When Is Degree Enough? Bounds on Degree-Eigenvector Misalignment in Assortative Structured Networks

TL;DR

This paper establishes bounds on how much the leading eigenvector of a network's adjacency matrix can deviate from the degree vector, especially in networks with assortativity and local structures, aiding understanding of node importance.

Contribution

It provides a constructive method to bound eigenvector-degree misalignment in complex networks with assortativity and local structures, extending spectral analysis tools.

Findings

Derived upper bounds on eigenvector-degree angle deviation.

Introduced degree-preserving rewiring algorithms to analyze network structures.

Validated bounds through numerical simulations.

Abstract

A tight alignment between the degree vector and the leading eigenvector arises naturally in networks with neutral degree mixing and the absence of local structures. Many real-world networks, however, violate both conditions. We derive bounds on the divergence between the degree vector and the eigenvector in networks with degree assortativity and local mesoscopic structures such as communities, core-peripheries, and cycles. Our approach is constructive. We design sufficiently general degree-preserving rewiring algorithms that start from a neutral benchmark and monotonically increase assortativity and the strength of local structures, with each step inducing a perturbation of the adjacency matrix. Using the Stewart--Sun Perturbation Bound, together with explicit spectral-norm control of the rewiring steps, we derive upper bounds on the angle between the eigenvector and the degree vector for modest levels of assortativity and local structures. Our analytical bounds delineate regions of `spectral safety' in which a node's degree can be used as a reliable measure of its systemic importance in real-world networks. We also substantiate our analytical bounds with numerical simulations that compute the exact angles of deviation.