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 $\gamma$, a well-known nonlocal operator. When $\gamma =1$, the fractional centrality reduces to the out-degree centrality. In contrast to undirected networks, where the measure remains predominantly local as $\gamma$ varies, we demonstrate that its intriguing behavior in directed networks reflects global effects beyond purely local contributions. We show that, in the limit $\gamma \to 0^{+}$, 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 $\gamma$ decreases from 1 to 0. These results establish the fractional centrality as a unifying framework integrating local and global influences in directed networks.

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• Dynamics of networks
• Network diffusion
• Random walks
• Complex networks
• Directed networks
• Centrality