Fault-tolerant properties of scale-free linear protocols for synchronization of homogeneous multi-agent systems

Originally, protocols were designed for multi-agent systems (MAS) using information about the network which might not be available. Recently, there has been a focus on scale-free synchronization where the protocol is designed without any prior information about the network. As long as the network contains a directed spanning tree, a scale-free protocol guarantees that the network achieves synchronization. If there is no directed spanning tree then synchronization cannot be achieved. But what happens when these scale-free protocols are applied to such a network where the directed spanning tree no longer exists? This paper establishes that the network decomposes into a number of basic bicomponents which achieves synchronization among all nodes in this basic bicomponent. On the other hand, nodes which are not part of any basic bicomponent converge to a weighted average of the synchronized trajectories of the basic bicomponents. The weights are independent of the initial conditions and are independent of the designed protocol.