# Robustness of Cyber-Physical Supply Networks in Cascading Failures
## 3 The Theoretical Model of Cyber-Physical Supply Networks  
This section is divided by subheadings and provides a precise description of the experimental results, their interpretation, and the experimental conclusions that can be drawn.  
As the interdependencies between the physical supply network and cyber supply network give rise to multiple possible failure spreading channels, a firm's failure can rashly influence its associated predecessor and successor firms and initiate a cascade of  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Failure** & **Cascading Failure** & \multicolumn{2}{c}{**Supply Network Domain**} & & \\ \hline
**Node Failure** & **Overload** & **Underload** & \
\begin{tabular}{c} **Cyber Supply** \\ **Network** \\ \end{tabular} &
\begin{tabular}{c} **Physical Supply** \\ **Network** \\ \end{tabular} & **Robustness Metrics** & **Work** \\ \hline ✓ & ✓ & & ✓ & ✓ & Comprehensive effectiveness index & [15] \\ ✓ & ✓ & & ✓ & ✓ & Network efficiency and percentage & [16] \\ ✓ & ✓ & & ✓ & ✓ & of unserved nodes & [17] \\ ✓ & ✓ & & ✓ & ✓ & Fraction of surviving nodes & [18] \\ ✓ & ✓ & & & ✓ & Number of surviving nodes & [18] \\ ✓ & & ✓ & & ✓ & Network efficiency & [14] \\ ✓ & & ✓ & & ✓ & Fraction of failed nodes & [13] \\ ✓ & ✓ & & & ✓ & Dynamic network load entropy & [20] \\ ✓ & ✓ & ✓ & ✓ & ✓ & Number of surviving nodes & This work \\ \hline \hline \end{tabular}
\end{table}
Table 1: Summary of the literature review.  
firm failures that can imperil the supply networks' operation. The load-capacity model in interdependent supply networks is built to duplicate the catastrophic propagation process. A comprehensive description of the proposed cyber-physical supply networks is presented in this section, as shown in Figure 1. The symbols used in the model are explained in Table 2.  
\begin{table}
\begin{tabular}{c c} \hline
**Notation** & **Meaning** \\ \hline \(G^{p}\) & Physical supply network \\ \(G^{c}\) & Cyber supply network \\ \(V^{p}\) & Set of nodes in the physical supply network \\ \(V^{c}\) & Set of nodes in the cyber supply network \\ \(p^{p}_{i}\) & A node \(i\) in the physical supply network \\ \(v^{c}_{i}\) & A node \(i\) in the cyber supply network \\ \(EP\) & Set of links in the physical supply network \\ \(e^{p}_{ij}\) & A directed connection from node \(v^{p}_{i}\) to node \(v^{p}_{j}\) in the physical supply network \\ \(e^{c}_{ij}\) & A connection from node \(v^{e}_{i}\) to node \(v^{e}_{j}\) in the cyber supply network \\ \(EC^{c}\) & Set of links in the cyber supply network \\ \(E^{pc}_{kj}\) & Set of dependency links connecting nodes between network \(G^{p}\) and network \(G^{c}\) \\ \(W^{p}\) & A weighted adjacency matrix of physical supply network \\ \(w^{p}_{ij}\) & The weight of the link \(e^{p}_{ij}\) \\ \(W^{c}\) & A weighted adjacency matrix of the cyber supply network \\ \(w^{c}_{ij}\) & Weight of link \(e^{c}_{ij}\) \\ \(a\) & Tunable parameter used to adjust the initial load in the physical supply network \\ \(\beta\) & Upper-bound capacity parameter of the node in the physical supply network \\ \(\gamma\) & Lower-bound capacity parameter of the node in the physical supply network \\ \(\theta\) & Weight parameter of the link in the physical supply network \\ \hline \end{tabular}
\end{table}
Table 2: Symbols used in this paper and their corresponding meanings.  
Figure 1: System model illustration for the cyber-physical supply networks, where network \(G^{p}\) is the physical supply network, and network \(G^{c}\) is the cyber supply network. Interdependence across the two networks is realized by the one-to-one support links shown by dash lines.