Graph Fourier Transform Centrality (GFT-C)
Graph Fourier Transform Centrality
(GFT-C) is a spectral measure that evaluates node importance based on the Laplacian eigendecomposition of a graph [2]. The eigenvalues of the Laplacian \(L\) capture the global variation of a graph signal across nodes: eigenvectors associated with large eigenvalues vary rapidly across adjacent nodes, whereas those corresponding to small eigenvalues vary slowly.
For node \(i\), the GFT-C centrality \(c_{GFT-C}(i)\) is defined as
\[
c_{GFT-C}(i) = \sum_{l=1}^N e^{k λ_l} \left| \sum_{j=1}^N f_i(j) u_l(j) \right|,
\]
where \(λ_l\) and \(u_l\) are the \(l\)-th eigenvalue and eigenvector of \(L\), \(k\) is a parameter (Singh et al. suggest \(k=0.1\)), and \(f_i(j)\) quantifies the relative importance of node \(j\) with respect to node \(i\):
\[
f_i(j) =
\begin{cases}
1, & i = j, \\
\dfrac{1/d_{ij}}{\sum_{k \neq i} 1/d_{ik}}, & i \neq j,
\end{cases}
\]
with \(d_{ij}\) denoting the shortest-path distance between nodes \(i\) and \(j\).
GFT-C of node \(i\) can be interpreted as the weighted sum of the graph Fourier transform coefficients of the importance signal \(f_i\). GFT-C is extended to the
fractional graph Fourier transform centrality
(FrGFTC) in [3], where the eigenvector matrix \(U = [u_1, \dots, u_N]\) is replaced by \(U^α\), with \(α\) denoting the fractional order.