Complex Brain Networks: A Graph-Theoretical Analysis

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fMRI data

Time series data

Adjacency

Matrix

Connectivity

Matrix

Functional Brain Network

Parcellation

Binarized

.. .

. .

. .

. . .^.

......

FIGURE 9.1

Functional brain network construction.

9.3

Analysis Parameters

We review basic parameters used in the analysis of brain networks which

provide information on the local or global network structures, and in many

cases, global network structures may be deduced from the local ones.

9.3.1

Density and Degree Distribution

The density of a graph shows how well it is connected; s sparse graph has very

few connections between its nodes in the order of O(n) where n is the number

of nodes. A dense graph on the other hand has edges in the order of O(n²).

Definition 9.1 (graph density) The density of a graph G denoted ρ(G) is

the ratio of the number of its edges to the maximum possible number of edges

in G as below.

ρ(G) =

2m

n(n −1)

(9.1)

where ρ(G) is between 0 and 1. The sum of degrees in an undirected graph

G is 2m, therefore, the average degree of G, deg(G), is 2m/n resulting in the

modification of Eqn. 9.1 as in Eqn. 9.2. The density of the graph of Figure 9.2

is 0.52 which means almost half of all possible edges exists in this graph.

ρ(G) = ^deg(G)

(n −1)

(9.2)

The degree distribution of a graph is another measure which shows the

percentage of vertices of a given degree.