Theoretical Analysis Theory

In the field of quantitative neuroimaging, graph theoretical analysis is part of methods oriented to study brain connectivity. In particular, graph theoretical analysis belongs to a group of methods based on covariance patterns (networks) identification, in that sense, it is similar to principal component analysis and seed analysis.
A primary concept of graph theoretical analysis is the notion of topology. Figure x illustrates this concept with a simple idea which is used when we travel on the subway of any large city, for example, the city of London. In figure x two maps of the London subway appear, the first map physically shows the rails (or lines) through which the trains travel (i.e., the subway topography), and the second shows the subway

By simply inspecting the topological maps of the subway of both cities, it is not easy to know if one subway is better organized than the other (i.e., subway efficiency). One way to simplify this problem is to use metrics that quantify or analyze the graph. Hence the concept of graphic theoretical analysis.
A first aspect to measure could be how easy is to travel between any two stations (e.g. the number of stations on average). 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 (or topological map). The metric of global efficiency is a way to quantify the global connectivity (network integration) of the subway network (or

For example, if this particular station only has a connection with another station, then if the station were to be out of service due to a fault, the neighbor station would be overloaded with passengers because in principle the passengers coming from the station out of service would be added. In contrast, if this particular station has connections with many of the other neighboring stations, the passengers have several options to reconnect. These two examples illustrate the concept of local efficiency. In the first case, the local efficiency of the station is low and in the second case is high. Thus, the metric of local efficiency is a way to quantify the local connectivity (network segregation) of the subway