Global Measures: Integration, Segregation and Complexity

In this section, we start by examining the Path Length as a measure of integration in brain networks and its difference across groups of healthy and schizophrenic patients. Figure3.4

suggests that Path Lengths are relatively larger for lower densities since in sparse networks, it is more likely that the shortest path between two nodes to be relatively large. Conversly, in highly connected networks (larger densities) it is more likely that two nodes are directly connected via a lower number of nodes and edges. In such networks, the level of integration is higher as the path lengths are relatively low. Complimentary to path length, we also measure the distance between the path length of each subject and its corresponding value in a randomised network. Therefore, values close to one suggest higher similarity between the empirical and randomised Path Lengths. The first row of Figure3.4suggests that the both non-normalised and normalised Path Lengths are identical across groups.

The results in the second row of Figure 3.4, suggest that in lower densities, the clustering coefficient is relatively low. A lower level of clustering coefficient means that networks exhibit a lower level of segregation. However, as the density grows, the clustering coefficient also increases. In higher densities, the networks are highly connected and it is more likely that three nodes form a triangle around each other. In addition to the clustering coefficient, we also show that the normalised clustering coefficient,Ω, which is analogous to the distance between the mean clustering coefficient of an empirical network and mean Clustering coefficients of the randomised networks. Normalised clustering coefficients, suggest a higher value in sparse networks as the randomised networks have greater free- dom in each realisation to dilute the segregated structures of the networks. As the density grows, the normalised clustering coefficient approaches one which suggests that the level of segregation in empirical networks is similar to randomised networks for higher densities.

For non-normlised clustering coefficients, the changes between the clustering coef- ficient of healthy and schizophrenic brain networks is not consistent over density. For low densities (5% to 20%) there is no significant difference between the two groups, however, above 20%, the Clustering coefficients in healthy brain networks suggest a higher value. Conversely, the normalised clustering coefficients suggest higher values in Schizophrenic brain networks which is statistically significantly across the majority of densities.

Another measure of the segregation which we discuss here is the modularity index

Q. We suffice to give a brief discussion of the modularity index, as a global measure of seg- regation, in this section, however, an in depth discussion regarding the difference between modular structures of two group is presented in section3.3.6. The Modularity index, similar to other segregation measures, shows a high value on sparse densities, however, as the den- sity grows the modularity index is decreased. For highly, or fully, connected networks the modularity index is expected to be approximately zero as it seems to be impossible to find a

modular structure in a fully connected lattice. It is notable that statistical inferences failed to provide any significant differences between modular structure of healthy and schizophrenic brain networks.

Eventually, we investigate the small-worldness as a measure of complexity. Results of small-worldness across groups suggest that the measure has larger values on sparse net- works. However, as the density grows the small-worldness is decreased until it settles to one where the segregation and integration of the empirical network is equal to the randomised network. It is worth noting, that one important condition for a network to be small-world is that its normalised path length stays close to one which is not the case for very sparse densities. For instance, for a 5% density, the normalised path length is≈ 1.4. Therefore, it is unlikely that the network with such a sparse density truly exhibits small-world be- haviour. However, the slope of decrease in the normalised path length is greater close to the cost-efficient density, the normalised path length across both groups reaches≈1.15.

Figure 3.4:Global measures of segregation, integration and complexity. Panel A. shows the global measures for the level of integration: path length (Left) and normalised path length (Right). Panel B. shows global measures for segregation: clustering coefficient (Left) and normalised clustering coefficient (Right) and maximised modularity index (Bottom) Panel C. shows the small-world index as a measure of complexity

Results of the small-worldness curve suggest that brain networks of Schizophrenic patients are more likely to exhibit small-world behaviour since for the majority of densi- ties, including the cost-efficient density, the small-world statistics are significantly larger in healthy subjects. These differences between the small-world index may have been caused by the difference, previously discussed, in the normalised clustering coefficients of healthy and Schizophrenia subjects which later were reflected in the results of the small-worldness.