Effects of superstrong coupling on the consensus convergence rate in directed networks

Collective consensus is a fundamental dynamical process in networked systems. The convergence rate, which indicates how quickly the system achieves consensus, is a critical metric and is characterized by the Fiedler value (the smallest real part of nonzero Laplacian eigenvalues). Recent studies on the consensus convergence rate in directed networks have mainly focused on edge addition, while it remains unclear how superstrong coupling, where some edge weights tend to infinity, affects consensus. Here, we develop a reach-based compression method suitable for any directed graph to reveal the superstrong coupling properties of the original system through a dimension-reduced system. We derive the analytic expression of consensus convergence rate for an arbitrary subset of edges under superstrong coupling. Furthermore, we graphically represent this analysis with all two-node and three-node motifs — the basic structural units — which provide essential insights into more complex structural patterns. These results offer guidance for optimizing the consensus convergence rate of networked systems.