273,925 research outputs found
Shock waves on complex networks
Power grids, road maps, and river streams are examples of infrastructural
networks which are highly vulnerable to external perturbations. An abrupt local
change of load (voltage, traffic density, or water level) might propagate in a
cascading way and affect a significant fraction of the network. Almost
discontinuous perturbations can be modeled by shock waves which can eventually
interfere constructively and endanger the normal functionality of the
infrastructure. We study their dynamics by solving the Burgers equation under
random perturbations on several real and artificial directed graphs. Even for
graphs with a narrow distribution of node properties (e.g., degree or
betweenness), a steady state is reached exhibiting a heterogeneous load
distribution, having a difference of one order of magnitude between the highest
and average loads. Unexpectedly we find for the European power grid and for
finite Watts-Strogatz networks a broad pronounced bimodal distribution for the
loads. To identify the most vulnerable nodes, we introduce the concept of
node-basin size, a purely topological property which we show to be strongly
correlated to the average load of a node
Topological Properties of Citation and Metabolic Networks
Topological properties of "scale-free" networks are investigated by
determining their spectral dimensions , which reflect a diffusion process
in the corresponding graphs. Data bases for citation networks and metabolic
networks together with simulation results from the growing network model
\cite{barab} are probed. For completeness and comparisons lattice, random,
small-world models are also investigated. We find that is around 3 for
citation and metabolic networks, which is significantly different from the
growing network model, for which is approximately 7.5. This signals a
substantial difference in network topology despite the observed similarities in
vertex order distributions. In addition, the diffusion analysis indicates that
whereas the citation networks are tree-like in structure, the metabolic
networks contain many loops.Comment: 11 pages, 3 figure
Holographic coherent states from random tensor networks
Random tensor networks provide useful models that incorporate various
important features of holographic duality. A tensor network is usually defined
for a fixed graph geometry specified by the connection of tensors. In this
paper, we generalize the random tensor network approach to allow quantum
superposition of different spatial geometries. We set up a framework in which
all possible bulk spatial geometries, characterized by weighted adjacent
matrices of all possible graphs, are mapped to the boundary Hilbert space and
form an overcomplete basis of the boundary. We name such an overcomplete basis
as holographic coherent states. A generic boundary state can be expanded on
this basis, which describes the state as a superposition of different spatial
geometries in the bulk. We discuss how to define distinct classical geometries
and small fluctuations around them. We show that small fluctuations around
classical geometries define "code subspaces" which are mapped to the boundary
Hilbert space isometrically with quantum error correction properties. In
addition, we also show that the overlap between different geometries is
suppressed exponentially as a function of the geometrical difference between
the two geometries. The geometrical difference is measured in an area law
fashion, which is a manifestation of the holographic nature of the states
considered.Comment: 33 pages, 8 figures. An error corrected on page 14. Reference update
Microwave Experiments on Graphs Simulating Spin-1/2 System
In this work I study the statistical properties of the Gaussian symplectic ensemble
(GSE) by means of microwave experiments on quantum graphs mimicking spin-1/2
systems. Additionally, the transport property of three terminal microwave graphs
with orthogonal, unitary and symplectic symmetry is investigated.
In the first part of this thesis, following the spirit of the idea proposed by Joyner
et al. we construct microwave quantum graphs to realize a antiunitary symmetry T
that squares to minus one, T^2 = -1. This symmetry induces degenerate eigenvalues,
which are called Kramers doublets. If the classical dynamics of the system is
chaotic, statistical features of the spectrum can be well described by the corresponding
statistics of random matrix Gaussian symplectic ensemble. Indeed, Kramers
doublets are observed in reflection spectrum as expected from the scattering properties
of a symplectic graph. The level spacing distribution of these doublets is compared
with the corresponding random matrix predictions. Since the level spacing
distribution accounts for the short range eigenvalue correlation, to study the spectral
long range correlation the spectral two point correlation function and its Fourier
transform, the spectral form factor are analyzed. In order to further examine the
fluctuation of the eigenvalues smoothed quantities such as number variance and
spectral rigidity are discussed. The graphs used in the experiment consist of two
subgraphs coupled via one pair of connecting bonds. Theoretical study shows that
the level spacing distribution for graphs with one pair of connecting bonds deviates
by few percents from the random matrix prediction. This difference is too small to
be resolved in the experiment. The one pair of bonds approximation is introduced
to better understand the symplectic graph we used in the experiment. This model is
extended to address more general cases of the symplectic graph. Finally, the parameter
dependent dynamical transition of the statistical features of the spectrum from
GSE via Gaussian unitary ensemble (GUE) to Gaussian orthogonal ensemble (GOE)
is studied.
In the second part of this thesis, the collaborative work with Dr. A. M. MartĂnez-
ArgĂĽello from Mexico is briefly presented. A three terminal setup is proposed to
study the universal transport properties of systems with orthogonal, unitary and
symplectic symmetry. The probability distribution for a transport related quantity is
predicted analytically and microwave graphs are constructed to test this prediction.
The absorption within the system is modeled by effective Hamiltonian approach.
The parameters of the absorption and coupling are extracted from the experimental
autocorrelation function. This allowed a comparison between experiment and theory
without any free parameters. Finally, a quantitative good agreement between
experiment and theory was found for all three symmetry classes
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