Hierarchical Bayesian models are increasingly used in large, inhomogeneous complex network dynamical systems by modeling parameters as draws from a hyperparameter-governed distribution. However, theoretical guarantees for these estimates as the system size grows have been lacking. A critical concern is that hyperparameter estimation may diverge for larger networks, undermining the model's reliability. Formulating the system's evolution in a measure transport perspective, we propose a theoretical framework for estimating hyperparameters with mean-type observations, which are prevalent in many scientific applications. Our primary contribution is a nonasymptotic bound for the deviation of estimate of hyperparameters in inhomogeneous complex network dynamical systems with respect to network population size, which is established for a general family of optimization algorithms within a fixed observation duration. While we firstly establish a consistency result for systems with independent nodes, our main result extends this guarantee to the more challenging and realistic setting of weakly-dependent nodes. We validate our theoretical findings with numerical experiments on two representative models: a Susceptible-Infected-Susceptible model and a Spiking Neuronal Network model. In both cases, the results confirm that the estimation error decreases as the network population size increases, aligning with our theoretical guarantees. This research proposes the foundational theory to ensure that hierarchical Bayesian methods are statistically consistent for large-scale inhomogeneous systems, filling a gap in this area of theoretical research and justifying their application in practice.

翻译：分层贝叶斯模型在大型非均匀复杂网络动力系统中日益得到应用，其方法是将参数建模为超参数控制分布中的抽样。然而，随着系统规模增长，这些估计的理论保证一直缺失。一个关键问题是超参数估计可能在更大规模的网络上发散，从而削弱模型的可靠性。通过从测度传输视角构建系统演化过程，我们提出了一个基于均值型观测的超参数估计理论框架，此类观测在众多科学应用中普遍存在。我们的主要贡献是建立了非均匀复杂网络动力系统中超参数估计偏差相对于网络群体规模的非渐近界，该结果针对固定观测时长内的一类通用优化算法族而确立。我们首先在节点独立的系统中建立了一致性结果，随后将这一保证推广至更具挑战性和现实意义的弱依赖节点场景。我们通过两个代表性模型的数值实验验证了理论发现：易感-感染-易感模型和脉冲神经元网络模型。两种情况下，实验结果均证实估计误差随网络群体规模增大而减小，与我们的理论保证一致。本研究提出了确保分层贝叶斯方法在大型非均匀系统中具有统计一致性的基础理论，填补了该领域理论研究的空白，并为其实际应用提供了理论依据。