Spectral Embeddings of Correlation Networks

Music City Center 

In many applications, weighted networks are constructed based on time series data in order to facilitate the application of network analysis tools. Most typically, a time series is associated with each vertex, and edge weights are given by correlations or other measures of dependence between times series. The result is a network that violates the additive, independent noise assumptions of most common network models. Nonetheless, it is common to apply embedding methods to networks built from correlations. In this work, we consider a setup in which a collection of time series are observed subject to noise, and a network is constructed based on correlations between the noisy series. We prove that, under suitable conditions, applying the adjacency spectral embedding to the network of noisily measured correlations recovers the embeddings of the true time series in the large-network limit. Additionally, we show that the resulting embedding encodes, up to orthogonal rotation, the Fourier coefficients of the true time series. This observation is novel to the networks literature, to the best of our knowledge.

Networks

Time series

Embeddings 

Section on Statistical Learning and Data Science