Fractional Graph Fourier Transform Centrality And Its Application to Social Network

Node centrality analysis is fundamental in studying complex networks as it allows for the identification of crucial nodes within a network. In recent years, the graph Fourier transform (GFT) centrality has been used for pinpointing influential individuals in social networks, key authors in citation networks, and pivotal stations in metro networks. The GFT centrality (GFTC) is a measure to identify the node's influence within a network. We introduced fractional graph Fourier transform centrality (FrGFTC) by extending the concept of GFTC to enable a versatile node importance measure. This allows for a nuanced analysis of network structures by adjusting the fractional order values in the fractional graph Fourier transform (FrGFT). The main tasks undertaken in this study are outlined below: Firstly, the FrGFT is defined and computed using a matrix function based on the eigenvalues and eigenvectors of the graph Fourier transform matrix. Secondly, FrGFTC is introduced by applying the fractional GFT to signals derived from the shortest path distance between network nodes. Notably, the conventional GFTC is revealed as a special case within the proposed FrGFTC framework. Finally, FrGFTC is employed to identify influential persons in the social network. The top two or three persons are determined by varying fractional orders of FrGFT. Comparative analyses with other node centrality metrics are conducted to showcase the efficacy of the proposed FrGFTC measures in network analysis.