We propose two multiscale comparisons of graphs using heat diffusion, allowing to compare graphs without node correspondence or even with different sizes. These multiscale comparisons lead to the definition of Lipschitz-continuous empirical processes indexed by a real parameter. The statistical properties of empirical means of such processes are studied in the general case. Under mild assumptions, we prove a functional central limit theorem, as well as a Gaussian approximation with a rate depending only on the sample size. Once applied to our processes, these results allow to analyze data sets of pairs of graphs. We design consistent confidence bands around empirical means and consistent two-sample tests, using bootstrap methods. Their performances are evaluated by simulations on synthetic data sets.

翻译：我们提出两种基于热扩散的多尺度图比较方法，允许在无需节点对应甚至图规模不同的情况下进行图比较。这些多尺度比较引出了由实参数索引的Lipschitz连续经验过程的定义。在一般情况下，我们研究了此类过程经验均值的统计性质。在温和假设下，我们证明了泛函中心极限定理，以及一个仅依赖于样本量的高斯近似速率。将这些结果应用于我们的过程后，我们能够分析包含图对的数据集。我们利用自助法设计了经验均值的一致置信带和一致两样本检验，并通过合成数据集的模拟评估了它们的性能。