  Investigating the network stability or synchronization dynamics of
multi-agent systems with time delays is of significant importance in numerous
real-world applications. Such investigations often rely on solving the
transcendental characteristic equations (TCEs) obtained from linearization of
the considered systems around specific solutions. While stability results based
on the TCEs with real-valued coefficients induced by symmetric networks in
time-delayed models have been extensively explored in the literature, there
remains a notable gap in stability analysis for the TCEs with complexvalued
coefficients arising from asymmetric networked dynamics with time delays. To
address this challenge comprehensively, we propose a rigorously geometric
approach. By identifying and studying the stability crossing curves in the
complex plane, we are able to determine the stability region of these systems.
This approach is not only suitable for analyzing the stability of models with
discrete time delays but also for models with various types of delays,
including distributed time delays. Additionally, it can also handle random
networks. We demonstrate the efficacy of this approach in designing delayed
control strategies for car-following systems, mechanical systems, and deep
brain stimulation modeling, where involved are complex-valued TCEs or/and
different types of delays. All these therefore highlight the broad
applicability of our approach across diverse domains.
