This paper studies a general class of stochastic population processes in which agents interact with one another over a network. Agents update their behaviors in a random and decentralized manner according to a policy that depends only on the agent's current state and an estimate of the macroscopic population state, given by a weighted average of the neighboring states. When the number of agents is large and the network is a complete graph (has all-to-all information access), the macroscopic behavior of the population can be well-approximated by a set of deterministic differential equations called a {\it mean-field approximation}. For incomplete networks such characterizations remained previously unclear, i.e., in general whether a suitable mean-field approximation exists for the macroscopic behavior of the population. The paper addresses this gap by establishing a generic theory describing when various mean-field approximations are accurate for \emph{arbitrary} interaction structures. Our results are threefold. Letting $W$ be the matrix describing agent interactions, we first show that a simple mean-field approximation that incorrectly assumes a homogeneous interaction structure is accurate provided $W$ has a large spectral gap. Second, we show that a more complex mean-field approximation which takes into account agent interactions is accurate as long as the Frobenius norm of $W$ is small. Finally, we compare the predictions of the two mean-field approximations through simulations, highlighting cases where using mean-field approximations that assume a homogeneous interaction structure can lead to inaccurate qualitative and quantitative predictions.

翻译：本文研究了一类广义的随机种群过程，其中个体通过网络相互交互。个体根据仅依赖于自身当前状态和宏观种群状态估计（由邻域状态的加权平均给出）的策略，以随机且去中心化的方式更新其行为。当个体数量较大且网络为完全图（具有全连接信息访问）时，种群的宏观行为可以通过一组称为平均场近似的确定性微分方程来良好近似。对于非完全网络，此类表征此前尚不明确，即一般而言，种群宏观行为是否存在合适的平均场近似尚不清楚。本文通过建立一套通用理论填补了这一空白，该理论描述了在任意交互结构下各种平均场近似的适用条件。我们的结果包含三个方面。设W为描述个体交互的矩阵，我们首先证明，当W具有大谱隙时，一种错误假设均匀交互结构的简单平均场近似是有效的。其次，我们证明一种考虑个体交互的复杂平均场近似在W的Frobenius范数较小时是有效的。最后，我们通过仿真比较了两种平均场近似的预测结果，突出了使用假设均匀交互结构的平均场近似可能导致定性及定量预测不准确的情况。