Consensus problem has been recognized as being of importance in distribution coordination of dynamic agent systems,which is widely applied in distributed computing,management science,flocking/swarming theory,distributed control,and sensor networks.In the past decade,the stability analysis of consensus algorithms has been one of focal points in control theory and mathematics.Besides static network topology,in many real-world applications,the agents are moving.In this case,one must consider time-varying topologies under link failure or creation.In this talk,I would like to present our recent results on the stability analysis of consensus algorithms of networks of multi-agent systems with randomly switching topologies.Here the time-varying topologies are modeled as an adaptive process,which is rather general and includes i.i.d.and Markov chain as special case,as well as non-stationary and non-ergodic processes.As I show,for directed topologies,the property of spanning trees acrossing a length of time interval is located at the core position to study consensus dynamics.Furthermore,when the transmission delays occur,the condition above can also guarantee consensus if self-links are instantaneous.Moreover,under self-linking delays and when only integer delays with certain patterns are possible,we observe and prove that the algorithm does not reach consensus but instead synchronizes at a periodic trajectory,whose period depends on the delay patterns.