Almost decouplability of any directed weighted network topology

• Almost any weighted directed network can be transformed into a similar form composed of uncoupled vertices.
• The analysis of many problems on dynamic networked systems can be simplified, especially the stability problems.
• The consensus problem of systems with time-varying network topologies is addressed as an example.

This paper introduces a conception that any weighted directed network topology is almost decouplable, which can help to transform the topology into a similar form composed of uncoupled vertices, and thus reduce the complexity of analysis for networked dynamical systems. As an example of its application, the consensus problem of linear multi-agent systems with time-varying network topologies is addressed. As a result, a necessary and sufficient condition for uniform consensus is proposed.