Braess's paradox for the spectral gap in random graphs and delocalization of eigenvectors1. Issue 4 (21st November 2016)

• Record Type: Journal Article
• Authors: Eldan, Ronen; Rácz, Miklós Z.; Schramm, Tselil
• Is Part Of: Random structures & algorithms. Volume 50:Issue 4(2017)
• Page Start: 584
• Page End: 611
• Publication Date: 2016-11-21
• Subjects: random graphs -- spectral gap -- Braess's paradox -- graph Laplacian; Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20696
• Languages: English
• ISSNs: 1042-9832

Abstract

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 counter‐intuitively, 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 Erdős‐Rényi random graphs G ( n, p ) with constant edge density p ∈ ( 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. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 584–611, 2017