This paper studies the matched network inference problem, where the goal is to determine if two networks, defined on a common set of nodes, exhibit a specific form of stochastic similarity. Two notions of similarity are considered: (i) equality, i.e., testing whether the networks arise from the same random graph model, and (ii) scaling, i.e., testing whether their probability matrices are proportional for some unknown scaling constant. We develop a testing framework based on a parametric bootstrap approach and a Frobenius norm-based test statistic. The proposed approach is highly versatile as it covers both the equality and scaling problems, and ensures adaptability under various model settings, including stochastic blockmodels, Chung-Lu models, and random dot product graph models. We establish theoretical consistency of the proposed tests and demonstrate their empirical performance through extensive simulations under a wide range of model classes. Our results establish the flexibility and computational efficiency of the proposed method compared to existing approaches. We also report a real-world application involving the Aarhus network dataset, which reveals meaningful sociological patterns across different communication layers.

翻译：本文研究匹配网络推断问题，其目标是判断定义在相同节点集上的两个网络是否表现出特定形式的随机相似性。我们考虑两种相似性概念：(i) 相等性检验，即检验网络是否源于相同的随机图模型；(ii) 比例性检验，即检验其概率矩阵是否在某个未知比例常数下成比例。我们开发了一种基于参数化Bootstrap方法和Frobenius范数检验统计量的检验框架。所提出的方法具有高度通用性，既涵盖相等性检验问题也涵盖比例性检验问题，并确保其在多种模型设置下的适应性，包括随机块模型、Chung-Lu模型和随机点积图模型。我们建立了所提出检验的理论一致性，并通过在广泛模型类别下的大量模拟实验证明了其经验性能。我们的结果表明，与现有方法相比，所提出方法具有灵活性和计算效率。我们还报告了一个涉及奥胡斯网络数据集的真实应用，该应用揭示了不同通信层间有意义的社会学模式。