40,486 research outputs found
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
A Spectral Graph Uncertainty Principle
The spectral theory of graphs provides a bridge between classical signal
processing and the nascent field of graph signal processing. In this paper, a
spectral graph analogy to Heisenberg's celebrated uncertainty principle is
developed. Just as the classical result provides a tradeoff between signal
localization in time and frequency, this result provides a fundamental tradeoff
between a signal's localization on a graph and in its spectral domain. Using
the eigenvectors of the graph Laplacian as a surrogate Fourier basis,
quantitative definitions of graph and spectral "spreads" are given, and a
complete characterization of the feasibility region of these two quantities is
developed. In particular, the lower boundary of the region, referred to as the
uncertainty curve, is shown to be achieved by eigenvectors associated with the
smallest eigenvalues of an affine family of matrices. The convexity of the
uncertainty curve allows it to be found to within by a fast
approximation algorithm requiring typically sparse
eigenvalue evaluations. Closed-form expressions for the uncertainty curves for
some special classes of graphs are derived, and an accurate analytical
approximation for the expected uncertainty curve of Erd\H{o}s-R\'enyi random
graphs is developed. These theoretical results are validated by numerical
experiments, which also reveal an intriguing connection between diffusion
processes on graphs and the uncertainty bounds.Comment: 40 pages, 8 figure
Spectral statistics of Erd\H{o}s-R\'{e}nyi graphs I: Local semicircle law
We consider the ensemble of adjacency matrices of Erd\H{o}s-R\'{e}nyi random
graphs, that is, graphs on vertices where every edge is chosen
independently and with probability . We rescale the matrix so
that its bulk eigenvalues are of order one. We prove that, as long as
(with a speed at least logarithmic in ), the density of
eigenvalues of the Erd\H{o}s-R\'{e}nyi ensemble is given by the Wigner
semicircle law for spectral windows of length larger than (up to
logarithmic corrections). As a consequence, all eigenvectors are proved to be
completely delocalized in the sense that the -norms of the
-normalized eigenvectors are at most of order with a very
high probability. The estimates in this paper will be used in the companion
paper [Spectral statistics of Erd\H{o}s-R\'{e}nyi graphs II: Eigenvalue spacing
and the extreme eigenvalues (2011) Preprint] to prove the universality of
eigenvalue distributions both in the bulk and at the spectral edges under the
further restriction that .Comment: Published in at http://dx.doi.org/10.1214/11-AOP734 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Step-By-Step Community Detection in Volume-Regular Graphs
Spectral techniques have proved amongst the most effective approaches to graph clustering. However, in general they require explicit computation of the main eigenvectors of a suitable matrix (usually the Laplacian matrix of the graph).
Recent work (e.g., Becchetti et al., SODA 2017) suggests that observing the temporal evolution of the power method applied to an initial random vector may, at least in some cases, provide enough information on the space spanned by the first two eigenvectors, so as to allow recovery of a hidden partition without explicit eigenvector computations. While the results of Becchetti et al. apply to perfectly balanced partitions and/or graphs that exhibit very strong forms of regularity, we extend their approach to graphs containing a hidden k partition and characterized by a milder form of volume-regularity. We show that the class of k-volume regular graphs is the largest class of undirected (possibly weighted) graphs whose transition matrix admits k "stepwise" eigenvectors (i.e., vectors that are constant over each set of the hidden partition). To obtain this result, we highlight a connection between volume regularity and lumpability of Markov chains. Moreover, we prove that if the stepwise eigenvectors are those associated to the first k eigenvalues and the gap between the k-th and the (k+1)-th eigenvalues is sufficiently large, the Averaging dynamics of Becchetti et al. recovers the underlying community structure of the graph in logarithmic time, with high probability
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