15 research outputs found
Vertices cannot be hidden from quantum spatial search for almost all random graphs
In this paper we show that all nodes can be found optimally for almost all
random Erd\H{o}s-R\'enyi graphs using continuous-time
quantum spatial search procedure. This works for both adjacency and Laplacian
matrices, though under different conditions. The first one requires
, while the seconds requires , where . The proof was made by analyzing the convergence
of eigenvectors corresponding to outlying eigenvalues in the norm. At the same time for , the property does
not hold for any matrix, due to the connectivity issues. Hence, our derivation
concerning Laplacian matrix is tight.Comment: 18 pages, 3 figur
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 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 with constant edge density , 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 , which
might be of independent interest.Comment: Version 2, minor change
Eigenvectors of random matrices: A survey
Eigenvectors of large matrices (and graphs) play an essential role in
combinatorics and theoretical computer science. The goal of this survey is to
provide an up-to-date account on properties of eigenvectors when the matrix (or
graph) is random.Comment: 64 pages, 1 figure; added Section 7 on localized eigenvector