Leader-Follower Consensus Over Finite Fields

In this paper, we investigate the leader-follower consensus of multi-agent systems over finite fields, which model agents with limited capacities for storing, processing, and transmitting the information, from the perspectives of the transition graph and the characteristic polynomial of the network matrix, respectively. By the features of dynamics over finite fields, we reveal that the transition graph of networks achieving the leader-follower consensus over finite fields is either a spanning in-tree topped at zero-state or is composed of spanning in-trees with the same structure, topped at steady states. To address the high time complexity associated with transition graphs and existing methods, we integrate the characteristic polynomial of the network matrix with the cycle and tree structures in the transition graph. As a result, a concise criterion is established only based on the characteristic polynomial, which merely requires polynomial computational complexity with respect to the number of network nodes. Finally, a numerical example is presented to validate the effectiveness of the obtained theoretical results.