Segregation Measures

The human brain is also known to be, functionally and structurally, a segregated network which leads the group of nodes to be highly intra-connected rather than inter-connected (Sporns,2011b). This group of intra-connected nodes leads to the formation of cliques and modules within the network. In this section, we use the metrics of the clustering coefficient and modularity as two commonly used tools for measuring the level of segregation in the human connectome.

Clustering Coefficient and Local Efficiency

Clustering coefficient measures, the prevalence of neighbours of two connected nodesni

andnj also happens to be connected with each other (Watts and Strogatz,1998;Newman, 2003a). In other words, the clustering coefficient is a ratio of the number of complete

triangles around a node over the number of all triangles that can be formed around a node. the number of triangles around nodeiin a binary network can be obtained as:

ti = 1 2 X j,h∈N ai jaihajh (3.2)

wherehand jare nodei’s neighbours. Therefore, the clustering coefficient for nodeican be mathematically quantified as:

Ci = 1 n X i∈N 2ti ki(ki−1) (3.3)

wherekiis number of connections to nodei. The mean clustering coefficient is the average

of clustering coefficient across all available nodes within a network which represents the level of segregation in a network. However, the mean clustering coefficient can be influ- enced by isolated nodes or leaves in a network (Kaiser, 2008). Although the likelihood of leave and isolated nodes is quite low in a human connectome, to ensure that the mean clustering coefficient is not underestimated, we discard the leave or isolated nodes in mean clustering coefficients.

Similar to the clustering coefficient,Latora and Marchiori(2001) proposed a mea- sure of local segregation, called local efficiency, which can be formulated as (Rubinov and Sporns,2010): Ei = 1 n X i∈N P i,h∈N,j_,iai jaih(djh(Ni))−1 ki(ki−1) (3.4) wheredjh(Ni) is number of edges of the shortest path between nodes jandhthat are only

one edge away from nodei. It is important to note that although the clustering coefficient and local efficiency are similar, they are not identical as the former measures the probability of formation of a triangle around nodei, while the latter measures the topological distance between two neighbours of nodei.

Modularity

Another sign of the presence of segregation in a network is modular structures. Group of nodes, which are tightly intra-connected with each other but, at the same time, relatively less inter-connected to the rest of the network are considered as a ‘module’ (Newman,2010). Detecting these modular structures in real-life networks is almost impossible without algo- rithmic methods. Recently, a large number of studies proposed a wider range of techniques to detect the modular structures in a network. These techniques mostly use heuristic meth- ods to find a combination of nodes where the group of nodes has greater intra-connected

edges than inter-connected edges (Fortunato, 2010). In this section, we discuss one of the most well-known methods for detecting non-overlapping modularity detection methods called multi-scaleModularity Maximisationmethod (Newman, 2006). This method mea- sures the level of segregation in a network by two factors. First, module assignments that indicate which module each of the nodes belongs to. Second, the modularity index which, ranging from -1 to 1, indicates to what extent a network is modular. Larger the absolute modularity index, closer the network is to the perfect modular network. The modularity index can be defined as:

Q=X

i j

Bijδ(σi, σj) (3.5)

where Bij is known as the ‘modular matrix’ andδ(σi, σj) is a delta function which is one

whenever module assignment of nodesiand jis the same, and zero otherwise. The most optimal combination of theδfunctions is obtained by Louvain method which performs a greedy search through the possibilities to find the one which maximisesQ(Blondelet al.,

2008). The modularity matrix, Bij, is defined as

Bij=fij− hkikj P iki i γ, (3.6)

wherefi jis one if an edge connects nodeito j(worth mentioning that fi jis binarised

form of the ai j), and zero otherwise. The last part of Eq. 3.6 is the probability of an

edge existing in an equivalent randomised network which preserves the degree of nodes but dilutes the modular structures. Eventually, a tunable scaler,γ, scales this probability which is called the resolution parameter. When the ‘resolution parameter’ is set to a value smaller than one, it is likely that ai j exceeds its probability which produce larger communities.

Conversely, when the resolution parameters is set to a value larger than one, it is less likely thatai jexceeds the probability and this consequently results in a greater number of smaller

modules (Betzelet al.,2013).

However, it was shown that maximised modularity is not always scalable by large resolutions as above a certain value the algorithm subdivides the network into singleton modules (Lancichinetti and Fortunato, 2011; Fortunato and Barthelemy, 2007). To find the optimal resolution parameters, we apply modularity detection technique with different resolution parameters which range from −log₁₀10−1.5 to −log₁₀100.5 by increments of −log₁₀10−1.3(Betzelet al.,2013). For each increment, we obtain the modularity index for the empirical network as well as a 1k random network. Eventually, we choose the resolution parameter which maximises the difference between the modular structure of the empirical and a random networks.