Random matrix analysis of localization properties of Gene co-expression network

TL;DR

This study applies random matrix theory to analyze gene co-expression networks, revealing spectral properties, eigenvalue localization, and key nodes, thereby providing insights into network structure and important gene modules.

Contribution

It introduces a spectral analysis framework using RMT to distinguish between different eigenvalue regimes and identify important nodes in gene co-expression networks.

Findings

Eigenvalue distribution follows RMT predictions in bulk.

Localized eigenvalues indicate important network nodes.

Spectral rigidity deviates at certain ranges, revealing network structure.

Abstract

We analyze gene co-expression network under the random matrix theory framework. The nearest neighbor spacing distribution of the adjacency matrix of this network follows Gaussian orthogonal statistics of random matrix theory (RMT). Spectral rigidity test follows random matrix prediction for a certain range, and deviates after wards. Eigenvector analysis of the network using inverse participation ratio (IPR) suggests that the statistics of bulk of the eigenvalues of network is consistent with those of the real symmetric random matrix, whereas few eigenvalues are localized. Based on these IPR calculations, we can divide eigenvalues in three sets; (A) The non-degenerate part that follows RMT. (B) The non-degenerate part, at both ends and at intermediate eigenvalues, which deviate from RMT and expected to contain information about {\it important nodes} in the network. (C) The degenerate part with $zero$ eigenvalue, which fluctuates around RMT predicted value. We identify nodes corresponding to the dominant modes of the corresponding eigenvectors and analyze their structural properties.