Fractional degree centrality for directed networks

Degree centrality is one of the most fundamental measures of local importance in directed networks. Its fractional version is defined via the fractional directed Laplacian with parameter γ, a well-known nonlocal operator. When γ=1, the fractional centrality reduces to the out-degree centrality. In contrast to undirected networks, where the measure remains predominantly local as γ varies, we demonstrate that its intriguing behavior in directed networks reflects global effects beyond purely local contributions. We show that, in the limit γ→0^{+}, the fractional centrality is characterized by rooted spanning trees, thereby quantifying how frequently each node acts as a broadcaster rather than a sink and reflecting its role in global information dissemination across the network. Through experiments on both real-world and random directed networks, we demonstrate the transition of the fractional centrality from a local measure to a global one as γ decreases from 1 to 0. These results establish the fractional centrality as a unifying framework integrating local and global influences in directed networks.