Impacts of Physical-Layer Information on Epidemic Spreading in Cyber-Physical Networked Systems

TL;DR

The paper addresses how physical-layer information influences epidemic spreading in cyber-physical networks by proposing a nonlinear multiplex with an information-propagation cyber layer (pairwise and 2-simplex interactions) and a physical-layer epidemic model (SIS). It derives the outbreak threshold using the Microscopic Markov Chain Approach (MMCA) and validates it against Monte Carlo simulations, demonstrating that physical-layer information raises the infection threshold $\beta_c$ and reduces infection density $\rho^I$. The study shows that 2-simplex information can mimic pairwise diffusion under higher simplex density, boosting awareness and suppressing outbreaks, and it confirms the model’s applicability to real networks, including the U.S. power grid. The findings highlight the pivotal role of integrating higher-order information diffusion and physical-layer sensing to control epidemic spread in cyber-physical systems, with practical implications for infrastructure like smart grids and security networks.

Abstract

Since Granell et al. proposed a multiplex network for information and epidemic propagation, researchers have explored how information propagation affects epidemic dynamics. However, the role of individuals acquiring information through physical interactions has received relatively less attention. In this work, we introduce a novel source of information: physical layer information, and derive the epidemic outbreak threshold using the Microscopic Markov Chain Approach (MMCA). Our simulation results indicate that the outbreak threshold derived from the MMCA is consistent with the Monte Carlo (MC) simulation results, thereby confirming the accuracy of the theoretical model. Furthermore, we find that the physical-layer information effectively increases the population's awareness density and the infection threshold $β_c$, while reducing the population's infection density, thereby suppressing the spreading of the epidemic. Another interesting finding is that when the density of 2-simplex information is relatively high, the 2-simplex plays a role similar to pairwise interaction, significantly enhancing the population's awareness density and effectively preventing large-scale epidemic outbreaks. In addition, our model works equally well for cyber physical systems with similar interaction mechanisms, while we simulate and validate it in a real grid system.

Figures (9)

• Figure 1: The proposed multiplex network. The upper layer is the cyber layer, simulating the propagation of epidemic information. Nodes can be in one of two states: Unaware (U) or Aware (A). The connections between nodes display the propagation of information, with purple lines indicating that unaware nodes can get informed by the aware nodes. The yellow triangular face denotes the propagation of information through the 2-simplex. The lower layer is the physical layer, describing the spreading of the epidemic, where nodes can be in one of two states: Susceptible (S) or Infected (I). The connections between nodes indicate physical contact, with red lines indicating contact between susceptible and infected nodes while black lines indicate contact between either susceptible or infected nodes. Dashed lines between the two layers indicate one-to-one matching of nodes, and the network itself is undirected and unweighted.
• Figure 2: Information propagation on the cyber layer. purple and green nodes indicate nodes in the A state and U state, respectively. In (a)-(d), node i receives information from its neighbors through pairwise interaction (link) with a probability of $\lambda$. In (e), node $i$ and its two neighbors form a 2-simplex (the triangular enclosure), but since only one neighbor is in the A state, the conditions for information propagation within the 2-simplex are not met. In (f), the 2-simplex is shown where node $i$ transitions to the A state with probability $\lambda$ as informed by its A-state neighbor, and under the influence of the 2-simplex, node $i$ may also become aware with probability $\lambda^{*}$. Additionally, at each time step, nodes in the A state may revert to the U state due to information loss, with a probability $\delta$, as shown in (g).
• Figure 3: Probability tree for transitions among four states (AS, AI, US, UI). The probability that an individual i in the A state does not get infected is denoted by $q_{i}^{A}$, and the probability that an individual in the U state does not get infected is denoted by $q_{i}^{U}$. $\delta$ denotes the probability that an individual in the A state forgets the information, and $\mu$ denotes the probability that an individual in the I state recovers. It is assumed that when individuals in the UI state contract the epidemic, they immediately transition to the AI state with probability 1.
• Figure 4: Comparison of the stationary density using MMCA and MC methods. The red solid line and the red dashed line illustrate the infection density $\rho^{I}$ and the awareness density $\rho^{A}$ obtained by the MMCA, respectively. The blue solid line with plus markers and the blue dashed line with plus markers display the infection density $\rho^{I}$ and the awareness density $\rho^{A}$ obtained by MC simulations, respectively. In the physical layer, the parameters are set as follows: the initial proportion of infected nodes is $1\%$, $\theta = 0.8$, $\alpha = 10$, and $\mu = 0.4$. In the cyber layer, the parameters are set as follows: the average number of 2-simplex each node is part of is $K_{S} = 2$, $\lambda = 0.1$, $\lambda^{*} = 0.1$, and $\delta = 0.8$. $\rho^{A} = \frac{\sum_{i}P_{i}^{A}}{N}, \quad \rho^{I} = \frac{\sum_{i}P_{i}^{I}}{N}$. The results are obtained from MC simulations, averaged over 100 iterations, with $\beta$ varying from 0 to 1 in increments of 0.02.
• Figure 5: Impact of the number of 2-simplex on awareness density and infection density. The red solid line and the red dashed line illustrate the infection density $\rho^{I}$ and the awareness density $\rho^{A}$ obtained by the MMCA, respectively. The blue solid line with plus markers and the blue dashed line with plus markers show the infection density $\rho^{I}$ and the awareness density $\rho^{A}$ obtained by MC simulations, respectively. In the physical layer, the initial proportion of infected nodes is 1%, $r_{i}^{(3)}(t)$ is not considered, and $\mu = 0.4$. In the cyber layer, $r_{i}^{(1)}(t)$ is not considered, $\lambda^{*} = 0.5$ and $\delta = 0.8$. In (a), (b), and (c), the average number of 2-simplex each node is part of is $K_{S} = 2$, $K_{S} = 4$, and $K_{S} = 12$, respectively. The results are obtained from MC simulations, averaged over 100 iterations, with $\beta$ varying from 0 to 1 in increments of 0.02.