A central limit theorem for an omnibus embedding of multiple random graphs and implications for multiscale network inference
A central limit theorem for an omnibus embedding of multiple random graphs and implications for multiscale network inference
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多个随机图综合嵌入的中心极限定理及其对多尺度网络推理的影响
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发表时间：
2017
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通讯作者：
C. Priebe
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作者：
Keith D. Levin;A. Athreya;M. Tang;V. Lyzinski;Youngser Park;C. Priebe
对图的集合进行统计分析对许多学科来说都是重要的，但是用于多样本图推理的原则性的、可扩展的方法很少。在这里，我们描述了一个“综合”嵌入，其中同一顶点集上的多个图被联合嵌入到一个空间中，每个图都有一个不同的表示。我们证明了这种嵌入的中心极限定理，并演示了它如何简化图形比较，避免了成对子空间对齐的需要。综合嵌入实现接近最佳的推理精度时，图形出现从一个共同的分布，但保留歧视性的权力作为一个测试程序比较不同的图形。此外，这种联合嵌入和伴随的中心极限定理对于回答多尺度图推理问题非常重要，例如识别负责网络之间相似性或差异性的特定子图或顶点。我们说明了这一点与一对来自dMRI和fMRI扫描人类受试者的连接体数据的分析。特别是，我们表明，这种嵌入允许识别与人口水平差异相关的特定大脑区域。最后，我们勾画了如何综合嵌入可以用来解决紧迫的开放问题，理论和实践，在多样本图推理。
Performing statistical analyses on collections of graphs is of import to many disciplines, but principled, scalable methods for multi-sample graph inference are few. Here we describe an "omnibus" embedding in which multiple graphs on the same vertex set are jointly embedded into a single space with a distinct representation for each graph. We prove a central limit theorem for this embedding and demonstrate how it streamlines graph comparison, obviating the need for pairwise subspace alignments. The omnibus embedding achieves near-optimal inference accuracy when graphs arise from a common distribution and yet retains discriminatory power as a test procedure for the comparison of different graphs. Moreover, this joint embedding and the accompanying central limit theorem are important for answering multiscale graph inference questions, such as the identification of specific subgraphs or vertices responsible for similarity or difference across networks. We illustrate this with a pair of analyses of connectome data derived from dMRI and fMRI scans of human subjects. In particular, we show that this embedding allows the identification of specific brain regions associated with population-level differences. Finally, we sketch how the omnibus embedding can be used to address pressing open problems, both theoretical and practical, in multisample graph inference.