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We present a rigorous and precise analysis of the maximum degree and the average degree in a dynamic duplication-divergence graph model introduced by Sol\'e, Pastor-Satorras et al. in which the graph grows according to a duplication-divergence mechanism, i.e. by iteratively creating a copy of some node and then randomly alternating the neighborhood of a new node with probability $p$. This model captures the growth of some real-world processes e.g. biological or social networks. In this paper, we prove that for some $0 < p < 1$ the maximum degree and the average degree of a duplication-divergence graph on $t$ vertices are asymptotically concentrated with high probability around $t^p$ and $\max\{t^{2 p - 1}, 1\}$, respectively, i.e. they are within at most a polylogarithmic factor from these values with probability at least $1 - t^{-A}$ for any constant $A > 0$.

This research investigates the numerical approximation of the two-dimensional convection-dominated singularly perturbed problem on square, circular, and elliptic domains. Singularly perturbed boundary value problems present a significant challenge due to the presence of sharp boundary layers in their solutions. Additionally, the considered domain exhibits characteristic points, giving rise to a degenerate boundary layer problem. The stiffness of the problem is attributed to the sharp singular layers, which can result in substantial computational errors if not appropriately addressed. Traditional numerical methods typically require extensive mesh refinements near the boundary to achieve accurate solutions, which can be computationally expensive. To address the challenges posed by singularly perturbed problems, we employ physics-informed neural networks (PINNs). However, PINNs may struggle with rapidly varying singularly perturbed solutions over a small domain region, leading to inadequate resolution and potentially inaccurate or unstable results. To overcome this limitation, we introduce a semi-analytic method that augments PINNs with singular layers or corrector functions. Through our numerical experiments, we demonstrate significant improvements in both accuracy and stability, thus demonstrating the effectiveness of our proposed approach.

Robust inferential methods based on divergences measures have shown an appealing trade-off between efficiency and robustness in many different statistical models. In this paper, minimum density power divergence estimators (MDPDEs) for the scale and shape parameters of the log-logistic distribution are considered. The log-logistic is a versatile distribution modeling lifetime data which is commonly adopted in survival analysis and reliability engineering studies when the hazard rate is initially increasing but then it decreases after some point. Further, it is shown that the classical estimators based on maximum likelihood (MLE) are included as a particular case of the MDPDE family. Moreover, the corresponding influence function of the MDPDE is obtained, and its boundlessness is proved, thus leading to robust estimators. A simulation study is carried out to illustrate the slight loss in efficiency of MDPDE with respect to MLE and, at besides, the considerable gain in robustness.

Longitudinal studies are often subject to missing data. The ICH E9(R1) addendum addresses the importance of defining a treatment effect estimand with the consideration of intercurrent events. Jump-to-reference (J2R) is one classically envisioned control-based scenario for the treatment effect evaluation using the hypothetical strategy, where the participants in the treatment group after intercurrent events are assumed to have the same disease progress as those with identical covariates in the control group. We establish new estimators to assess the average treatment effect based on a proposed potential outcomes framework under J2R. Various identification formulas are constructed under the assumptions addressed by J2R, motivating estimators that rely on different parts of the observed data distribution. Moreover, we obtain a novel estimator inspired by the efficient influence function, with multiple robustness in the sense that it achieves $n^{1/2}$-consistency if any pairs of multiple nuisance functions are correctly specified, or if the nuisance functions converge at a rate not slower than $n^{-1/4}$ when using flexible modeling approaches. The finite-sample performance of the proposed estimators is validated in simulation studies and an antidepressant clinical trial.

In applications such as remote estimation and monitoring, update packets are transmitted by power-constrained devices using short-packet codes over wireless networks. Therefore, networks need to be end-to-end optimized using information freshness metrics such as age of information under transmit power and reliability constraints to ensure support for such applications. For short-packet coding, modelling and understanding the effect of block codeword length on transmit power and other performance metrics is important. To understand the above optimization for short-packet coding, we consider the optimal tradeoff problem between age of information and transmit power under reliability constraints for short packet point-to-point communication model with an exogenous packet generation process. In contrast to prior work, we consider scheduling policies that can possibly adapt the block-length or transmission time of short packet codes in order to achieve the optimal tradeoff. We characterize the tradeoff using a semi-Markov decision process formulation. We also obtain analytical upper bounds as well as numerical, analytical, and asymptotic lower bounds on the optimal tradeoff. We show that in certain regimes, such as high reliability and high packet generation rate, non-adaptive scheduling policies (fixed transmission time policies) are close-to-optimal. Furthermore, in a high-power or in a low-power regime, non-adaptive as well as state-independent randomized scheduling policies are order-optimal. These results are corroborated by numerical and simulation experiments. The tradeoff is then characterized for a wireless point-to-point channel with block fading as well as for other packet generation models (including an age-dependent packet generation model).

Due to their intrinsic capabilities on parallel signal processing, optical neural networks (ONNs) have attracted extensive interests recently as a potential alternative to electronic artificial neural networks (ANNs) with reduced power consumption and low latency. Preliminary confirmation of the parallelism in optical computing has been widely done by applying the technology of wavelength division multiplexing (WDM) in the linear transformation part of neural networks. However, inter-channel crosstalk has obstructed WDM technologies to be deployed in nonlinear activation in ONNs. Here, we propose a universal WDM structure called multiplexed neuron sets (MNS) which apply WDM technologies to optical neurons and enable ONNs to be further compressed. A corresponding back-propagation (BP) training algorithm is proposed to alleviate or even cancel the influence of inter-channel crosstalk on MNS-based WDM-ONNs. For simplicity, semiconductor optical amplifiers (SOAs) are employed as an example of MNS to construct a WDM-ONN trained with the new algorithm. The result shows that the combination of MNS and the corresponding BP training algorithm significantly downsize the system and improve the energy efficiency to tens of times while giving similar performance to traditional ONNs.

Data science and artificial intelligence have become an indispensable part of scientific research. While such methods rely on high-quality and large quantities of machine-readable scientific data, the current scientific data infrastructure faces significant challenges that limit effective data curation and sharing. These challenges include insufficient return on investment for researchers to share quality data, logistical difficulties in maintaining long-term data repositories, and the absence of standardized methods for evaluating the relative importance of various datasets. To address these issues, this paper presents the Lennard Jones Token, a blockchain-based proof-of-concept solution implemented on the Ethereum network. The token system incentivizes users to submit optimized structures of Lennard Jones particles by offering token rewards, while also charging for access to these valuable structures. Utilizing smart contracts, the system automates the evaluation of submitted data, ensuring that only structures with energies lower than those in the existing database for a given cluster size are rewarded. The paper explores the details of the Lennard Jones Token as a proof of concept and proposes future blockchain-based tokens aimed at enhancing the curation and sharing of scientific data.

We propose an innovative and generic methodology to analyse individual and collective behaviour through individual trajectory data. The work is motivated by the analysis of GPS trajectories of fishing vessels collected from regulatory tracking data in the context of marine biodiversity conservation and ecosystem-based fisheries management. We build a low-dimensional latent representation of trajectories using convolutional neural networks as non-linear mapping. This is done by training a conditional variational auto-encoder taking into account covariates. The posterior distributions of the latent representations can be linked to the characteristics of the actual trajectories. The latent distributions of the trajectories are compared with the Bhattacharyya coefficient, which is well-suited for comparing distributions. Using this coefficient, we analyse the variation of the individual behaviour of each vessel during time. For collective behaviour analysis, we build proximity graphs and use an extension of the stochastic block model for multiple networks. This model results in a clustering of the individuals based on their set of trajectories. The application to French fishing vessels enables us to obtain groups of vessels whose individual and collective behaviours exhibit spatio-temporal patterns over the period 2014-2018.

The relationship between the thermodynamic and computational characteristics of dynamical physical systems has been a major theoretical interest since at least the 19th century, and has been of increasing practical importance as the energetic cost of digital devices has exploded over the last half century. One of the most important thermodynamic features of real-world computers is that they operate very far from thermal equilibrium, in finite time, with many quickly (co-)evolving degrees of freedom. Such computers also must almost always obey multiple physical constraints on how they work. For example, all modern digital computers are periodic processes, governed by a global clock. Another example is that many computers are modular, hierarchical systems, with strong restrictions on the connectivity of their subsystems. This properties hold both for naturally occurring computers, like brains or Eukaryotic cells, as well as digital systems. These features of real-world computers are absent in 20th century analyses of the thermodynamics of computational processes, which focused on quasi-statically slow processes. However, the field of stochastic thermodynamics has been developed in the last few decades - and it provides the formal tools for analyzing systems that have exactly these features of real-world computers. We argue here that these tools, together with other tools currently being developed in stochastic thermodynamics, may help us understand at a far deeper level just how the fundamental physical properties of dynamic systems are related to the computation that they perform.

The remarkable practical success of deep learning has revealed some major surprises from a theoretical perspective. In particular, simple gradient methods easily find near-optimal solutions to non-convex optimization problems, and despite giving a near-perfect fit to training data without any explicit effort to control model complexity, these methods exhibit excellent predictive accuracy. We conjecture that specific principles underlie these phenomena: that overparametrization allows gradient methods to find interpolating solutions, that these methods implicitly impose regularization, and that overparametrization leads to benign overfitting. We survey recent theoretical progress that provides examples illustrating these principles in simpler settings. We first review classical uniform convergence results and why they fall short of explaining aspects of the behavior of deep learning methods. We give examples of implicit regularization in simple settings, where gradient methods lead to minimal norm functions that perfectly fit the training data. Then we review prediction methods that exhibit benign overfitting, focusing on regression problems with quadratic loss. For these methods, we can decompose the prediction rule into a simple component that is useful for prediction and a spiky component that is useful for overfitting but, in a favorable setting, does not harm prediction accuracy. We focus specifically on the linear regime for neural networks, where the network can be approximated by a linear model. In this regime, we demonstrate the success of gradient flow, and we consider benign overfitting with two-layer networks, giving an exact asymptotic analysis that precisely demonstrates the impact of overparametrization. We conclude by highlighting the key challenges that arise in extending these insights to realistic deep learning settings.