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占据核心地位，统计推断问题在此背景下自然产生。本文基于最大熵原理，研究统计网络的拟合优度与双样本检验方法。我们的方法在网络空间上构建了受预定结构约束的熵最大化优化问题。所得检验统计量通过约束优化问题的拉格朗日乘子定义，据我们所知，这在统计网络文献中具有创新性。我们在顶点数固定的经典体系下证明了方法的一致性。随后考虑图规模随样本量增长的渐近体系，针对稠密与稀疏两种场景分别构建检验方法。在稠密情形中，我们分析了指数随机图模型（包括Erdös-Rènyi模型）；在稀疏体系下，我们的理论适用于Erd{ö}s-R{è}nyi图。研究运用了随机图非线性大偏差理论的最新进展。我们进一步证明，所提出的拉格朗日乘子框架能与约束极大似然估计的经典得分检验自然衔接。该成果为不同增长体系下的网络模型评估提供了统一的熵基理论框架。