1,520 research outputs found
Hamiltonian System Approach to Distributed Spectral Decomposition in Networks
Because of the significant increase in size and complexity of the networks,
the distributed computation of eigenvalues and eigenvectors of graph matrices
has become very challenging and yet it remains as important as before. In this
paper we develop efficient distributed algorithms to detect, with higher
resolution, closely situated eigenvalues and corresponding eigenvectors of
symmetric graph matrices. We model the system of graph spectral computation as
physical systems with Lagrangian and Hamiltonian dynamics. The spectrum of
Laplacian matrix, in particular, is framed as a classical spring-mass system
with Lagrangian dynamics. The spectrum of any general symmetric graph matrix
turns out to have a simple connection with quantum systems and it can be thus
formulated as a solution to a Schr\"odinger-type differential equation. Taking
into account the higher resolution requirement in the spectrum computation and
the related stability issues in the numerical solution of the underlying
differential equation, we propose the application of symplectic integrators to
the calculation of eigenspectrum. The effectiveness of the proposed techniques
is demonstrated with numerical simulations on real-world networks of different
sizes and complexities
Utilizing Graph Structure for Machine Learning
The information age has led to an explosion in the size and availability of data. This data often exhibits graph-structure that is either explicitly defined, as in the web of a social network, or is implicitly defined and can be determined by measuring similarity between objects. Utilizing this graph-structure allows for the design of machine learning algorithms that reflect not only the attributes of individual objects but their relationships to every other object in the domain as well. This thesis investigates three machine learning problems and proposes novel methods that leverage the graph-structure inherent in the tasks. Quantum walk neural networks are classical neural nets that use quantum random walks for classifying and regressing on graphs. Asymmetric directed node embeddings are another neural network architecture designed to embed the nodes of a directed graph into a vector space. Filtered manifold alignment is a novel two-step approach to domain adaptation
Dynamical localization and eigenstate localization in trap models
The one-dimensional random trap model with a power-law distribution of mean
sojourn times exhibits a phenomenon of dynamical localization in the case where
diffusion is anomalous: The probability to find two independent walkers at the
same site, as given by the participation ratio, stays constant and high in a
broad domain of intermediate times. This phenomenon is absent in dimensions two
and higher. In finite lattices of all dimensions the participation ratio
finally equilibrates to a different final value. We numerically investigate
two-particle properties in a random trap model in one and in three dimensions,
using a method based on spectral decomposition of the transition rate matrix.
The method delivers a very effective computational scheme producing numerically
exact results for the averages over thermal histories and initial conditions in
a given landscape realization. Only a single averaging procedure over disorder
realizations is necessary. The behavior of the participation ratio is compared
to other measures of localization, as for example to the states' gyration
radius, according to which the dynamically localized states are extended. This
means that although the particles are found at the same site with a high
probability, the typical distance between them grows. Moreover the final
equilibrium state is extended both with respect to its gyration radius and to
its Lyapunov exponent. In addition, we show that the phenomenon of dynamical
localization is only marginally connected with the spectrum of the transition
rate matrix, and is dominated by the properties of its eigenfunctions which
differ significantly in dimensions one and three.Comment: 10 pages, 10 figures, submitted to EPJ
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