Correlation networks from random walk time series

TL;DR

This paper introduces a network model where nodes are random walks, connected based on Pearson correlation, revealing properties like small-world behavior and a phase transition at a critical threshold.

Contribution

The study proposes a novel network model linking random walks via correlation, providing insights into their structural properties and potential as a null hypothesis in time series network analysis.

Findings

Networks exhibit high clustering and small-world properties.

Degree distribution varies with the correlation threshold.

A phase transition occurs at a critical threshold H_c.

Abstract

Stimulated by the growing interest in the applications of complex networks framework on time series analysis, we devise a network model in which each of $N$ nodes is associated with a random walk of length $L$. Connectivity between any two nodes is established when the Pearson correlation coefficient(PCC) of the corresponding time series is greater than or equal to a threshold $H$, resulting in similarity networks with interesting properties. In particular, these networks can have high average clustering coefficients, "small world" property, and their degree distribution can vary from scale-free to quasi-constant depending on $H$. A giant component of size $N$ exists until a critical threshold $H_c$ is crossed, at which point relatively rare walks begin to detach from it, and remain isolated. This model can be used as a first step for building a null hypothesis for networks constructed from time series.