Analysing degeneracies in networks spectra

TL;DR

This paper investigates why certain eigenvalues in real-world networks have high degeneracy by analyzing the adjacency matrix and eigenvectors, revealing structural contributions to spectral multiplicities.

Contribution

It introduces a simple transformation of the adjacency matrix to understand the origins of spectral degeneracies in networks.

Findings

Eigenvectors associated with degenerate eigenvalues reveal structural features.

Degeneracies are rare in model graphs but common in real-world networks.

The approach helps identify structures contributing to spectral degeneracy.

Abstract

Many real-world networks exhibit a high degeneracy at few eigenvalues. We show that a simple transformation of the network's adjacency matrix provides an understanding of the origins of occurrence of high multiplicities in the networks spectra. We find that the eigenvectors associated with the degenerate eigenvalues shed light on the structures contributing to the degeneracy. Since these degeneracies are rarely observed in model graphs, we present results for various cancer networks. This approach gives an opportunity to search for structures contributing to degeneracy which might have an important role in a network.