Adjacency Spectral Embeddings of Correlation Networks

TL;DR

This paper provides a theoretical foundation for using spectral embeddings on correlation networks derived from time series data, showing they recover latent Fourier basis coefficients even with noise.

Contribution

It establishes that correlation networks from time series with Fourier structure can be modeled as latent space networks with dependent edges, and spectral embeddings recover true latent variables under noise.

Findings

Spectral embeddings recover Fourier coefficients from noisy correlation networks.

Correlation networks with Fourier structure are linked to latent space models with dependent edges.

Embedding methods are justified for correlation networks under certain conditions.

Abstract

In many applications, weighted networks are constructed based on time series data: each time series is associated to a vertex and edge weights are given by pairwise correlations. The result is a network whose edge dependency structure violates the assumptions of most common network models. Nonetheless, it is common to analyze these "correlation networks" using embedding methods derived from edge-independent network models, based on a belief that the edges are approximately independent. In this work, we put this modeling choice on firm theoretical ground. We show that when the time series are expressible in terms of a small number of Fourier basis elements (or in some other suitably-chosen basis), correlation networks correspond to latent space networks with dependent edge noise in which the vertex-level latent variables encode the basis coefficients. Further, we show that when time series are observed subject to noise, spectral embedding of the resulting noisy correlation network still recovers these true vertex-level latent representations under suitable assumptions. This characterization of embeddings as learning Fourier coefficients appears to be folklore in the signal processing community in the context of principal component analysis, but is, to the best of our knowledge, new to the statistical network analysis literature.