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

## TL;DR

This paper introduces a new method to measure and improve consensus convergence rates in complex multi-agent systems using a novel metric called the Hermitian Kirchhoff index.

## Contribution

The paper introduces the Hermitian Kirchhoff index as a novel metric for quantifying consensus convergence rates in multi-agent systems.

## Key findings

- The Hermitian Kirchhoff index is theoretically justified and relates to algebraic connectivity and eigenvectors.
- Adding or removing edges/arcs impacts the index, as shown through eigenvalue perturbation analysis.
- Numerical simulations confirm the effectiveness of the proposed approach.

## Abstract

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.