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The Airy line ensemble is a positive-integer indexed system of random continuous curves whose finite dimensional distributions are given by the multi-line Airy process. It is a natural object in the KPZ universality class: for example, its highest curve, the Airy In this paper, we employ the Brownian Gibbs property to make a close comparison between the Airy line ensemble’s curves after affine shift and Brownian bridge, proving the finiteness of a superpolynomially growing moment bound on Radon-Nikodym derivatives. We also determine the value of a natural exponent describing in Brownian last passage percolation the decay in probability for the existence of several near geodesics that are disjoint except for their common endpoints, where the notion of ‘near’ refers to a small deficit in scaled geodesic energy, with the parameter specifying this nearness tending to zero. To prove both results, we introduce a technique that may be useful elsewhere for finding upper bounds on probabilities of events concerning random systems of curves enjoying the Brownian Gibbs property. Several results in this article play a fundamental role in a further study of Brownian last passage percolation in three companion papers (Hammond 2017a,b,c), in which geodesic coalescence and geodesic energy profiles are investigated in scaled coordinates.

We consider the problem of estimating the graph associated with a binary Ising Markov random field. We describe a method based on ℓ₁-regularized logistic regression, in which the neighborhood of any given node is estimated by performing logistic regression subject to an ℓ₁-constraint. The method is analyzed under high-dimensional scaling in which both the number of nodes p and maximum neighborhood size d are allowed to grow as a function of the number of observations n. Our main results provide sufficient conditions on the triple (n, p, d) and the model parameters for the method to succeed in consistently estimating the neighborhood of every node in the graph simultaneously. With coherence conditions imposed on the population Fisher information matrix, we prove that consistent neighborhood selection can be obtained for sample sizes n = Ω(d³ log p) with exponentially decaying error. When these same conditions are imposed directly on the sample matrices, we show that a reduced sample size of n = Ω(d² log p) suffices for the method to estimate neighborhoods consistently. Although this paper focuses on the binary graphical models, we indicate how a generalization of the method of the paper would apply to general discrete Markov random fields.

There are many factors leading to construction safety accident. The rule presented under the influence of these factors should be a statistical random rule. To reveal those random rules and study the probability prediction method of construction safety accident, according to stochastic process theory, general stochastic process, Markov process and normal process are respectively used to simulate the risk-accident process in this paper. First, in the general-random-process-based analysis the probability of accidents in a period of time is calculated. Then, the Markov property of the construction safety risk evolution process is illustrated, and the analytical expression of probability density function of first-passage time of Markov-based risk-accident process is derived to calculate the construction safety probability. In the normal-process-based analysis, the construction safety probability formulas in cases of stationary normal risk process and non-stationary normal risk process with zero mean value are derived respectively. Finally, the number of accidents that may occur on construction site in a period is studied macroscopically based on Poisson process, and the probability distribution of time interval between adjacent accidents and the time of the nth accident are calculated respectively. The results provide useful reference for the prediction and management of construction accidents.

Communication of signals among nodes in a complex network poses fundamental problems of efficiency and cost. Routing of messages along shortest paths requires global information about the topology, while spreading by diffusion, which operates according to local topological features, is informationally \"cheap\" but inefficient. We introduce a stochastic model for network communication that combines local and global information about the network topology to generate biased random walks on the network. The model generates a continuous spectrum of dynamics that converge onto shortest-path and random-walk (diffusion) communication processes at the limiting extremes. We implement the model on two cohorts of human connectome networks and investigate the effects of varying the global information bias on the network's communication cost. We identify routing strategies that approach a (highly efficient) shortest-path communication process with a relatively small global information bias on the system's dynamics. Moreover, we show that the cost of routing messages from and to hub nodes varies as a function of the global information bias driving the system's dynamics. Finally, we implement the model to identify individual subject differences from a communication dynamics point of view. The present framework departs from the classical shortest paths vs. diffusion dichotomy, unifying both models under a single family of dynamical processes that differ by the extent to which global information about the network topology influences the routing patterns of neural signals traversing the network.

Stochastic models of biomolecular reaction networks are commonly employed in systems and synthetic biology to study the effects of stochastic fluctuations emanating from reactions involving species with low copy-numbers. For such models, the Kolmogorov’s forward equation is called the chemical master equation (CME), and it is a fundamental system of linear ordinary differential equations (ODEs) that describes the evolution of the probability distribution of the random state-vector representing the copy-numbers of all the reacting species. The size of this system is given by the number of states that are accessible by the chemical system, and for most examples of interest this number is either very large or infinite. Moreover, approximations that reduce the size of the system by retaining only a finite number of important chemical states (e.g. those with non-negligible probability) result in high-dimensional ODE systems, even when the number of reacting species is small. Consequently, accurate numerical solution of the CME is very challenging, despite the linear nature of the underlying ODEs. One often resorts to estimating the solutions via computationally intensive stochastic simulations. The goal of the present paper is to develop a novel deep-learning approach for computing solution statistics of high-dimensional CMEs by reformulating the stochastic dynamics using Kolmogorov’s backward equation. The proposed method leverages superior approximation properties of Deep Neural Networks (DNNs) to reliably estimate expectations under the CME solution for several user-defined functions of the state-vector. This method is algorithmically based on reinforcement learning and it only requires a moderate number of stochastic simulations (in comparison to typical simulation-based approaches) to train the “policy function”. This allows not just the numerical approximation of various expectations for the CME solution but also of its sensitivities with respect to all the reaction network parameters (e.g. rate constants). We provide four examples to illustrate our methodology and provide several directions for future research.

This study presents a novel approach to modeling breast cancer dynamics, one of the most significant health threats to women worldwide. Utilizing a piecewise mathematical framework, we incorporate both deterministic and stochastic elements of cancer progression. The model is divided into three distinct phases: (1) initial growth, characterized by a constant-order Caputo proportional operator (CPC), (2) intermediate growth, modeled by a variable-order CPC, and (3) advanced stages, capturing stochastic fluctuations in cancer cell populations using a stochastic operator. Theoretical analysis, employing fixed-point theory for the fractional-order phases and Ito calculus for the stochastic phase, establishes the existence and uniqueness of solutions. A robust numerical scheme, combining the nonstandard finite difference method for fractional models and the Euler-Maruyama method for the stochastic system, enables simulations of breast cancer progression under various scenarios. Critically, the model is validated against real breast cancer data from Saudi Arabia spanning 2004-2016. Numerical simulations accurately capture observed trends, demonstrating the model’s predictive capabilities. Further, we investigate the impact of chemotherapy and its associated cardiotoxicity, illustrating different treatment response scenarios through graphical representations. This piecewise fractional-stochastic model offers a powerful tool for understanding and predicting breast cancer dynamics, potentially informing more effective treatment strategies.

The inherent stochasticity in metabolic reactions is a potent source of phenotypic heterogeneity in cell populations, with potentially fundamental implications for cancer research. Natural instability in cellular metabolism Molecular fluctuations in individual metabolic reactions are widely thought to have little effect on cell growth, because of averaging over the many biochemical reactions involved. Now Sander Tans and colleagues use time-lapse microscopy to accurately determine the growth rate of single bacterial cells while also monitoring individual enzyme levels, and find that elemental molecular noise does propagate and cause variation in growth. Conversely, they observe that growth fluctuations propagate back to perturb gene expression, so that molecular noise in one gene can influence unrelated genes via growth. The results demonstrate that the inherent variability in metabolic reactions is a potent source of phenotypic heterogeneity in cell populations, with fundamental implications for cancer research. Elucidating the role of molecular stochasticity 1 in cellular growth is central to understanding phenotypic heterogeneity 2 and the stability of cellular proliferation 3 . The inherent stochasticity of metabolic reaction events 4 should have negligible effect, because of averaging over the many reaction events contributing to growth. Indeed, metabolism and growth are often considered to be constant for fixed conditions 5 , 6 . Stochastic fluctuations in the expression level 1 , 7 , 8 , 9 of metabolic enzymes could produce variations in the reactions they catalyse. However, whether such molecular fluctuations can affect growth is unclear, given the various stabilizing regulatory mechanisms 10 , 11 , 12 , the slow adjustment of key cellular components such as ribosomes 13 , 14 , and the secretion 15 and buffering 16 , 17 of excess metabolites. Here we use time-lapse microscopy to measure fluctuations in the instantaneous growth rate of single cells of Escherichia coli , and quantify time-resolved cross-correlations with the expression of lac genes and enzymes in central metabolism. We show that expression fluctuations of catabolically active enzymes can propagate and cause growth fluctuations, with transmission depending on the limitation of the enzyme to growth. Conversely, growth fluctuations propagate back to perturb expression. Accordingly, enzymes were found to transmit noise to other unrelated genes via growth. Homeostasis is promoted by a noise-cancelling mechanism that exploits fluctuations in the dilution of proteins by cell-volume expansion. The results indicate that molecular noise is propagated not only by regulatory proteins 18 , 19 but also by metabolic reactions. They also suggest that cellular metabolism is inherently stochastic, and a generic source of phenotypic heterogeneity.

The rich-get-richer rule reinforces actions that have been frequently chosen in the past. What happens to the evolution of individuals’ inclinations to choose an action when agents interact? Interaction tends to homogenize, while each individual dynamics tends to reinforce its own position. Interacting stochastic systems of reinforced processes have recently been considered in many papers, in which the asymptotic behavior is proven to exhibit almost sure synchronization. In this paper we consider models where, even if interaction among agents is present, absence of synchronization may happen because of the choice of an individual nonlinear reinforcement. We show how these systems can naturally be considered as models for coordination games or technological or opinion dynamics.

In this paper, we begin our study by exploring a hypothetical model of stochastic growth of a population, using a single-valued stochastic integral equation that incorporates the control of feeding and harvest. Taking into account the inaccuracies and uncertainties in the measurements, we are led to a nonlinear bilateral multivalued stochastic integral equation that contains multivalued stochastic integrals on both sides of the equation. Due to the possibility of absence of an element opposite to a fixed set, such an equation cannot be reduced to classical unilateral notation with the sign of sum of sets only on one side. The fundamental question arises: Is there a solution to the equation under consideration, and is it the only one? By imposing on the coefficients of the equation the condition of satisfying a certain integral inequality, we prove the existence and uniqueness of solution of the considered equation. The result is preceded by a few lemmas with the sequence of approximate solutions. We also show that solutions have the property of stability. Finally, it has been demonstrated that the results obtained can be applied to establish corresponding theorems for deterministic bilateral multivalued integral equations.

Phenotypic plasticity contributes significantly to treatment failure in many cancers. Despite the increased prevalence of experimental studies that interrogate this phenomenon, there remains a lack of applicable quantitative tools to characterise data, and importantly to distinguish between resistance as a discrete phenotype and a continuous distribution of phenotypes. To address this, we develop a stochastic individual-based model of plastic phenotype adaptation through a continuously-structured phenotype space in low-cell-count proliferation assays. That our model corresponds probabilistically to common partial differential equation models of resistance allows us to formulate a likelihood that captures the intrinsic noise ubiquitous to such experiments. We apply our framework to assess the identifiability of key model parameters in several population-level data collection regimes; in particular, parameters relating to the adaptation velocity and cell-to-cell heterogeneity. Significantly, we find that cell-to-cell heterogeneity is practically non-identifiable from both cell count and proliferation marker data, implying that population-level behaviours may be well characterised by homogeneous ordinary differential equation models. Additionally, we demonstrate that population-level data are insufficient to distinguish resistance as a discrete phenotype from a continuous distribution of phenotypes. Our results inform the design of both future experiments and future quantitative analyses that probe phenotypic plasticity in cancer.