Two-sample hypothesis testing for large graphs is popular in cognitive science, probabilistic machine learning and artificial intelligence. While numerous methods have been proposed in the literature to address this problem, less attention has been devoted to scenarios involving graphs of unequal size or situations where there are only one or a few samples of graphs. In this article, we propose a Frobenius test statistic tailored for small sample sizes and unequal-sized random graphs to test whether they are generated from the same model or not. Our approach involves an algorithm for generating bootstrapped adjacency matrices from estimated community-wise edge probability matrices, forming the basis of the Frobenius test statistic. We derive the asymptotic distribution of the proposed test statistic and validate its stability and efficiency in detecting minor differences in underlying models through simulations. Furthermore, we explore its application to fMRI data where we are able to distinguish brain activity patterns when subjects are exposed to sentences and pictures for two different stimuli and the control group.

翻译：大图的双样本假设检验在认知科学、概率机器学习及人工智能领域应用广泛。尽管文献中已提出众多方法解决该问题，但针对不等规模图或仅有一/少量图样本的场景研究较少。本文提出一种适用于小样本量和不等规模随机图的Frobenius检验统计量，用于检验这些图是否来自同一生成模型。该方法通过从估计的社区间边概率矩阵生成自举邻接矩阵的算法，构成Frobenius检验统计量的基础。我们推导了所提统计量的渐近分布，并通过仿真验证其在检测基础模型细微差异时的稳定性和有效性。进一步地，我们探索该方法在fMRI数据中的应用——当受试者分别接受句子和图片两种不同刺激时，以及对照组中，成功区分其脑活动模式差异。