733,167 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
Efficient Tree Tensor Network States (TTNS) for Quantum Chemistry: Generalizations of the Density Matrix Renormalization Group Algorithm
We investigate tree tensor network states for quantum chemistry. Tree tensor
network states represent one of the simplest generalizations of matrix product
states and the density matrix renormalization group. While matrix product
states encode a one-dimensional entanglement structure, tree tensor network
states encode a tree entanglement structure, allowing for a more flexible
description of general molecules. We describe an optimal tree tensor network
state algorithm for quantum chemistry. We introduce the concept of
half-renormalization which greatly improves the efficiency of the calculations.
Using our efficient formulation we demonstrate the strengths and weaknesses of
tree tensor network states versus matrix product states. We carry out benchmark
calculations both on tree systems (hydrogen trees and \pi-conjugated
dendrimers) as well as non-tree molecules (hydrogen chains, nitrogen dimer, and
chromium dimer). In general, tree tensor network states require much fewer
renormalized states to achieve the same accuracy as matrix product states. In
non-tree molecules, whether this translates into a computational savings is
system dependent, due to the higher prefactor and computational scaling
associated with tree algorithms. In tree like molecules, tree network states
are easily superior to matrix product states. As an ilustration, our largest
dendrimer calculation with tree tensor network states correlates 110 electrons
in 110 active orbitals.Comment: 15 pages, 19 figure
Riemann-Theta Boltzmann Machine
A general Boltzmann machine with continuous visible and discrete integer
valued hidden states is introduced. Under mild assumptions about the connection
matrices, the probability density function of the visible units can be solved
for analytically, yielding a novel parametric density function involving a
ratio of Riemann-Theta functions. The conditional expectation of a hidden state
for given visible states can also be calculated analytically, yielding a
derivative of the logarithmic Riemann-Theta function. The conditional
expectation can be used as activation function in a feedforward neural network,
thereby increasing the modelling capacity of the network. Both the Boltzmann
machine and the derived feedforward neural network can be successfully trained
via standard gradient- and non-gradient-based optimization techniques.Comment: 29 pages, 11 figures, final version published in Neurocomputin
Atomistic models of hydrogenated amorphous silicon nitride from first principles
We present a theoretical study of hydrogenated amorphous silicon nitride (a-SiNx:H), with equal concentrations of Si and N atoms (x=1), for two considerably different densities (2.0 and 3.0 g/cm3). Densities and hydrogen concentration were chosen according to experimental data. Using first-principles molecular-dynamics within density-functional theory the models were generated by cooling from the liquid. Where both models have a short-range order resembling that of crystalline Si3N4 because of their different densities and hydrogen concentrations they show marked differences at longer length scales. The low-density nitride forms a percolating network of voids with the internal surfaces passivated by hydrogen. Although some voids are still present for the high-density nitride, this material has a much denser and uniform space filling. The structure factors reveal some tendency for the nonstoichiometric high-density nitride to phase separate into nitrogen rich and poor areas. For our slowest cooling rate (0.023 K/fs) we obtain models with a modest number of defect states, where the low (high) density nitride favors undercoordinated (overcoordinated) defects. Analysis of the structural defects and electronic density of states shows that there is no direct one-to-one correspondence between the structural defects and states in the gap. There are several structural defects that do not contribute to in-gap states and there are in-gap states that do only have little to no contributions from (atoms in) structural defects. Finally an estimation of the size and cooling rate effects on the amorphous network is reported.
1/f Noise in Electron Glasses
We show that 1/f noise is produced in a 3D electron glass by charge
fluctuations due to electrons hopping between isolated sites and a percolating
network at low temperatures. The low frequency noise spectrum goes as
\omega^{-\alpha} with \alpha slightly larger than 1. This result together with
the temperature dependence of \alpha and the noise amplitude are in good
agreement with the recent experiments. These results hold true both with a
flat, noninteracting density of states and with a density of states that
includes Coulomb interactions. In the latter case, the density of states has a
Coulomb gap that fills in with increasing temperature. For a large Coulomb gap
width, this density of states gives a dc conductivity with a hopping exponent
of approximately 0.75 which has been observed in recent experiments. For a
small Coulomb gap width, the hopping exponent approximately 0.5.Comment: 8 pages, Latex, 6 encapsulated postscript figures, to be published in
Phys. Rev.
Spectral properties of the trap model on sparse networks
One of the simplest models for the slow relaxation and aging of glasses is
the trap model by Bouchaud and others, which represents a system as a point in
configuration-space hopping between local energy minima. The time evolution
depends on the transition rates and the network of allowed jumps between the
minima. We consider the case of sparse configuration-space connectivity given
by a random graph, and study the spectral properties of the resulting master
operator. We develop a general approach using the cavity method that gives
access to the density of states in large systems, as well as localisation
properties of the eigenvectors, which are important for the dynamics. We
illustrate how, for a system with sparse connectivity and finite temperature,
the density of states and the average inverse participation ratio have
attributes that arise from a non-trivial combination of the corresponding mean
field (fully connected) and random walk (infinite temperature) limits. In
particular, we find a range of eigenvalues for which the density of states is
of mean-field form but localisation properties are not, and speculate that the
corresponding eigenvectors may be concentrated on extensively many clusters of
network sites.Comment: 41 pages, 15 figure
Controlling chaos in a chaotic neural network
The chaotic neural network constructed with chaotic neuron shows the associative memory function, but its memory searching process cannot be stabilized in a stored state because of the chaotic motion of the network. In this paper, a pinning control method focused on the chaotic neural network is proposed. The computer simulation proves that the chaos in the chaotic neural network can be controlled with this method and the states of the network can converge in one of its stored patterns if the control strength and the pinning density are chosen suitable. It is found that in general the threshold of the control strength of a controlled network is smaller at higher pinned density and the chaos of the chaotic neural network can be controlled more easily if the pinning control is added to the variant neurons between the initial pattern and the target pattern
Multi-dimensional Density of States by Multicanonical Monte Carlo
Multi-dimensional density of states provides a useful description of complex
frustrated systems. Recent advances in Monte Carlo methods enable efficient
calculation of the density of states and related quantities, which renew the
interest in them. Here we calculate density of states on the plane (energy,
magnetization) for an Ising Model with three-spin interactions on a random
sparse network, which is a system of current interest both in physics of glassy
systems and in the theory of error-correcting codes. Multicanonical Monte Carlo
algorithm is successfully applied, and the shape of densities and its
dependence on the degree of frustration is revealed. Efficiency of
multicanonical Monte Carlo is also discussed with the shape of a projection of
the distribution simulated by the algorithm.Comment: Presented at SPDSA 2004, Hayama, Japa
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