30,410 research outputs found
Network Density of States
Spectral analysis connects graph structure to the eigenvalues and
eigenvectors of associated matrices. Much of spectral graph theory descends
directly from spectral geometry, the study of differentiable manifolds through
the spectra of associated differential operators. But the translation from
spectral geometry to spectral graph theory has largely focused on results
involving only a few extreme eigenvalues and their associated eigenvalues.
Unlike in geometry, the study of graphs through the overall distribution of
eigenvalues - the spectral density - is largely limited to simple random graph
models. The interior of the spectrum of real-world graphs remains largely
unexplored, difficult to compute and to interpret.
In this paper, we delve into the heart of spectral densities of real-world
graphs. We borrow tools developed in condensed matter physics, and add novel
adaptations to handle the spectral signatures of common graph motifs. The
resulting methods are highly efficient, as we illustrate by computing spectral
densities for graphs with over a billion edges on a single compute node. Beyond
providing visually compelling fingerprints of graphs, we show how the
estimation of spectral densities facilitates the computation of many common
centrality measures, and use spectral densities to estimate meaningful
information about graph structure that cannot be inferred from the extremal
eigenpairs alone.Comment: 10 pages, 7 figure
Distributed Adaptive Learning of Graph Signals
The aim of this paper is to propose distributed strategies for adaptive
learning of signals defined over graphs. Assuming the graph signal to be
bandlimited, the method enables distributed reconstruction, with guaranteed
performance in terms of mean-square error, and tracking from a limited number
of sampled observations taken from a subset of vertices. A detailed mean square
analysis is carried out and illustrates the role played by the sampling
strategy on the performance of the proposed method. Finally, some useful
strategies for distributed selection of the sampling set are provided. Several
numerical results validate our theoretical findings, and illustrate the
performance of the proposed method for distributed adaptive learning of signals
defined over graphs.Comment: To appear in IEEE Transactions on Signal Processing, 201
Nonparametric spectral analysis with applications to seizure characterization using EEG time series
Understanding the seizure initiation process and its propagation pattern(s)
is a critical task in epilepsy research. Characteristics of the pre-seizure
electroencephalograms (EEGs) such as oscillating powers and high-frequency
activities are believed to be indicative of the seizure onset and spread
patterns. In this article, we analyze epileptic EEG time series using
nonparametric spectral estimation methods to extract information on
seizure-specific power and characteristic frequency [or frequency band(s)].
Because the EEGs may become nonstationary before seizure events, we develop
methods for both stationary and local stationary processes. Based on penalized
Whittle likelihood, we propose a direct generalized maximum likelihood (GML)
and generalized approximate cross-validation (GACV) methods to estimate
smoothing parameters in both smoothing spline spectrum estimation of a
stationary process and smoothing spline ANOVA time-varying spectrum estimation
of a locally stationary process. We also propose permutation methods to test if
a locally stationary process is stationary. Extensive simulations indicate that
the proposed direct methods, especially the direct GML, are stable and perform
better than other existing methods. We apply the proposed methods to the
intracranial electroencephalograms (IEEGs) of an epileptic patient to gain
insights into the seizure generation process.Comment: Published in at http://dx.doi.org/10.1214/08-AOAS185 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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