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In this paper, we introduce a novel model for random hypergraphs based on weighted random connection models. In accordance with the standard theory for hypergraphs, this model is constructed from a bipartite graph. In our stochastic model, both vertex sets of this bipartite graph form marked Poisson point processes and the connection radius is inversely proportional to a product of suitable powers of the marks. Hence, our model is a common generalization of weighted random connection models and AB random geometric graphs. For this hypergraph model, we investigate the limit theory of a variety of graph-theoretic and topological characteristics such as higher-order degree distribution, Betti numbers of the associated Dowker complex as well as simplex counts. In particular, for the latter quantity we identify regimes of convergence to normal and to stable distribution depending on the heavy-tailedness of the weight distribution. We conclude our investigation by a simulation study and an application to the collaboration network extracted from the arXiv dataset., Comment: 29 pages. 11 figures

We present a process-level Poisson-approximation result for the degree-k vertices in a high-density weighted random connection model with preferential-attachment kernel in the unit volume. Our main focus lies on the impact of the left tails of the weight distribution for which we establish general criteria based on their small-weight quantiles. To illustrate that our conditions are broadly applicable, we verify them for weight distributions with polynomial and stretched exponential left tails. The proofs rest on truncation arguments and a recently established quantitative Poisson approximation result for functionals of Poisson point processes.

In this paper, we investigate the potential of the age-dependent random connection model (ADRCM) with the aim of representing higher-order networks. A key contribution of our work are probabilistic limit results in large domains. More precisely, we first prove that the higher-order degree distributions have a power-law tail. Second, we establish central limit theorems for the edge counts and Betti numbers of the ADRCM in the regime where the degree distribution is light tailed. Moreover, in the heavy-tailed regime, we prove that asymptotically, the recentered and suitably rescaled edge counts converge to a stable distribution. We also propose a modification of the ADRCM in the form of a thinning procedure that enables independent adjustment of the power-law exponents for vertex and edge degrees. To apply the derived theorems to finite networks, we conduct a simulation study illustrating that the power-law degree distribution exponents approach their theoretical limits for large networks. It also indicates that in the heavy-tailed regime, the limit distribution of the recentered and suitably rescaled Betti numbers is stable. We demonstrate the practical application of the theoretical results to real-world datasets by analyzing scientific collaboration networks based on data from arXiv., Comment: 28 pages

Ultracold gases of highly magnetic lanthanide atoms have enabled the realisation of dipolar quantum droplets and supersolids. However, future studies could be limited by the achievable atom numbers and hindered by high three-body loss rates. Here we study density-dependent atom loss in an ultracold gas of ${}^{166}$Er for magnetic fields below $4\ \textrm{G}$, identifying six previously unknown features which display both a strong temperature dependence and also sensitivity to the polarisation and intensity of the light used to optically trap the atoms. This detailed knowledge of the loss landscape allows us to optimise the production of dipolar BECs with more than $2 \times 10^5$ atoms and points towards optimal strategies for the study of large-atom-number dipolar gases in the droplet and supersolid regimes., Comment: 6 pages, 4 figures

Homogeneous quantum gases open up new possibilities for studying many-body phenomena and have now been realised for a variety of systems. For gases with short-range interactions the way to make the cloud homogeneous is, predictably, to trap it in an ideal (homogeneous) box potential. We show that creating a close to homogeneous dipolar gas in the roton regime, when long-range interactions are important, actually requires trapping particles in soft-walled (inhomogeneous) box-like potentials. In particular, we numerically explore a dipolar gas confined in a pancake trap which is harmonic along the polarisation axis and a cylindrically symmetric power-law potential $r^p$ radially. We find that intermediate $p$'s maximise the proportion of the sample that can be brought close to the critical density required to reach the roton regime, whereas higher $p$'s trigger density oscillations near the wall even when the bulk of the system is not in the roton regime. We characterise how the optimum density distribution depends on the shape of the trapping potential and find it is controlled by the trap wall steepness., Comment: 7 pages, 3 figures

We use a deep neural network to detect and place region-of-interest boxes around ultracold atom clouds in absorption and fluorescence images---with the ability to identify and bound multiple clouds within a single image. The neural network also outputs segmentation masks that identify the size, shape and orientation of each cloud from which we extract the clouds' Gaussian parameters. This allows 2D Gaussian fits to be reliably seeded thereby enabling fully automatic image processing.

In this paper, a network-based stochastic information propagation model is developed. The information flow is modeled by a probabilistic differential equation system. The numerical solution of these equations leads to the expected number of informed nodes as a function of time and reveals the relationship between the degrees of the nodes and their reception time. The validity of the model is justified by Monte Carlo network simulation through the analysis of information propagation in scale-free and Erdos-Renyi networks. It has been found that the developed model provides more accurate results compared to the widely used network-based SI mean-field model, especially in sparse networks.

Bitcoin is a digital currency and electronic payment system operating over a peer-to-peer network on the Internet. One of its most important properties is the high level of anonymity it provides for its users. The users are identified by their Bitcoin addresses, which are random strings in the public records of transactions, the blockchain. When a user initiates a Bitcoin-transaction, his Bitcoin client program relays messages to other clients through the Bitcoin network. Monitoring the propagation of these messages and analyzing them carefully reveal hidden relations. In this paper, we develop a mathematical model using a probabilistic approach to link Bitcoin addresses and transactions to the originator IP address. To utilize our model, we carried out experiments by installing more than a hundred modified Bitcoin clients distributed in the network to observe as many messages as possible. During a two month observation period we were able to identify several thousand Bitcoin clients and bind their transactions to geographical locations.