Fractional centralities on networks: Consolidating the local and the global

We propose a new centrality incorporating two classical node-level centralities, the degree centrality and the information centrality, which are considered as local and global centralities, respectively. These two centralities have expressions in terms of the graph Laplacian $L$, which motivates us to exploit its fractional analog $L^{gamma}$ with a fractional parameter $gamma$. As $gamma$ varies from $0$ to $1$, the proposed fractional version of the information centrality makes intriguing changes in the node centrality rankings. These changes could not be generated by the fractional degree centrality since it is mostly influenced by the local aspect. We prove that these two fractional centralities behave similarly when $gamma$ is close to $0$. This result provides its complete understanding of the boundary of the interval in which $gamma$ lies since the fractional information centrality with $gamma=1$ is the usual information centrality. Moreover, our computation for the correlation coefficients between the fractional information centrality and the degree centrality reveals that the fractional information centrality is transformed from a local centrality into being a global one as $gamma$ changes from $0$ to $1$.