4.4. Introduction to graph theoretical analysis. The case of brain perfusion SPECT.
In the field of quantitative neuroimaging, graph theoretical analysis is one of the methods to study brain connectivity [93-95, 100, 101]. A key concept of this method is the notion of topology. This concept can be illustrated with a simple idea which is used when we travel in the subway of any large city. In figure 3 two maps of the London subway appear, the first map shows a precise spatial description of the railways (or lines) through which trains travel (i.e., the subway topography), whereas the second one is only concerned with the relative locations of subway stations and connecting lines (i.e., the subway topology). These two maps do not coincide with
By simply inspecting the topological maps, it is not easy to know if one subway is better organized than the other (e.g., subway efficiency). One way to simplify this problem is to use metrics that quantify or analyze the subway network using graph theory [94, 101]. Hence the name of graph theoretical analysis.
A first aspect to measure could be how easy it is to travel between any two stations (e.g. the number of stations on average, between the start and end of the trip). This aspect is relevant, especially if the traveler wants to visit different parts of the city on the same day. This example illustrates the concept of global efficiency of a graph [94, 101]. The metric of global efficiency is a way to quantify the global connectivity (integration) of the network. In this example, it is assumed that the number of stations (nodes) and lines (connectors) are the same in the two subways that are
they are also topological neighbor), but without direct connections with distant topographical nodes (i.e., a network with a high local but a low global efficiency) (Table 1). At the other extreme would be a random network, where direct connections between any two nodes are random. Thus, in a random network, the local efficiency is low and the global efficiency is high as compared to an ordered network. In the middle of these two extremes would be what is known as a complex network, in which there is a balance between global and local efficiencies so that both are relatively high [94, 101] (Table 1). This kind of networks is said to have a ‘small-world’ topology. The concept of 'small world' comes from the social sciences, and reflects the fact that two persons (nodes) who do not know each other are nevertheless connected by a relatively short chain of persons known to each other. A complex network is also characterized by the presence of hubs and high modularity [94,