Correlation networks from random walk time series

Stimulated by the growing interest in the applications of complex networks framework for 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) between 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 and the "small world" property, and their degree distribution can vary from scale-free to quasiconstant 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.