6,157 research outputs found
Betti number signatures of homogeneous Poisson point processes
The Betti numbers are fundamental topological quantities that describe the
k-dimensional connectivity of an object: B_0 is the number of connected
components and B_k effectively counts the number of k-dimensional holes.
Although they are appealing natural descriptors of shape, the higher-order
Betti numbers are more difficult to compute than other measures and so have not
previously been studied per se in the context of stochastic geometry or
statistical physics.
As a mathematically tractable model, we consider the expected Betti numbers
per unit volume of Poisson-centred spheres with radius alpha. We present
results from simulations and derive analytic expressions for the low intensity,
small radius limits of Betti numbers in one, two, and three dimensions. The
algorithms and analysis depend on alpha-shapes, a construction from
computational geometry that deserves to be more widely known in the physics
community.Comment: Submitted to PRE. 11 pages, 10 figure
Fast maximum likelihood estimation via equilibrium expectation for large network data
This is the final version. Available from the publisher via the DOI in this record.A major line of contemporary research on complex networks is based on the development of statistical models that specify the local motifs associated with macro-structural properties observed in actual networks. This statistical approach becomes increasingly problematic as network size increases. In the context of current research on efficient estimation of models for large network data sets, we propose a fast algorithm for maximum likelihood estimation (MLE) that affords a significant increase in the size of networks amenable to direct empirical analysis. The algorithm we propose in this paper relies on properties of Markov chains at equilibrium, and for this reason it is called equilibrium expectation (EE). We demonstrate the performance of the EE algorithm in the context of exponential random graph models (ERGMs) a family of statistical models commonly used in empirical research based on network data observed at a single period in time. Thus far, the lack of efficient computational strategies has limited the empirical scope of ERGMs to relatively small networks with a few thousand nodes. The approach we propose allows a dramatic increase in the size of networks that may be analyzed using ERGMs. This is illustrated in an analysis of several biological networks and one social network with 104,103 nodes.Swiss National Science Foundatio
Bayesian Exponential Random Graph Models with Nodal Random Effects
We extend the well-known and widely used Exponential Random Graph Model
(ERGM) by including nodal random effects to compensate for heterogeneity in the
nodes of a network. The Bayesian framework for ERGMs proposed by Caimo and
Friel (2011) yields the basis of our modelling algorithm. A central question in
network models is the question of model selection and following the Bayesian
paradigm we focus on estimating Bayes factors. To do so we develop an
approximate but feasible calculation of the Bayes factor which allows one to
pursue model selection. Two data examples and a small simulation study
illustrate our mixed model approach and the corresponding model selection.Comment: 23 pages, 9 figures, 3 table
Statistical Mechanics of Steiner trees
The Minimum Weight Steiner Tree (MST) is an important combinatorial
optimization problem over networks that has applications in a wide range of
fields. Here we discuss a general technique to translate the imposed global
connectivity constrain into many local ones that can be analyzed with cavity
equation techniques. This approach leads to a new optimization algorithm for
MST and allows to analyze the statistical mechanics properties of MST on random
graphs of various types
Quantum projection noise limited interferometry with coherent atoms in a Ramsey type setup
Every measurement of the population in an uncorrelated ensemble of two-level
systems is limited by what is known as the quantum projection noise limit.
Here, we present quantum projection noise limited performance of a Ramsey type
interferometer using freely propagating coherent atoms. The experimental setup
is based on an electro-optic modulator in an inherently stable Sagnac
interferometer, optically coupling the two interfering atomic states via a
two-photon Raman transition. Going beyond the quantum projection noise limit
requires the use of reduced quantum uncertainty (squeezed) states. The
experiment described demonstrates atom interferometry at the fundamental noise
level and allows the observation of possible squeezing effects in an atom
laser, potentially leading to improved sensitivity in atom interferometers.Comment: 8 pages, 8 figures, published in Phys. Rev.
Bosenova and three-body loss in a Rb-85 Bose-Einstein condensate
Collapsing Bose-Einstein condensates are rich and complex quantum systems for
which quantitative explanation by simple models has proved elusive. We present
new experimental data on the collapse of high density Rb-85 condensates with
attractive interactions and find quantitative agreement with the predictions of
the Gross-Pitaevskii equation. The collapse data and measurements of the decay
of atoms from our condensates allow us to put new limits on the value of the
Rb-85 three-body loss coefficient K_3 at small positive and negative scattering
lengths.Comment: 6 pages, 5 figure
Cold atom gravimetry with a Bose-Einstein Condensate
We present a cold atom gravimeter operating with a sample of Bose-condensed
Rubidium-87 atoms. Using a Mach-Zehnder configuration with the two arms
separated by a two-photon Bragg transition, we observe interference fringes
with a visibility of 83% at T=3 ms. We exploit large momentum transfer (LMT)
beam splitting to increase the enclosed space-time area of the interferometer
using higher-order Bragg transitions and Bloch oscillations. We also compare
fringes from condensed and thermal sources, and observe a reduced visibility of
58% for the thermal source. We suspect the loss in visibility is caused partly
by wavefront aberrations, to which the thermal source is more susceptible due
to its larger transverse momentum spread. Finally, we discuss briefly the
potential advantages of using a coherent atomic source for LMT, and present a
simple mean-field model to demonstrate that with currently available
experimental parameters, interaction-induced dephasing will not limit the
sensitivity of inertial measurements using freely-falling, coherent atomic
sources.Comment: 6 pages, 4 figures. Final version, published PR
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