# Robustness of Cyber-Physical Supply Networks in Cascading Failures
## 1 Introduction  
The physical and digital worlds are becoming continuously more intertwined, bringing about cyber-physical supply networks with emergent interactions [1]. The physical supply network depends on the cyber supply network for its control, and the cyber supply network depends on the physical network for flow information. While the management of cyber-physical supply networks is a challenging problem [2], these interdependent systems tend to be fragile against failures, hazards, and attacks [3]. Due to functional interdependency, a failure of firms in one network results in a failure of dependent firms in other networks, which may induce further damage to the first network and so on [4]. The failures can trigger multiple parts of supply networks, influencing the performance and viability of the components of the entire network [5]. The cyber-physical supply network can be modeled as an interdependent network to indicate the complex interdependencies of its subsystems and components [5, 6]. The subsystems and components can be signified as nodes, and the dependencies can be represented as links. For example, nodes can denote firms in the physical supply network, and links can denote conveyance mechanisms [7]. With the context of CPS and complex networks, failure can be defined as a form of deadlock, where all firms of the network are halted while waiting for products, and the complex interdependencies between the firms of the network make freeing the deadlock difficult [8, 9]. A cascading failure process is one in which the failure of one or more properties in a network (links/nodes) can trigger the failure of other parts of the network [10]. Overloading failure will prevent the transmission of data package information and lead to a decrease in the efficiency of the cyber supply network [11]. The overload phenomenon implies that data flow exceeds the node's capacity in the cyber supply network. In this case, data will be transmitted by the node closest to the affected node with sufficientcapacity [12]. If the adjacent load does not exceed the capacity, the cyber supply network usually operates. If not, the adjacent nodes will overload, leading to further redistribution of load and the accumulation of cascading failures. Unlike overload failures, failures of firms in the physical supply network result from underloading. When firms cannot fulfill the expected production requirement to overcome the fixed production costs, they will fail to gain profit and possibly exit the market [13]. A firm's failure can decrease the product demand from upstream firms, which may force the upstream firms to stop production. Moreover, a firm's failure can decrease material supply for downstream firms, which may force downstream firms to shut down. For example, during the COVID-19 pandemic, the inability of suppliers to provide a diverse set of resources to complex networks of organizations led to large parts of the supply chain becoming deadlocked [8]. Therefore, an underload cascading failure model is more suitable for the physical supply network [14]. The robustness of the cyber-physical supply network is usually defined as the relative size of the firms that survive the cascading failures [11]. Our goal was to construct a cascading failure model that can quantify the robustness of the cyber-physical supply network to provide a scientific basis for the development of network protection. Considering that the cascading failure is triggered by overloading in the cyber supply network and provoked by underloading in the physical supply network, this paper tried to answer the following questions:  
RQ1: How can we find a certain region of the parameter space where cascade failures occur under cyber node failure? RQ2: How can we find a certain region of the parameter space where cascade failures occur under physical node failure? This study makes two main contributions.