Stochastic network models play a central role across a wide range of scientific disciplines, and questions of statistical inference arise naturally in this context. In this paper we investigate goodness-of-fit and two-sample testing procedures for statistical networks based on the principle of maximum entropy (MaxEnt). Our approach formulates a constrained entropy-maximization problem on the space of networks, subject to prescribed structural constraints. The resulting test statistics are defined through the Lagrange multipliers associated with the constrained optimization problem, which, to our knowledge, is novel in the statistical networks literature. We establish consistency in the classical regime where the number of vertices is fixed. We then consider asymptotic regimes in which the graph size grows with the sample size, developing tests for both dense and sparse settings. In the dense case, we analyze exponential random graph models (ERGM) (including the Erdös-Rènyi models), while in the sparse regime our theory applies to Erd{ö}s-R{è}nyi graphs. Our analysis leverages recent advances in nonlinear large deviation theory for random graphs. We further show that the proposed Lagrange-multiplier framework connects naturally to classical score tests for constrained maximum likelihood estimation. The results provide a unified entropy-based framework for network model assessment across diverse growth regimes.

翻译：随机网络模型在众多科学领域中扮演着核心角色，这一背景下自然衍生出统计推断问题。本文基于最大熵原理，针对统计网络提出拟合优度检验和双样本检验方法。该方法在网络空间上构建了约束熵最大化问题，并施加预设的结构约束。由此产生的检验统计量通过约束优化问题相关的拉格朗日乘子定义——据我们所知，这在统计网络文献中尚属首次。在顶点数固定的经典情形下，我们证明了该方法的一致性。随后考虑图规模随样本量增长的渐近情形，分别针对稠密网络和稀疏网络开发了检验方法。在稠密情形中，我们分析了指数随机图模型（包括Erdős–Rényi模型）；在稀疏情形中，理论适用于Erdős–Rényi图。分析过程利用了随机图非线性大偏差理论的最新进展。进一步研究表明，所提拉格朗日乘子框架与约束极大似然估计的经典得分检验存在自然关联。这些结果为不同规模增长模式下的网络模型评估提供了统一的基于熵的框架。