1. Measures

Several measure and indices can be used to analyze the network efficiency. Many of them were initially developed by Kansky [Kansky, 1963] and can be used for:

• Expressing the relationship between values and the network structures they represent.
• Comparing different transportation networks at a specific point in time.
• Comparing the evolution of a transport network at different points in time.

Outside the number of nodes and edges, three basic measures are used to define the structural attributes of a graph; the diameter, the number of cycles and the order of a node.

Diameter (d). The length of the shortest path between the most distanced nodes of a graph is the diameter. d measures the extent of a graph and the topological length between two nodes.

The diameter enables to measure the development of a network in time. The higher diameter, the less linked a network tends to be. In the case of a complex graph, the diameter can be found with a topological distance matrix (Shimbel distance), which computes for each node pair its minimal topological distance. Graphs which extent remains constant, but with a higher connectivity, have lower diameter values.

Number of Cycles (u). The maximum number of independent cycles in a graph. This number (u) is estimated through the number of nodes (v), links (e) and of sub-graphs (p); u = e-v+p.

Trees and simple networks have a value of 0 since they have no cycles. The more complex a network is, the higher the value of u, so it can be used as an indicator of the level of development and complexity of a transport system.

Order (degree) of a Node (o). The number of its attached links and is a simple, but effective measure of nodal importance. The higher its value, the more a node is important in a graph as many links converge to it. Hub nodes have a high order, while terminal points have an order that can be as low as 1. A perfect hub would have its order equal to the summation of all the orders of the other nodes in the graph and a perfect spoke would have an order of 1.

2. Indexes

Indexes are more complex methods to represent the structural properties of a graph since they involve the comparison of a measure over another.

Detour Index. A measure of the efficiency of a transport network in terms of how well it overcomes distance or the friction of distance. The closer the detour index gets to 1, the more the network is spatially efficient. Networks having a detour index of 1 are rarely, if ever, seen and most networks would fit on an asymptotic curve getting close to 1, but never reaching it.

For instance, the straight distance (DD) between two nodes may be 40 km but the transport distance (TD; real distance) is 50 km. The detour index is thus 0.8 (40 / 50). The complexity of the topography is often a good indicator of the level of detour.

Network Density. Measures the territorial handhold of a transport network in terms of km of links (L) per square kilometers of surface (S). The higher it is, the more a network is developed.

Pi Index. The relationship between the total length of the graph L(G) and the distance along its diameter D(d). It is labeled as Pi because of its similarity with the real Pi (3.14), which is expressing the ratio between the circumference and the diameter of a circle. A high index shows a developed network. It is a measure of distance per units of diameter and an indicator of the shape of a network.

Eta Index. Average length per link. Adding new nodes will cause a decrease of Eta as the average length per link declines.

Theta Index. Measures the function of a node, that is the average amount of traffic per intersection. The higher theta is, the greater the load of the network.

Beta Index. Measures 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). Trees and simple networks have Beta value of less than one. A connected network with one cycle has a value of 1. More complex networks have a value greater than 1. In a network with a fixed number of nodes, the higher the number of links, the higher the number of paths possible in the network. Complex networks have a high value of Beta.

Alpha Index. A measure of connectivity which evaluates the number of cycles in a graph in comparison with the maximum number of cycles. The higher the alpha index, the more a network is connected. Trees and simple networks will have a value of 0. A value of 1 indicates a completely connected network. Measures the level of connectivity independently of the number of nodes. It is very rare that a network will have an alpha value of 1, because this would imply very serious redundancies.

Gamma Index (g). A measure of connectivity that considers the relationship between the number of observed links and the number of possible links. The value of gamma is between 0 and 1 where a value of 1 indicates a completely connected network and would be extremely unlikely in reality. Gamma is an efficient value to measure the progression of a network in time.