Asynchronous implementation of distributed coordination algorithms: Conditions using partially scrambling and essentially cyclic matrices

Given a distributed coordination algorithm (DCA) for agents coupled by a network, which can be characterized by a stochastic matrix, we say that the DCA can be asynchronously implemented if the consensus property is preserved when the agents are activated to update their states according to their own clocks. This paper focuses on two central problems in asynchronous implementation of DCA: Which class of DCA can be asynchronously implemented, and which other cannot. We identify two types of stochastic matrices, called partially scrambling and essentially cyclic matrices, for which we prove that DCA associated with a partially scrambling matrix can be asynchronously implemented, and there exists at least one asynchronous implementation sequence, which fails to realize consensus for DCA associated with an essentially cyclic matrix.