In this paper, we develop an inferential framework with sharp asymptotic optimality guarantees for Ising models on inhomogeneous random graphs in the subcritical parameter regime. We begin by characterizing the asymptotic distribution of the maximum likelihood (ML) estimate of the natural parameter, based on a single sample from the underlying model, covering both sparse and dense network regimes. Next, to overcome the computational intractability of the ML method, we propose a simple closed-form estimate obtained from a one-step approximation to the likelihood equation. We show that this estimate attains the same asymptotic distribution and variance as the ML estimate, thereby yielding a computationally efficient and asymptotically valid confidence interval for the natural parameter. We complement these inferential results by establishing a Hájek--Le Cam-type local asymptotic minimax theorem, showing that the proposed estimate achieves the smallest possible asymptotic maximum risk, both in rate and in leading constant, over shrinking neighborhoods of the true parameter. We also derive the corresponding limit of experiments. To the best of our knowledge, these are among the first sharp asymptotic optimality results for network-dependent data. Finally, we study goodness-of-fit testing for the natural parameter, deriving the local power of the likelihood ratio test and minimax detection rates. Our analysis relies on new fluctuation results for the sufficient statistic (Hamiltonian) and for the random partition function of Ising models on inhomogeneous random graphs, which are of independent interest.

翻译：本文针对亚临界参数区域内的非均匀随机图上的伊辛模型，建立了具有渐近最优性严格保证的推断框架。首先，我们基于单次观测样本刻画了自然参数最大似然估计的渐近分布，该结果同时覆盖稀疏与稠密网络情形。其次，为克服最大似然方法的计算困难，我们提出一种基于似然方程一步近化的简单闭式估计量，并证明该估计量具有与最大似然估计相同的渐近分布与方差，从而为自然参数生成计算高效且渐近有效的置信区间。通过建立Hájek--Le Cam型局部渐近极小极大定理，我们进一步补充上述推断结论：在真实参数的收缩邻域内，所提估计量在收敛速度与首项常数两个维度上均达到最小可能渐近最大风险。文中同时推导了相应的实验极限。据我们所知，这是网络依赖数据领域首批严格的渐近最优性结果之一。最后，我们研究自然参数的拟合优度检验问题，导出似然比检验的局部势函数与极小极大检测率。本文分析依赖于非均匀随机图上伊辛模型充分统计量（哈密顿量）与随机配分函数的新波动结果，这些结果本身具有独立研究价值。