A distributed influence measurement algorithm in leader–follower networks

This study proposes a vector-wise step-sized consensus dynamics (VSCD) for distributed networks represented by positively weighted leader–follower graphs. Unlike traditional discrete consensus dynamics, VSCD employs node-specific vector step sizes, enabling faster convergence. We define an influence matrix in continuous consensus dynamics and extend it to a discrete influence matrix in VSCD, demonstrating equivalent convergence properties under specific vector step size conditions. To facilitate the application of VSCD in distributed networks, we analyze the maximum boundary vector step size conditions using graph-theoretic methods. Building on this formulation, we propose a fully distributed influence measurement algorithm (DIMA), which enables each node in a distributed network to determine its valid influence nodes and their corresponding influence using only local information, without requiring global parameters. The effectiveness and scalability of the proposed methods are validated through simulations.