Numerical Investigation of Graph Spectra and Information Interpretability of Eigenvalues

TL;DR

This paper conducts a comprehensive numerical analysis of graph spectra across various network types to understand the significance of eigenvalues in relation to network topology and information content, introducing novel methods for identifying key eigenvalues.

Contribution

It introduces a new technique for identifying the most informative eigenvalues in evolving networks by comparing spectral behavior to algorithmic complexity.

Findings

Identified key eigenvalues in biological and synthetic networks.

Demonstrated the method's effectiveness on tissue-specific regulatory networks.

Provided insights into the relationship between eigenvalues and network information content.

Abstract

We undertake an extensive numerical investigation of the graph spectra of thousands regular graphs, a set of random Erd\"os-R\'enyi graphs, the two most popular types of complex networks and an evolving genetic network by using novel conceptual and experimental tools. Our objective in so doing is to contribute to an understanding of the meaning of the Eigenvalues of a graph relative to its topological and information-theoretic properties. We introduce a technique for identifying the most informative Eigenvalues of evolving networks by comparing graph spectra behavior to their algorithmic complexity. We suggest that extending techniques can be used to further investigate the behavior of evolving biological networks. In the extended version of this paper we apply these techniques to seven tissue specific regulatory networks as static example and network of a na\"ive pluripotent immune cell in the process of differentiating towards a Th17 cell as evolving example, finding the most and least informative Eigenvalues at every stage.