We show how graphons can be used to model and analyze open multi-agent systems, which are multi-agent systems subject to arrivals and departures, in the specific case of linear consensus. First, we analyze the case of replacements, where under the assumption of a deterministic interval between two replacements, we derive an upper bound for the disagreement in expectation. Then, we study the case of arrivals and departures, where we define a process for the evolution of the number of agents that guarantees a minimum and a maximum number of agents. Next, we derive an upper bound for the disagreement in expectation, and we establish a link with the spectrum of the expected graph used to generate the graph topologies. Finally, for stochastic block model (SBM) graphons, we prove that the computation of the spectrum of the expected graph can be performed based on a matrix whose dimension depends only on the graphon and it is independent of the number of agents.

翻译：本文展示了如何利用图元对开放多智能体系统——即存在智能体加入与退出的多智能体系统——进行建模与分析，并聚焦于线性一致性问题。首先，我们分析了智能体替换场景：在假设两次替换间隔时间确定的前提下，推导了期望分歧度的上界。随后，我们研究了智能体加入与退出场景：通过定义智能体数量演化过程，确保系统维持最小与最大智能体数量边界。接着，我们推导了该场景下期望分歧度的上界，并建立了该上界与生成图拓扑的期望图谱之间的理论联系。最后针对随机分块模型图元，我们证明了期望图谱的计算可基于一个维度仅取决于图元结构、且与智能体数量无关的矩阵来实现。