Computing Graph Fourier Transform Centrality with Graph Filter Designed by Hermite Polynomial

In this paper, we introduce the Hermite polynomial as a tool for computing graph Fourier transform centrality (GFTC), facilitating the identification of crucial nodes within a network. First, we begin by providing the definition of GFTC and delve into the computation process using conventional eigen-decomposition method. Second, recognizing the computational complexity inherent in eigen-decomposition, the paper explores the application of a graph filter method. This method transforms the spectral-domain task into a more manageable vertex-domain task. Third, to achieve an optimal spectral response, the Hermite polynomial is employed in designing the graph filter. This ensures that the desired spectral characteristics are met. The closed-form solution for filter coefficients is then derived by leveraging the orthogonal relation of the Hermite polynomial. Fourth, an implementation structure is derived by using the recursion relation of Hermite polynomial. This structure facilitates the distributed computation of GFTC. Finally, the proposed method's efficacy is exemplified through its application to identify important stations within the Taipei metro network. This practical demonstration serves to underscore the utility and effectiveness of the presented method.