Research on the Consensus Convergence Rate of Multi-Agent Systems Based on Hermitian Kirchhoff Index Measurement

Multi-agent systems (MAS) typically model interaction topologies using directed or undirected graphs when analyzing consensus convergence rates. However, as system complexity increases, purely directed or undirected networks may be insufficient to capture interaction heterogeneity. This paper adopts hybrid networks as interaction topology to investigate strategies for improving consensus convergence rates. We propose the Hermitian Kirchhoff index, a novel metric based on resistance distance, to quantify the consensus convergence rates and establish its theoretical justification. We then examine how adding or removing edges/arcs affects the Hermitian Kirchhoff index, employing first-order eigenvalue perturbation analysis to relate these changes to algebraic connectivity and its associated eigenvectors. Numerical simulations corroborate the theoretical findings and demonstrate the effectiveness of the proposed approach.