# Graph Theory Approach to the Vulnerability of Transportation Networks
## 2 Materials and Methods
### Current Indicators for Transportation Network Description  
[PERSON] proposed in the 1960s [20; 26]--through using graph theory to describe transport networks--several indicators describing transport networks with two levels of detail. The first one is related to determining parameters or indicators about the whole network, and the others concerning the nodes. The main objective of their applicability is that they provide methods to [20]* Express of the relationship between the values of these parameters and the structure of the network;
* Comparing the different transportation networks at certain points in time;
* Comparing the development of transport networks at different times.  
According to the above, the well-known graph theory parameters describing the transportation networks are as follows [20; 26]:  
1. Network level:  
* Cost, which represents the total length of the network measured in real transport distances.
* Detour index is a measure of the efficiency of a transport network in terms of how well it overcomes distance or the friction of distance.
* Network density measures the territorial occupation of a transport network in terms of km of links per square kilometers of surface.
* Pi index describes the relationship between the total length of the graph and the distance along its diameter.
* Eta index describes average length per link.
* Theta Index measures the function of a node, which is the average amount of traffic per intersection.
* Alpha index is a measure of connectivity which evaluates the number of cycles in a graph in comparison with the maximum number of cycles.
* Beta index is a measure of the level of connectivity in a graph and is expressed by the relationship between the number of links \(e\) over the number of nodes \(v\).
* Gamma index is a measure of connectivity that considers the relationship between the number of observed links and the number of possible links.
2. Node level:  
* Order (degree) of a node (o). The number of its attached links; it is a simple, but effective measure of nodal importance.
* Koenig number (or associated number, eccentricity) is a measure of fairness based on the number of links needed to reach the most distant node in the graph.
* Hub dependence is a measure of node vulnerability that is the share of the highest traffic link in total traffic (weighted degree).
* Average nearest neighbors degree is a measure of neighborhoods indicating the type of environment in which the node situates.
* Cohesion index (ci) for a given edge \(\{u,v\}\): this index measures the ratio between the number of common neighbors connected to nodes \(u\) and \(v\) and the total number of their neighbors.
* Within-module degree (z-score) describes how well connected a node is to other nodes in the same module (or cluster, community).
* The participation coefficient allows one to compare the number of links (order, degree) of node \(u\) to nodes in all clusters with its number of links within its own cluster.
* Shimbel index (or Shimbel distance, nodal accessibility, nodality) is a measure of accessibility representing the sum of the length of all shortest paths connecting all other nodes in the graph.
* Betweenness centrality (or shortest-path betweenness) is a measure of accessibility that is the number of times a node is crossed by shortest paths in the graph.