Correlation matrices of Gaussian Markov random fields over cycle graphs

Gaussian Markov Random Fields over graphs have been widely used in many fields of application. Here, we address the matrix construction problem that arises in the study of Gaussian Markov Random Fields with uniform correlation, i.e., those in which all correlations between adjacent nodes in the graph are equal. We provide a characterization of the correlation matrix of a Gaussian Markov Random Field with uniform correlation over a cycle graph, which is circulant and has a sparse inverse matrix, and study the relationship with the stationary Gaussian Markov Process on the circle. Two methods for computing the correlation matrix are also provided. Ultimately, asymptotic results for cycle graphs of large order point out the relation between Gaussian Markov Random Fields with uniform correlation over cycle and path graphs. (c) 2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).