基于低阶特征向量的网络空间信息提取
Extracting spatial information from networks with low-order eigenvectors
摘要：
我们考虑的问题，推断有意义的空间信息在网络中的网络节点之间的连接强度的不完全信息。我们考虑两个空间分布的网络：美国境内的人口迁移流动网络，以及比利时城市之间的移动电话网络。对于这两个网络，我们使用从链接强度构造的拉普拉斯矩阵的特征向量来获得信息可视化和捕获自然地理细分。我们观察到，一些低阶特征向量本地化非常好，似乎揭示了小的地理凝聚力的地区，匹配非常好的政治和行政边界。我们讨论了可能的解释，这种观察通过描述扩散映射和本地化的本征函数。此外，我们讨论了一个可能的连接与加权图切割问题，并提供数值证据支持的想法，低阶特征向量指出局部削减网络。然而，我们没有为我们的观察提供正式和严格的理由。
英文摘要：
We consider the problem of inferring meaningful spatial information in networks from incomplete information on the connection intensity between the nodes of the network. We consider two spatially distributed networks: a population migration flow network within the US, and a network of mobile phone calls between cities in Belgium. For both networks we use the eigenvectors of the Laplacian matrix constructed from the link intensities to obtain informative visualizations and capture natural geographical subdivisions. We observe that some low order eigenvectors localize very well and seem to reveal small geographically cohesive regions that match remarkably well with political and administrative boundaries. We discuss possible explanations for this observation by describing diffusion maps and localized eigenfunctions. In addition, we discuss a possible connection with the weighted graph cut problem, and provide numerical evidence supporting the idea that lower order eigenvectors point out local cuts in the network. However, we do not provide a formal and rigorous justification for our observations.
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