Braess's Paradox for the Spectral Gap in Random Graphs and Delocalization of Eigenvectors

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 Erdos-Renyi 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. (C) 2016 Wiley Periodicals, Inc.