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Six degrees of separation in multi-agent systems

Networked control systems pose the problem of choosing the communication structure of the controller with the aim to get an overall system with satisfactory performance. This paper shows that in multi-agent systems with leader-follower structure a small percentage p of all possible communication links suffice to ensure a quick reaction of all agents to set-point changes of the leader. It considers agents that are connected in a single path and choose additional communication links randomly with the probability p. The main result shows that for any p > 0 there is an upper bound of the delay with which any agent in a network of arbitrary size reacts on leader commands. This result appears as a consequence of the property to have short paths from the leader to all followers, which is known as “six degrees of separation” in large networks and which is extended here to networked control systems in which the nodes represent dynamical systems.