In this paper, the synchronization problem for networks over finite fields is investigated, which is a generalization of consensus and provides a new perspective for networks of agents with limited capacities of memory and communication. It is assumed that the states and communication weights can only attain values from a finite alphabet equipped with a prime number of integers, termed finite fields, and operations are processed relying on modular arithmetic. For this synchronization problem, necessary and sufficient conditions are derived based on the transition graph of the studied network. The large number of nodes in the transition graph, dependent on the numbers of integers in finite fields and the agents, may lead to high computational cost and difficulties in verifying synchronization. To avoid this, an equivalent condition for synchronization of networks is provided by the characteristic polynomial of the studied network matrix. Furthermore, in a synchronized network over finite fields, the periodic behavior can be determined by the network matrix and the initial state.

本文研究了有限域网络的同步问题，它是对共识的一个概括，为存储和通信能力有限的智能体网络提供了新的视角。假定状态和通信权重只能从配备有素数整数的有限字母（称为有限域）中获取值，并且依赖于模块化算术来处理运算。对于此同步问题，根据研究网络的过渡图得出了必要条件和充分条件。依赖于有限域中的整数和代理，过渡图中的大量节点可能导致较高的计算成本和验证同步的难度。为了避免这种情况，研究的网络矩阵的特征多项式提供了网络同步的等效条件。此外，在有限域上的同步网络中，周期性行为可以由网络矩阵和初始状态确定。