A Geometric Approach to Stability Analysis of Asymmetric or Random Delayed Network Dynamics

Investigating network stability and synchronization of multi-agent systems (MASs) with time delays is crucial in real-world applications. This often involves solving transcendental characteristic equations (TCEs) from system linearization. While stability results for TCEs with real-valued coefficients induced by symmetric networks are well-studied, there is a gap for complex-valued coefficients arising from asymmetric networks. To bridge this gap, we propose a geometric approach by studying stability crossing curves in the complex plane. This approach applies to various delay types, including distributed delays, and is demonstrated effective in stability analysis of MASs with both deterministic and random networks.