Spectral Embedding of Weighted Graphs

When analyzing weighted networks using spectral embedding, a judicious transformation of the edge weights may produce better results. To formalize this idea, we consider the asymptotic behavior of spectral embedding for different edge-weight representations, under a generic low rank model. We measure the quality of different embeddings—which can be on entirely different scales—by how easy it is to distinguish communities, in an information-theoretical sense. For common types of weighted graphs, such as count networks or <i>p</i>-value networks, we find that transformations such as tempering or thresholding can be highly beneficial, both in theory and in practice. Supplementary materials for this article are available online.

在利用谱嵌入（spectral embedding）分析加权网络时，对边权重进行审慎的变换往往能获得更优的分析结果。为将该思路形式化，我们在通用低秩模型框架下，探究了不同边权重表示方式下谱嵌入的渐近行为。我们从信息论视角出发，以社区区分难度作为衡量不同嵌入（其尺度可能存在显著差异）质量的标准。针对计数网络、p值网络等常见加权图类型，我们发现调温变换（tempering）或阈值化（thresholding）等变换手段在理论与实践层面均具备显著的应用价值。本文的补充材料可在线获取。