Effect on normalized graph Laplacian spectrum by motif attachment and duplication

TL;DR

This paper studies how motif doubling and attachment, biologically relevant graph operations, influence the spectrum of the normalized graph Laplacian, explaining observed eigenvalue multiplicities in real networks.

Contribution

It provides a detailed analysis of how motif attachment and duplication affect the normalized Laplacian spectrum, offering explanations for spectral features in real networks.

Findings

Motif doubling and attachment alter the spectrum significantly.

Eigenvalue multiplicities at 1 and related values are explained.

Results shed light on spectral patterns in biological networks.

Abstract

To some extent, graph evolutionary mechanisms can be explained by its spectra. Here, we are interested in two graph operations, namely, motif (subgraph) doubling and attachment that are biologically relevant. We investigate how these two processes affect the spectrum of the normalized graph Laplacian. A high (algebraic) multiplicity of the eigenvalues $1, 1\pm 0.5, 1\pm \sqrt{0.5}$ and others has been observed in the spectrum of many real networks. We attempt to explain the production of distinct eigenvalues by motif doubling and attachment. Results on the eigenvalue $1$ are discussed separately.