# GFT - Graph Fourier Transform Centrality

#### Definition

This method considers local (from importance signal) as well as global (from importance spectrum) properties of a node in a complex network. It uses $GFT$ coefficients of an importance signal corresponding to the reference node.It relies on the global smoothness (or variations) of the carefully defined importance signal corresponding to a reference node. The importance signal for a reference node is the indicator of how remaining nodes in a network are seeing the reference node individually.

The importance signal describes the relation of a reference node to the rest of the nodes in a network.The importance signal corresponding to reference node $n$ can be calculated as:

$$\hat f_n(\lambda_l)={\underset{i=1}{\overset{N}{\sum}}} f_n(i) u_l^*(i)$$

where $f_n(i)$ is the importance of the reference node $n$ with respect to node $i$.

$GFT-C$ of node $n$ indicated as $I_n$, which is given as:

$$I_n={\underset{l=0}{\overset{N-1}{\sum}}} w(\lambda_l)|\hat f_n(\lambda_l)|$$

where $w(\lambda_l)$ is the weight assigned to the $GFT$ coefficient corresponding to frequency $\lambda_l$.

The importance signal describes the relation of a reference node to the rest of the nodes in a network.The importance signal corresponding to reference node $n$ can be calculated as:

$$\hat f_n(\lambda_l)={\underset{i=1}{\overset{N}{\sum}}} f_n(i) u_l^*(i)$$

where $f_n(i)$ is the importance of the reference node $n$ with respect to node $i$.

$GFT-C$ of node $n$ indicated as $I_n$, which is given as:

$$I_n={\underset{l=0}{\overset{N-1}{\sum}}} w(\lambda_l)|\hat f_n(\lambda_l)|$$

where $w(\lambda_l)$ is the weight assigned to the $GFT$ coefficient corresponding to frequency $\lambda_l$.

#### References

- Singh R., Chakraborty A., Manoj B., 2017. GFT centrality: A new node importance measure for complex networks. Physica A: Statistical Mechanics and its Applications, 487, pp.185-195. DOI: 10.1016/j.physa.2017.06.018