# Graph Theory Approach to the Vulnerability of Transportation Networks
## 2 Materials and Methods
### Basic Concepts of Vulnerability in Transport  
The term _vulnerablity_ is defined as a physical feature or operational attribute that makes an individual open to using or vulnerable to a given threat [40; 41; 42; 43; 44; 45; 46; 47; 48; 49; 50; 51]. However, in terms of reliability, it can be measured as the probability that the system can be in a critical state or worse in less time than expected. This can be determined by certain external factors that cause large adverse effects on other sensitive systems (consequences above a fixed level). In other words, this approach can be expressed as the inherent properties of the system or part of its assets, community, and environment that make them vulnerable to the harmful effects of natural and anthropogenic threats. In the literature in the field of transport [41; 42; 43; 44; 45; 46; 47; 48; 49; 50; 51], the concept of vulnerability is used to describe the consequences for travel in the transport system or transportation network, which may extend the travel time, alter the length of the road, or limit the availability of services. In the case of transport networks or transport systems in general, the restriction of access to services may be due to natural disasters or harmful human activities. When we consider the transportation network as a complex system, it is possible to use the methods for characterizing the vulnerability of this type of system. One way is to use topological indices derived from the algebraic connectivity of graphs; e.g., a graph's spectrum. This spectral approach is demonstrated and discussed, among others, in the paper [52]. Another successful method applicable to quantifying the degree of vulnerability risk is based on the introduction of the general vulnerability index \(V[S,D]\), presented in [53]. This method allows identifying the points of a network that are crucial to the functioning of the infrastructure network. The authors in [53] applied this method to a transportation system that represents the Boston subway, and they got the nodes and connections whose protection must be assumed as the first concern of any level of policy. The concept of vulnerability is naturally associated with the idea of _resilience_ of a system or network. The essence of this concept is the inherent ability of the system (network) to maintain or recover a dynamically stable state that allows it to continue operating after a severe accident caused by natural or anthropotechnical factors while continuing to occur [54; 55]. Hence, transport network resilience is defined as its ability to resume operations at a level similar to those before disruption occurred in nodes, arches, or in both sets of elements simultaneously. The less interference in capacity and fluidity, and the faster the network resumes regular operations, the higher its resilience. The strength of a transport network is influenced by its structure; in particular, its redundancy. The studies of vulnerability can be performed concerning two levels of detail of the graph representing the considered transportation network. Assuming a simple, undirected graph, it is possible analyzing the overall vulnerability of its topological structure to notice the occurrence of various types of disturbances in operation. When the model of a transport network is given by a vertex-weighted or edge-weighted graph, the level of details increases.