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There are many methods to estimate the quasi-stationary infected fraction of the SIS process on (random) graphs. A challenge is to adequately incorporate correlations, which is especially important in sparse graphs. Methods typically are either significantly biased in sparse graphs, or computationally very demanding already for small network sizes. In this paper we present a new method to determine the infected fraction in sparse graphs, which we test on Erd\H{o}s-R\'enyi graphs. Our method does take into account correlations and gives accurate predictions. At the same time, computations are very feasible and can easily be done even for large networks., Comment: 44 pages, 22 figures

Optimal frequency identification in astronomical datasets is crucial for variable star studies, exoplanet detection, and asteroseismology. Traditional period-finding methods often rely on specific parametric assumptions, employ binning procedures, or overlook the regression nature of the problem, limiting their applicability and precision. We aim to introduce a universal, nonparametric kernel regression method for optimal frequency determination that is generalizable, efficient, and robust across various astronomical data types. FINKER uses nonparametric kernel regression on folded datasets at different frequencies, selecting the optimal frequency by minimizing squared residuals. This technique inherently incorporates a weighting system that accounts for measurement uncertainties and facilitates multiband data analysis. We evaluate our method's performance across a range of frequencies pertinent to diverse data types and compare it with an established period-finding algorithm, conditional entropy. The method demonstrates superior performance in accuracy and robustness compared to existing algorithms, requiring fewer observations to identify significant frequencies reliably. It exhibits resilience against noise and adapts well to datasets with varying complexity.

Aims. Traditional star-galaxy classification techniques often rely on feature estimation from catalogues, a process susceptible to introducing inaccuracies, thereby potentially jeopardizing the classification's reliability. Certain galaxies, especially those not manifesting as extended sources, can be misclassified when their shape parameters and flux solely drive the inference. We aim to create a robust and accurate classification network for identifying stars and galaxies directly from astronomical images. By leveraging convolutional neural networks (CNN) and additional information about the source position, we aim to accurately classify all stars and galaxies within a survey, particularly those with a signal-to-noise ratio (S/N) near the detection limit. Methods. The AutoSourceID-Classifier (ASID-C) algorithm developed here uses 32x32 pixel single filter band source cutouts generated by the previously developed ASID-L code. ASID-C utilizes CNNs to distinguish these cutouts into stars or galaxies, leveraging their strong feature-learning capabilities. Subsequently, we employ a modified Platt Scaling calibration for the output of the CNN. This technique ensures that the derived probabilities are effectively calibrated, delivering precise and reliable results. Results. We show that ASID-C, trained on MeerLICHT telescope images and using the Dark Energy Camera Legacy Survey (DECaLS) morphological classification, outperforms similar codes like SourceExtractor. ASID-C opens up new possibilities for accurate celestial object classification, especially for sources with a S/N near the detection limit. Potential applications of ASID-C, like real-time star-galaxy classification and transient's host identification, promise significant contributions to astronomical research.

We study random walks on $\mathbb{Z}$ which have a linear (or almost linear) drift towards 0 in a range around 0. This drift leads to a metastable Gaussian distribution centered at zero. We give specific, fast growing, time windows where we can explicitely bound the distance of the distribution of the walk to an appropriate Gaussian. In this way we give a solid theoretical foundation to the notion of metastability. We show that the supercritical contact process on the complete graph has a drift towards its equilibrium point which is locally linear and that our results for random walks apply. This leads to the conclusion that the infected fraction of the population in metastability (when properly scaled) converges in distribution to a Gaussian, uniformly for all times in a fast growing interval.

Aims. In astronomy, machine learning has been successful in various tasks such as source localisation, classification, anomaly detection, and segmentation. However, feature regression remains an area with room for improvement. We aim to design a network that can accurately estimate sources' features and their uncertainties from single-band image cutouts, given the approximated locations of the sources provided by the previously developed code AutoSourceID-Light (ASID-L) or other external catalogues. This work serves as a proof of concept, showing the potential of machine learning in estimating astronomical features when trained on meticulously crafted synthetic images and subsequently applied to real astronomical data. Methods. The algorithm presented here, AutoSourceID-FeatureExtractor (ASID-FE), uses single-band cutouts of 32x32 pixels around the localised sources to estimate flux, sub-pixel centre coordinates, and their uncertainties. ASID-FE employs a two-step mean variance estimation (TS-MVE) approach to first estimate the features and then their uncertainties without the need for additional information, for example the point spread function (PSF). For this proof of concept, we generated a synthetic dataset comprising only point sources directly derived from real images, ensuring a controlled yet authentic testing environment. Results.We show that ASID-FE, trained on synthetic images derived from the MeerLICHT telescope, can predict more accurate features with respect to similar codes such as SourceExtractor and that the two-step method can estimate well-calibrated uncertainties that are better behaved compared to similar methods that use deep ensembles of simple MVE networks. Finally, we evaluate the model on real images from the MeerLICHT telescope and the Zwicky Transient Facility (ZTF) to test its transfer learning abilities.

Heterogeneous Markovian Susceptible-Infected-Susceptible (SIS) epidemics with a general infection rate matrix $\widetilde{A}$ are considered. Using a non-negative matrix factorization to approximate $\widetilde{A}$, we are able to identify when a metastable state can be expected, and that the metastable distribution, under certain conditions, will feature a normal distribution with known expectation and covariance. Furthermore, we model a heterogeneous Markovian SIS epidemic, that starts with a fraction of initially infected nodes different from that in the metastable state, by approximating its behaviour by a standard linear stochastic differential equation (SDE) in sufficiently high dimensions. By exploiting the knowledge of the covariance matrix from the SDE, we demonstrate significant accuracy improvements over the first-order mean-field approximation NIMFA., Comment: 20 pages and 5 figures

We use a Bayesian approach to derive the distance probability distribution for one object from its parallax with measurement uncertainty for two spatial distribution priors, viz. a homogeneous spherical distribution and a galactocentric distribution - applicable for radio pulsars - observed from Earth. We investigate the dependence on measurement uncertainty, and show that a parallax measurement can underestimate or overestimate the actual distance, depending on the spatial distribution prior. We derive the probability distributions for distance and luminosity combined, and for each separately, when a flux with measurement error for the object is also available, and demonstrate the necessity of and dependence on the luminosity function prior. We apply this to estimate the distance and the radio and gamma-ray luminosities of PSR J0218+4232. The use of realistic priors improves the quality of the estimates for distance and luminosity, compared to those based on measurement only. Use of a wrong prior, for example a homogeneous spatial distribution without upper bound, may lead to very wrong results., Comment: 10 pages, 9 figures, accepted 27-04-2016 to Astronomy and Astrophysics

We consider two dependent Brownian motions with (possibly) different drift, and apply a result by le Gall on cone points of two dimensional Brownian motion to show that with probability one, there will not be a time that is a local maximum for both processes., Comment: 2 pages

In this paper we will prove a shape theorem for the last passage percolation model on a two dimensional $F$-compound Poisson process, called the Hammersley model with random weights. We will also provide diffusive upper bounds for shape fluctuations. Finally we will indicate how these results can be used to prove existence and coalescence of semi-infinite geodesics in some fixed direction $\alpha$, following an approach developed by Newman and co-authors, and applied to the classical Hammersley process by W\"uthrich. These results will be crucial in the development of an upcoming paper on the relation between Busemann functions and equilibrium measures in last passage percolation models., Comment: 12 pages

In arXiv:0907.0079 by Cator and Lopuhaa, an asymptotic expansion for the MCD estimators is established in a very general framework. This expansion requires the existence and non-singularity of the derivative in a first-order Taylor expansion. In this paper, we prove the existence of this derivative for multivariate distributions that have a density and provide an explicit expression. Moreover, under suitable symmetry conditions on the density, we show that this derivative is non-singular. These symmetry conditions include the elliptically contoured multivariate location-scatter model, in which case we show that the minimum covariance determinant (MCD) estimators of multivariate location and covariance are asymptotically equivalent to a sum of independent identically distributed vector and matrix valued random elements, respectively. This provides a proof of asymptotic normality and a precise description of the limiting covariance structure for the MCD estimators., Comment: 21 pages