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Bioinformatics of the Brain

shortest paths through node v.

V C(v) =



s̸=t̸=v

σst(v)

σst

(9.6)

9.3.4.3

Edge Betweenness Centrality

Analogous to vertex centrality, the importance of an edge e in a graph may be

determined by calculating the percentage of shortest paths that run through

e called betweenness centrality BC(e). A relatively higher BC(e) value than

other edges in a graph for an edge e means that this edge has a significant role

in carrying data among all nodes in the network. Formally, BC(e) is defined

as in Eqn. 9.7 where σst shows the number of shortest paths between all nodes

s and t other than node v, and σst(e) is the number of shortest paths through

edge e. This parameter may be used for clustering in biological networks since

clusters have a good probability of being connected with edges that have high

BC values and removing these edges provides a method of isolating clusters.

BC(e) =



s̸=t̸=v

σst(e)

σst

(9.7)

9.3.5

Network Models

Complex networks can be classified based on their topological properties as

follows:

• Random Networks: The basic assumption in forming these networks is

that an edge (u, v) between the vertices u and v has the probability

p = 2m/(n(n −1)). A random network is identified by a short average

path length and a clustering coefficient that is inversely proportional to the

size of the network [3].

• Small-world Networks: The distance between any two nodes in such a net-

work is small, providing fast data transfer among all the nodes [4]. The

diameter of a small-world network is proportional to log n where n is the

number of nodes in the network. Brain networks exhibit this property pro-

viding efficient data transfer between separate brain regions.

• Scale-free networks: This kind of complex networks is characterized by few

nodes with high degrees with the rest of the nodes having relatively low

degrees. Many biological networks such as protein interaction networks,

metabolic pathways and also complex brain networks exhibit this property.

The degree distribution of these networks can be specified in Eqn. 9.8 where

γ is known as the power-law exponent.

P(k) ≈k−γ, γ > 1

(9.8)