Bayesian network models (Erdos Renyi, stochastic block models, random dot product graphs, graphons) are widely used in neuroscience, epidemiology, and the social sciences, yet real networks are sparse, heterogeneous, and exhibit higher-order dependence. How stable are network-based decisions, model selection, and policy recommendations to small model misspecification? We study local decision-theoretic robustness by allowing the posterior to vary within a small Kullback-Leibler neighborhood and choosing actions that minimize worst-case posterior expected loss. Exploiting low-dimensional functionals available under exchangeability, we (i) adapt decision-theoretic robustness to exchangeable graphs via graphon limits and derive sharp small-radius expansions of robust posterior risk; under squared loss the leading inflation is controlled by the posterior variance of the loss, and for robustness indices that diverge at percolation/fragmentation thresholds we obtain a universal critical exponent describing the explosion of decision uncertainty near criticality. (ii) Develop a nonparametric minimax theory for robust model selection between sparse Erdos-Renyi and block models, showing-via robustness error exponents-that no Bayesian or frequentist method can uniformly improve upon the decision-theoretic limits over configuration models and sparse graphon classes for percolation-type functionals. (iii) Propose a practical algorithm based on entropic tilting of posterior or variational samples, and demonstrate it on functional brain connectivity and Karnataka village social networks.

翻译：贝叶斯网络模型（Erdos Renyi模型、随机块模型、随机点积图、图函数）在神经科学、流行病学和社会科学中广泛应用，然而真实网络具有稀疏性、异质性并呈现高阶依赖性。基于网络的决策、模型选择与政策建议对于微小的模型误设具有何种稳定性？我们通过允许后验分布在较小的Kullback-Leibler邻域内变动，并选择最小化最坏情况后验期望损失的行动，来研究局部决策理论鲁棒性。利用可交换性下的低维泛函，我们（i）通过图函数极限将决策理论鲁棒性适配至可交换图，并推导出鲁棒后验风险的尖锐小半径展开式：在平方损失下，主要膨胀项由损失的后验方差控制；对于在渗流/破碎阈值处发散的鲁棒性指标，我们获得描述临界点附近决策不确定性爆发的普适临界指数。（ii）针对稀疏Erdo-Renyi模型与块模型之间的鲁棒模型选择，发展非参数极小极大理论，通过鲁棒性误差指数证明：对于渗流型泛函，在配置模型与稀疏图函数类上，任何贝叶斯或频率学派方法均无法一致地超越决策理论极限。（iii）提出基于后验或变分样本熵倾斜的实用算法，并在功能性脑连接网络与卡纳塔克邦乡村社交网络上进行验证。