Braess's paradox for the spectral gap in random graphs and delocalization of eigenvectors

TL;DR

This paper investigates how adding edges to Erdős-Rényi random graphs can counterintuitively decrease their spectral gap, revealing a phenomenon similar to Braess's paradox and establishing new eigenvector delocalization results.

Contribution

It demonstrates that in Erdős-Rényi graphs, adding a random edge often reduces the spectral gap and introduces a novel delocalization result for Laplacian eigenvectors.

Findings

Adding a random edge decreases the spectral gap with positive probability.

Spectral gap reduction occurs in typical Erdős-Rényi graphs.

New eigenvector delocalization results for $G(n,p)$ Laplacians.

Abstract

We study how the spectral gap of the normalized Laplacian of a random graph changes when an edge is added to or removed from the graph. There are known examples of graphs where, perhaps counterintuitively, adding an edge can decrease the spectral gap, a phenomenon that is analogous to Braess's paradox in traffic networks. We show that this is often the case in random graphs in a strong sense. More precisely, we show that for typical instances of Erd\H{o}s-R\'enyi random graphs $G(n,p)$ with constant edge density $p \in (0,1)$, the addition of a random edge will decrease the spectral gap with positive probability, strictly bounded away from zero. To do this, we prove a new delocalization result for eigenvectors of the Laplacian of $G(n,p)$, which might be of independent interest.