model \(\textbf{F}(\cdot)\) and the random perturbation will provide the dynamic dependency as well as the randomness to secure the dynamic irrelevant wireless communications, and is hard to guess by brute force, unless the total underlying network dynamics are hacked. We will then elaborate how to pursue information encryption and decryption using the graph layer dynamics.