The spectral degree exponent of a graph

TL;DR

This paper introduces the spectral degree exponent as a new graph metric, generalizes previous work to weighted graphs, and explores its relation to degree assortativity across various graph types.

Contribution

It extends the spectral degree exponent to weighted graphs, provides efficient computation methods, and reveals its strong correlation with degree assortativity.

Findings

High correlation between spectral degree exponent and degree assortativity.

Efficient iterative formulas and bounds for the spectral degree exponent.

Asymptotic expansions for various graph families.

Abstract

We propose the spectral degree exponent as a novel graph metric. Although Hofmeister \cite{HofmeisterThesis} has studied the same metric, we generalise Hofmeister's work to weighted graphs. We provide efficient iterative formulas and bounds for the spectral degree exponent and provide highly accurate asymptotic expansions for the spectral degree exponent for several families of graphs. Furthermore, we uncover a close relation between the spectral degree exponent and the well-known degree assortativity, by showing high correlations between the two metrics in all small graphs, several random graph models and many real-world graphs.