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The long-time behavior of many complex molecular systems can often be described by Markovian dynamics in a slow subspace spanned by a few reaction coordinates referred to as collective variables (CVs). However, determining CVs poses a fundamental challenge in chemical physics. Depending on intuition or trial and error to construct CVs can lead to non-Markovian dynamics with long memory effects, hindering analysis. To address this problem, we continue to develop a recently introduced deep-learning technique called spectral map [J. Rydzewski, J. Phys. Chem. Lett. 14, 5216–5220 (2023)]. Spectral map learns slow CVs by maximizing a spectral gap of a Markov transition matrix describing anisotropic diffusion. Here, to represent heterogeneous and multiscale free-energy landscapes with spectral map, we implement an adaptive algorithm to estimate transition probabilities. Through a Markov state model analysis, we validate that spectral map learns slow CVs related to the dominant relaxation timescales and discerns between long-lived metastable states. [ABSTRACT FROM AUTHOR]

In a celebrated paper of Marcus and Ree [M. Marcus, R. Ree. Diagonals of doubly stochastic matrices. Quart J Math Oxford Ser. 1959; 10(1):296–302], it was shown that if $ A=[a_{ij}] $ A = [ a ij ] is an $ n \times n $ n × n doubly stochastic matrix, then there is a permutation $ \sigma \in S_n $ σ ∈ S n such that $ \sum _{i,j=1}^{n} a_{i,j}^{2} \leq \sum _{i=1}^{n} a_{i,\sigma (i)} $ ∑ i , j = 1 n a i , j 2 ≤ ∑ i = 1 n a i , σ (i) . Erdős asked for which doubly stochastic matrices the inequality is saturated. Although Marcus and Ree provided some insight for the set of solutions, the question appears to have fallen into oblivion. Our goal is to provide a complete answer in the particular, yet non-trivial, case when n = 3. [ABSTRACT FROM AUTHOR]

The article "Commentary: Accelerating spiking neural network simulations with PymoNNto and PymoNNtorch" published in Frontiers in Neuroinformatics compares different simulators for spiking neural networks. The study reveals that discrepancies in firing rates between simulators are due to the use of an unsuitable numerical integration method. The analysis confirms that only NEST simulator accurately simulates the model mathematically. The study also uncovers a subtle bug in NESTML and emphasizes the importance of precise benchmarks in simulation studies. [Extracted from the article]

Featured Application: This study can be used for navigation operations of agricultural equipment or vehicles in farm environments. Due to the limitations of low coverage, high repetition rate, and slow convergence speed of the basic genetic algorithm (GA) in robot complete coverage path planning, the state transition matrix of the Markov chain is introduced to guide individual mutation based on the genetic mutation path planning algorithm, which can improve the quality of population individuals, enhancing the search ability and convergence speed of the genetic algorithm. The proposed improved genetic algorithm is used for complete coverage path planning simulation analysis in different work areas. The analysis results show that compared to traditional genetic algorithms, the improved genetic algorithm proposed in this paper reduces the average path length by 21.8%, the average number of turns by 6 times, the repetition rate by 83.8%, and the coverage rate by 7.76% in 6 different work areas. The results prove that the proposed improved genetic algorithm is applicable in complete coverage path planning. To verify whether the Markov chain genetic algorithm (MCGA) proposed is suitable for agricultural robot path tracking and operation, it was used to plan the path of an actual land parcel. An automatic navigation robot can track the planned path, which can verify the feasibility of the MCGA proposed. [ABSTRACT FROM AUTHOR]

Despite the availability of electronic navigators and automated vehicles, static wayfinding instructions remain widely used due to their resistance to signal disturbances, as well as their economic and environmental advantages over electronic signs. To investigate the impact of static wayfinding on the network cost and flow distribution, this paper presents an efficient method for updating the incident matrix and extends the stochastic user equilibrium (SUE) framework to incorporate static wayfinding instructions by using the generalized nested logit (GNL) choice model to represent user behavior. The SUE principle relaxes the assumption that users possess perfect knowledge of traffic conditions and always choose the optimal link to minimize their costs. The GNL model improves generalization performance of the cross-nested logit (CNL) model while solving the overlap problem of the multinomial logit (MNL) model. The disaggregate simplicial decomposition (DSD) algorithm is applied to solve proposed user equilibrium by iteratively finding decent directions through an auxiliary solution and determining step size using different methods. The results indicate that the self-regulated averaging (SRA) method can solve the proposed model efficiently. Additionally, increasing travel time cost on guided links and even outer links can be potential influences caused by static wayfinding instructions. The study results can assist decision-makers in quantitatively assessing the value of placing static wayfinding instructions at certain locations and choosing effective layout information. [ABSTRACT FROM AUTHOR]

We analyze trade specialization dynamics in Botswana, applying the Galtonian regression and Markov matrix on the Lafay index for the period of 1998 to 2019. Galtonian regressions reveal de-specialization – Botswana gained a comparative advantage in industries for which it was initially not specialized and became less competitive in industries for which it was initially specialized. The industry hierarchy was altered only slightly due to persistence in trade specialization patterns. The Markov matrix reveals stronger upward industry mobility than downward industry mobility. De-specialization suggests slow-paced economic diversification, induced by import substitution in de-specialized industries, rather than by an expanding export base. [ABSTRACT FROM AUTHOR]

In recent years, the non-invasive load monitoring (NILM) method based on sparse coding has shown promising research prospects. This type of method learns a sparse dictionary for each monitoring target device, and it expresses load decomposition as a problem of signal reconstruction using dictionaries and sparse vectors. The existing NILM methods based on sparse coding have problems such as inability to be applied to multi-state and time-varying devices, single-load characteristics, and poor recognition ability for similar devices in distributed manners. Using the analysis above, this paper focuses on devices with similar features in households and proposes a distributed non-invasive load monitoring method using Karhunen–Loeve (KL) feature extraction and an improved deep dictionary. Firstly, Karhunen–Loeve expansion (KLE) is used to perform subspace expansion on the power waveform of the target device, and a new load feature is extracted by combining singular value decomposition (SVD) dimensionality reduction. Afterwards, the states of all the target devices are modeled as super states, and an improved deep dictionary based on the distance separability measure function (DSM-DDL) is learned for each super state. Among them, the state transition probability matrix and observation probability matrix in the hidden Markov model (HMM) are introduced as the basis for selecting the dictionary order during load decomposition. The KL feature matrix of power observation values and improved depth dictionary are used to discriminate the current super state based on the minimum reconstruction error criterion. The test results based on the UK-DALE dataset show that the KL feature matrix can effectively reduce the load similarity of devices. Combined with DSM-DDL, KL has a certain information acquisition ability and acceptable computational complexity, which can effectively improve the load decomposition accuracy of similar devices, quickly and accurately estimating the working status and power demand of household appliances. [ABSTRACT FROM AUTHOR]

Accurate forecasting of high-resolution particulate matter 2.5 (PM2.5) levels is essential for the development of public health policy. However, datasets used for this purpose often contain missing observations. This study presents a two-stage approach to handle this problem. The first stage is a multivariate spatial time series (MSTS) model, used to generate forecasts for the sampled spatial units and to impute missing observations. The MSTS model utilizes the similarities between the temporal patterns of the time series of the spatial units to impute the missing data across space. The second stage is the high-resolution prediction model, which generates predictions that cover the entire study domain. The second stage faces the big N problem giving rise to complex memory and computational problems. As a solution to the big N problem, we propose a Gaussian Markov random field (GMRF) for innovations with the Matérn covariance matrix obtained from the corresponding Gaussian field (GF) matrix by means of the stochastic partial differential equation (SPDE) method and the finite element method (FEM). For inference, we propose Bayesian statistics and integrated nested Laplace approximation (INLA) in the R-INLA package. The above approach is demonstrated using daily data collected from 13 PM2.5 monitoring stations in Jakarta Province, Indonesia, for 1 January–31 December 2022. The first stage of the model generates PM2.5 forecasts for the 13 monitoring stations for the period 1–31 January 2023, imputing missing data by means of the MSTS model. To capture temporal trends in the PM2.5 concentrations, the model applies a first-order autoregressive process and a seasonal process. The second stage involves creating a high-resolution map for the period 1–31 January 2023, for sampled and non-sampled spatiotemporal units. It uses the MSTS-generated PM2.5 predictions for the sampled spatiotemporal units and observations of the covariate's altitude, population density, and rainfall for sampled and non-samples spatiotemporal units. For the spatially correlated random effects, we apply a first-order random walk process. The validation of out-of-sample forecasts indicates a strong model fit with low mean squared error (0.001), mean absolute error (0.037), and mean absolute percentage error (0.041), and a high R² value (0.855). The analysis reveals that altitude and precipitation negatively impact PM2.5 concentrations, while population density has a positive effect. Specifically, a one-meter increase in altitude is linked to a 7.8% decrease in PM2.5, while a one-person increase in population density leads to a 7.0% rise in PM2.5. Additionally, a one-millimeter increase in rainfall corresponds to a 3.9% decrease in PM2.5. The paper makes a valuable contribution to the field of forecasting high-resolution PM2.5 levels, which is essential for providing detailed, accurate information for public health policy. The approach presents a new and innovative method for addressing the problem of missing data and high-resolution forecasting. [ABSTRACT FROM AUTHOR]

We investigate the convex polytope Ω n (123 ‾) of doubly stochastic matrices spanned by the n × n permutation matrices that avoid an increasing pattern of length 3, the 123 ‾ -permutation matrices. We determine some of its facets and other faces, including faces of small dimension, and their connection to facets and faces of the polytope Ω n of all n × n doubly stochastic matrices. The paper concludes with some relations concerning the frequencies of the positions of the 1's in n × n 123 ‾ -permutation matrices, and the vertex-edge graph of Ω n (123 ‾). [ABSTRACT FROM AUTHOR]

Background: Traditional recombinant inbred lines (RILs) are generated from repeated self-fertilization or brother-sister mating from the F1 hybrid of two inbred parents. Compared with the F2 population, RILs cumulate more crossovers between loci and thus increase the number of recombinants, resulting in an increased resolution of genetic mapping. Since they are inbred to the isogenic stage, another consequence of the heterozygosity reduction is the increased genetic variance and thus the increased power of QTL detection. Self-fertilization is the primary form of developing RILs in plants. Brother-sister mating is another way to develop RILs but in small laboratory animals. To ensure that the RILs have at least 98% of homozygosity, we need about seven generations of self-fertilization or 20 generations of brother-sister mating. Prior to homozygosity, these lines are called pre-recombinant inbred lines (PRERIL). Phenotypic values of traits in PRERILs are often collected but not used in QTL mapping. To perform QTL mapping in PRERILs, we need the recombination fraction between two markers at generation t for t < 7 (selfing) or t < 20 (brother-sister mating) so that the genotypes of QTL flanked by the markers can be inferred. Results: In this study, we developed formulas to calculate the recombination fractions of PRERILs at generation t in self-fertilization, brother-sister mating, and random mating. In contrast to existing works in this topic, we used computer code to construct the transition matrix to form the Markov chain of genotype array between consecutive generations, the so-called recurrent equations. Conclusions: We provide R functions to calculate the recombination fraction using the newly developed recurrent equations of ordered genotype array. With the recurrent equations and the R code, users can perform QTL mapping in PRERILs. Substantial time and effort can be saved compared with QTL mapping in RILs. [ABSTRACT FROM AUTHOR]

This article examines the stability properties of linear stochastic difference equations with delays. For this purpose, a novel approach is used that combines the theory of inverse-positive matrices and the asymptotic methods developed by N.V. Azbelev and his students for deterministic functional differential equations. Several efficient conditions for p-stability and exponential p-stability (2 ≤ p < ∞) of systems of linear Itô-type difference equations with delays and random coefficients are found. All results are conveniently formulated in terms of the coefficients of the equations. The suggested examples illustrate the feasibility of the approach. [ABSTRACT FROM AUTHOR]

The purpose of this article is to develop the receding horizon (RH) estimation for multi‐rate sampled systems with measurement delays and packet losses in the measurements. An event‐triggered transmission scheme is introduced to remove measurements that are unnecessary for the design of the estimator. The stochastic diagonal matrixes are introduced to represent the phenomenon of packet losses, where each component is subject to an individual Bernoulli process. The original system is firstly transformed into a delay‐free one by using the reorganized observation method. Further, a batch form and an iterative form of RH estimation are proposed by minimizing a given cost function that includes some terminal weighting terms based on the new system. The stability of the proposed RH estimation is guaranteed by the natural assumptions and a simulation instance is given to verify the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

Random walk based stochastic simulation methods for solving a nonlinear system of coupled transient diffusion and drift-diffusion equations governing a two-component chemotaxis process are developed. The nonlinear system is solved by linearization, the system is evolved in time, by small time steps, where on each step a linear system of equations is solved by using the solution from the previous time step. Three different stochastic algorithms are suggested, (1) the global random walk on grid (GRWG), (2) a randomized vector algorithm (RVA) based on a special transformation of the original matrix to a stochastic matrix, and (3) a stochastic projection algorithm (SPA). To get high precision results, these methods are combined with an iterative refinement method. [ABSTRACT FROM AUTHOR]

Denote by $ \omega _n $ ω n the convex polytope of $ n\times n $ n × n doubly substochastic matrices and by $ \Omega _n $ Ω n the convex polytope of $ n\times n $ n × n doubly stochastic matrices. We generalize the Bruhat order for permutation matrices and doubly stochastic matrices to the subpermutation matrices and doubly substochastic matrices, respectively. Moreover we study the Bruhat shadow of a subpermutation matrix and the Bruhat faces of $ \omega _n $ ω n . The relations between the Bruhat faces in $ \omega _n $ ω n and those in $ \Omega _n $ Ω n are also given. [ABSTRACT FROM AUTHOR]

In this work, we characterize the fixed points of the doubly stochastic operators defined on discrete spaces $ \ell ^p(I) , $ ℓ p (I) , where I is a nonempty set and $ p\in [1,+\infty). $ p ∈ [ 1 , + ∞). We use our results to characterize all Markov chains with infinite state space and doubly stochastic transition probability matrix that have at least one stationary distribution vector. [ABSTRACT FROM AUTHOR]

How to analyze flocking behaviors of a multi-agent system with local interaction functions is a challenging problem in theory. Motsch and Tadmor in 2011 also stressed the significance to assume that the interaction function is rapidly decaying or cut-off at a finite distance (cf. Motsch and Tadmor in J. Stat. Phys. 2011). In this paper, we study the flocking behavior of a Cucker–Smale type model with compactly supported interaction functions. Using properties of a connected stochastic matrix, together with an elaborate analysis on perturbations of a linearized system, we obtain a sufficient condition imposed only on model parameters and initial data to guarantee flocking. Moreover, it is shown that the system achieves flocking at an exponential rate. [ABSTRACT FROM AUTHOR]

The prevailing environmental, social and governance (ESG) framework is currently based on micro‐ESG indicators. Research on national ESG is often limited to theory building and policy analysis. Based on previous scholars, this paper constructs a national ESG index framework consisting of 39 indices and updates the national ESG indices for 121 countries worldwide from 1990 to 2021 using the entropy weight method, aiming to provide a set of instrumental indices that capture the status and evolution of national ESG performance. The research findings are as follows: First, the Gini coefficient shows that the gap between national ESG performance has gradually widened over time. Second, the kernel density distribution suggests that global ESG performance is on the rise. High‐income countries are placing greater emphasis on ESG growth. Third, the results of the Markov transformation matrix suggest that there is a "club convergence" in ESG performance across countries. [ABSTRACT FROM AUTHOR]

We analyze connections between two low rank modeling approaches from the last decade for treating dynamical data. The first one is the coherence problem (or coherent set approach), where groups of states are sought that evolve under the action of a stochastic transition matrix in a way maximally distinguishable from other groups. The second one is a low rank factorization approach for stochastic matrices, called direct Bayesian model reduction (DBMR), which estimates the low rank factors directly from observed data. We show that DBMR results in a low rank model that is a projection of the full model, and exploit this insight to infer bounds on a quantitative measure of coherence within the reduced model. Both approaches can be formulated as optimization problems, and we also prove a bound between their respective objectives. On a broader scope, this work relates the two classical loss functions of nonnegative matrix factorization, namely the Frobenius norm and the generalized Kullback–Leibler divergence, and suggests new links between likelihood-based and projection-based estimation of probabilistic models. [ABSTRACT FROM AUTHOR]

We propose a graph theoretic approach to determine the trace of the product of two permutation matrices through a weighted digraph representation for a pair of permutation matrices. Consequently, we derive trace-zero doubly stochastic (DS) matrices of order 5 whose k -th power is also a trace-zero DS matrix for k ∈ { 2 , 3 , 4 , 5 }. Then, we determine necessary conditions for the coefficients of a generic polynomial of degree 5 to be realizable as the characteristic polynomial of a trace-zero DS matrix of order 5. Finally, we approximate the eigenvalue region of trace-zero DS matrices of order 5. [ABSTRACT FROM AUTHOR]

The region in the complex plane containing the eigenvalues of all n × n stochastic matrices was described by Karpelevič in 1951, and it is since then known as the Karpelevič region. The boundary of the Karpelevič region is the union of arcs called the Karpelevič arcs. We provide a complete characterization of the Karpelevič arcs that are powers of some other Karpelevič arc. Furthermore, we find the necessary and sufficient conditions for a sparsest stochastic matrix associated with the Karpelevič arc of order n to be a power of another stochastic matrix. [ABSTRACT FROM AUTHOR]

Given a stochastic matrix P partitioned in four blocks P i j , i , j = 1 , 2 , Kemeny's constant κ (P) is expressed in terms of Kemeny's constants of the stochastic complements P 1 = P 11 + P 12 (I − P 22) − 1 P 21 , and P 2 = P 22 + P 21 (I − P 11) − 1 P 12. Specific cases concerning periodic Markov chains and Kronecker products of stochastic matrices are investigated. Bounds to Kemeny's constant of perturbed matrices are given. Relying on these theoretical results, a divide-and-conquer algorithm for the efficient computation of Kemeny's constant of graphs is designed. Numerical experiments performed on real world problems show the high efficiency and reliability of this algorithm. • Expression of Kemeny's constant employing the constants of stochastic complements. • New recursive algorithms for computing Kemeny's constant. • Application of the new expression to structured transition matrices. [ABSTRACT FROM AUTHOR]

We propose a general mixture of Markov jump processes. The key novel feature of the proposed mixture is that the generator matrices of the Markov processes comprising the mixture are entirely unconstrained. The Markov processes are mixed with distributions that depend on the initial state of the mixture process. The maximum likelihood (ML) estimates of the mixture's parameters are obtained from continuous realizations of the mixture process and their standard errors from an explicit form of the observed Fisher information matrix, which simplifies the Louis (Journal of the Royal Statistical Society Series B, 44:226–233, 1982) general formula for the same matrix. The asymptotic properties of the ML estimators are also derived. A simulation study verifies the estimates' accuracy. The proposed mixture provides an exploratory tool for identifying the homogeneous subpopulations in a heterogeneous population. This is illustrated with an application to a medical dataset. [ABSTRACT FROM AUTHOR]

The event‐triggered control of Markov jump systems has attracted more and more interest in field control. However, the problem of how to design a transition probability‐dependent event‐triggered mechanism and controller has not been fully considered. This paper investigates the problem of event‐triggered control for Lipschitz nonlinear Markov jump systems. Through Taylor series expansion, a linear auxiliary system is constructed to obtain the approximate state, whose system matrices are described by the probability‐weighted matrices of nonlinear Markov jump systems. By redefining the measurement error as the difference between the current state and the approximate state, a probability‐dependent event‐triggered mechanism is designed for Markov jump systems. The effectiveness of the developed approach is illustrated by two comparison examples. [ABSTRACT FROM AUTHOR]

In this paper, a new micromechanical hysteresis loop constitutive model of C/SiC composites with different interphases was developed considering the probabilistic‐statistical matrix fragmentation process. The lengths of matrix fragmentation were divided into three types, that is, long matrix fragments (LMFs), medium matrix fragments (MMFs), and short matrix fragments (SMFs). The distributions of the LMFs, MMFs, and SMFs with increasing tensile stress were determined using the probabilistic‐stochastic model by assuming the two‐parameter matrix strength distribution. The micro stress field of the LMFs, MMFs, and SMFs upon unloading and reloading was obtained and adopted to determine the corresponding stress‐strain relations. The interaction of matrix fragmentation lengths, especially for the LMFs with large debonding energy (LDE) and SMFs, was considered in the closed‐form constitutive model and hysteresis‐based inverse tangent modulus (ITMs) damage parameter. Synergistic effects of the fiber volumes, peak stresses, and interface debonding energy on the interface damage state, mechanical hysteresis loops, and related ITMs with small debonding energy and LDE were also analyzed. Comparisons of the mechanical hysteresis loops using the new hysteresis models considering matrix stochastic fragmentation and hysteresis models considering constant matrix fragmentation were also discussed. Experimental cyclic tensile hysteresis loops and unloading/reloading ITMs of C/(PyC)/SiC and C/(PyC+SiC)/SiC composites with different interphase thickness (i.e., t = 300, 600, 1000, and 2000 nm) were predicted using the developed constitutive model. Evolution of the unloading/reloading interface slip ratio was analyzed for different tensile peak stresses. [ABSTRACT FROM AUTHOR]

Existing methods for detecting communities in attributed social networks often rely solely on network topology, which leads to suboptimal accuracy in community detection, inefficient use of available data, and increased time required for identifying groups. This paper introduces the Dual Embedding-based Graph Convolution Network (DEGCN) to address these challenges. This new method uses graph embedding techniques in a new deep learning framework to improve accuracy and speed up community detection by combining the nodes' content with the network's topology. Initially, we compute the modularity and Markov matrices of the input graph. Each matrix is then processed through a graph embedding network with at least two layers to produce a condensed graph representation. As a result, a multilayer perceptron neural network classifies each node's community based on these generated embeddings. We tested the suggested method on three standard datasets: Cora, CiteSeer, and PubMed. Then, we compared the outcomes to many basic and advanced approaches using five important metrics: F1-score, adjusted rand index (ARI), normalized mutual information (NMI), and accuracy. The findings demonstrate that the DEGCN accurately captures community structure, achieves superior precision, and has higher ARI, NMI, and F1 scores, significantly outperforming existing algorithms for identifying community structures in medium-scale networks. [ABSTRACT FROM AUTHOR]

The statement of the Karpelevič theorem concerning the location of the eigenvalues of stochastic matrices in the complex plane (known as the Karpelevič region) is long and complicated and his proof methods are, at best, nebulous. Fortunately, an elegant simplification of the statement was provided by Ito—in particular, Ito's theorem asserts that the boundary of the Karpelevič region consists of arcs whose points satisfy a polynomial equation that depends on the endpoints of the arc. Unfortunately, Ito did not prove his version and only showed that it is equivalent. More recently, Johnson and Paparella showed that points satisfying Ito's equation belong to the Karpelevič region. Although not the intent of their work, this initiated the process of proving Ito's theorem and hence providing another proof of the Karpelevič theorem. The purpose of this work is to continue this effort by showing that an arc appears in the prescribed sector. To this end, it is shown that there is a continuous function λ : [ 0 , 1 ] ⟶ C such that P I (λ (α)) = 0 , ∀ α ∈ [ 0 , 1 ] , where P I is a Type I reduced Ito polynomial. It is also shown that these arcs are simple. Finally, an elementary argument is given to show that points on the boundary of the Karpelevič region are extremal whenever n > 3. [ABSTRACT FROM AUTHOR]

Background: This article applies a variant of the Markov chain that explicitly incorporates spatial effects. It is an extension of the Markov class allowing a more complete analysis of the spatial dimensions of transition dynamics. The aim is to provide a methodology for applying the explicit model to spatial dependency analysis. Methods: Here, the question is to study and quantify whether neighborhood context affects transitional dynamics. Rather than estimating a homogeneous law, the model requires the estimation of k transition laws each dependent on spatial neighbor state. This article used published data on confirmed cases of Covid'19 in the 22 regions of Madagascar. These data were discretized to obtain a discrete state of propagation intensity. Results: The analysis gave us the transition probabilities between Covid'19 intensity states knowing the context of neighboring regions, and the propagation time laws knowing the spatial contexts. The results showed that neighboring regions had an effect on the propagation of Covid'19 in Madagascar. Conclusion: After analysis, we can say that there is spatial dependency according to these spatial transition matrices. [ABSTRACT FROM AUTHOR]

Cell trajectory inference is very important to understand the details of tissue cell development, state differentiation and gene dynamic regulation. However, due to the high noise and heterogeneity of the single-cell data, it is challenging to infer cell trajectory in complex biological processes. Here, we proposed a new trajectory inference method, called Metric Learning Bhattacharyya Kernel Feature Decomposition (MLBKFD). In MLBKFD, a statistical model was used to infer cell trajectory by calculating the Markov transition matrix and Bhattacharyya kernel matrix between cells. Before that, to expedite the matrix calculation in the statistical model, a deep feedforward neural network was used to perform dimensionality reduction on single-cell data. The MLBKFD was evaluated on four typical datasets as well as seven recent human fetal lung datasets. Comparisons with the two outstanding methods (i.e., DTFLOW and MARGARET) demonstrate that the MLBKFD is capable of accurately inferring cell development and differentiation trajectories from single-cell data with different sizes and sources. Notably, MLBKFD exhibits nearly twice the speed of DTFLOW while maintaining high precision, particularly when dealing with large datasets. MLBKFD provides accurate and efficient trajectory inference, empowering researchers to gain deeper insights into the complex dynamics of cell development and differentiation. 1. MLBKFD is a powerful probabilistic model to effectively infer cell pseudo trajectories from single-cell data by calculating the Markov transition matrix and Bhattacharyya kernel matrix between cells. 2. MLBKFD exhibits fast speed and excellent visualization effects on cell trajectory inference, by utilizing a deep feedforward neural networks for dimensionality reduction of single-cell data. 3. By iteratively refining cell labels in the deep neural networks, MLBKFD can achieve excellent cell clustering results, enabling it to infer cell trajectories without relying on true cell labels. [ABSTRACT FROM AUTHOR]

The fact that the famous basic reproduction number R 0 , i.e., the largest eigenvalue of the next generation matrix F V − 1 , sometimes has a probabilistic interpretation is not as well known as it deserves to be. It is well understood that half of this formula, − V , is a Markovian generating matrix of a continuous-time Markov chain (CTMC) modeling the evolution of one individual on the compartments. It has also been noted that the not well-enough-known rank-one formula for R 0 of Arino et al. (2007) may be interpreted as an expected final reward of a CTMC, whose initial distribution is specified by the rank-one factorization of F. Here, we show that for a large class of ODE epidemic models introduced in Avram et al. (2023), besides the rank-one formula, we may also provide an integral renewal representation of R 0 with respect to explicit "age kernels" a (t) , which have a matrix exponential form.This latter formula may be also interpreted as an expected reward of a probabilistic continuous Markov chain (CTMC) model. Besides the rather extensively studied rank one case, we also provide an extension to a case with several susceptible classes. [ABSTRACT FROM AUTHOR]

The Shaanxi section is the central region of the Loess Plateau. Its unique wetland environment plays an indispensable role in regional ecological environment security. Clarifying the characteristics of wetland changes in the region is an important prerequisite for wetland management and protection. This study, based on the remote sensing data of the Shaanxi section of the Loess Plateau, analyzed the changes in the wetland area and type transfer in this region in 1980, 1990, 2000, 2010 and 2020 using the wetland dynamic degree model, the Markov transfer matrix, the landscape pattern index, and centroid analysis. The results showed that, from 1980 to 2020, the total wetland area and natural wetland area in the Shaanxi section of the Loess Plateau continued to shrink, decreasing by 79.35 km2 and 80.50 km2, respectively, while the artificial wetland area increased by 1.14 km2. Among the regions, Xi'an experienced the most significant reduction, with a total decrease of 83.04 km2 over 40 years, followed by Xianyang City, where the wetland area decreased by 6.50 km2. In contrast, the wetland areas of Yulin City, Weinan City, Yan'an City, Baoji City and Tongchuan City increased slightly. From 1980 to 2020, the change in the wetland types in the Shaanxi section of the Loess Plateau was mainly characterized by transfers between beach lands and river canals. River canals are the primary type of wetland in this region. The degree of fragmentation is the highest in reservoir potholes, while marshes have the largest clumpiness index. Over the same period, the centroid of the wetlands in the Shaanxi section of the Loess Plateau moved from south to north as a whole, although, between 1990 and 2000, the centroid position remained relatively stable. These results provide a theoretical basis and data support for wetland monitoring and protection in the Shaanxi section of the Loess Plateau and also provide a reference for the protection and sustainable development of other inland wetland resources in arid and semi-arid regions. [ABSTRACT FROM AUTHOR]

Understanding the regional disparities, dynamic evolution, and convergence–divergence characteristics of global agricultural energy efficiency is crucial for enhancing agricultural energy efficiency, ensuring food security, and responding to global green development trends. This paper utilizes 2002–2021 panel data from 144 countries globally, employing the epsilon-based measure–global Malmquist–Luenberger (EBM-GML) model to estimate agricultural energy efficiency, considering unexpected output. The Dagum Gini coefficient, kernel density estimation, spatial Markov matrix, and spatial convergence model are employed to explain the spatial patterns and evolving trends of global and regional agricultural energy efficiency at three levels: regional disparities, dynamic evolution, and convergence. The results indicate significant spatial heterogeneity in global agricultural energy efficiency, with Europe exhibiting the highest efficiency, followed by Asia and the Americas, while Oceania and Africa demonstrate the lowest efficiency. Agricultural energy efficiency globally and in each region continues to improve, with increasing regional disparities, and difficulties in grade transitions in agricultural energy efficiency across regions. Each region exhibits β-convergence characteristics, but the convergence rates vary, and various factors influence growth rates of agricultural energy efficiency differently across regions. Therefore, countries should tailor their strategies based on local conditions, considering their own resource endowments and developmental stages, and strengthen international exchanges and cooperation. [ABSTRACT FROM AUTHOR]

This study introduces a new numerical approach named flatlet oblique multiwavelets (FOMW) to solve fractional‐order stochastic integro‐differential equation (FSI‐DE). The FOMW is used to create an operational matrix of the stochastic integral, which helps transform the FSI‐DE into a linear system of algebraic equations. This method requires only a small number of basis functions to achieve satisfactory results. The paper also includes a comprehensive discussion on error estimation and convergence analysis, which provides an upper bound of error for the proposed method. Furthermore, numerical experiments have been conducted and compared with other methods, demonstrating the superiority of FOMW in terms of accuracy and efficiency. [ABSTRACT FROM AUTHOR]

Simple Summary: In scRNA-seq analysis, diffusion-based approaches help identify the connections between cells, allowing us to observe the progression of individual cells as they change phenotypes within a mathematical space known as a manifold. Recently, these approaches have been used as a reference for imputation, a technique that addresses missing data, a common challenge in scRNA-seq analysis. For example, MAGIC is a popular diffusion-based imputation method, and it has shown success in uncovering gene–gene interactions related to phenotypic transitions that would not be possible without imputation. However, previous evaluations have not adequately compared the impact of different parameter settings on MAGIC, particularly over-smoothing issues. To address this, we developed sc-PHENIX, which utilizes a similar diffusion approach as MAGIC but incorporates a PCA-UMAP initialization step, whereas MAGIC only uses PCA. We compared sc-PHENIX and MAGIC in terms of imputation accuracy, visualization, biological insights, and preservation of data structure. Our findings show that sc-PHENIX outperforms MAGIC across various common parameters such as "diffusion time" (t), the number of nearest neighbors (knn), and PCA dimensions. It effectively captures and preserves the global, local, and continuous data structures, leading to more reliable imputation and potentially uncovering new biological insights in diverse datasets. Single-cell transcriptomics (scRNA-seq) is revolutionizing biological research, yet it faces challenges such as inefficient transcript capture and noise. To address these challenges, methods like neighbor averaging or graph diffusion are used. These methods often rely on k-nearest neighbor graphs from low-dimensional manifolds. However, scRNA-seq data suffer from the 'curse of dimensionality', leading to the over-smoothing of data when using imputation methods. To overcome this, sc-PHENIX employs a PCA-UMAP diffusion method, which enhances the preservation of data structures and allows for a refined use of PCA dimensions and diffusion parameters (e.g., k-nearest neighbors, exponentiation of the Markov matrix) to minimize noise introduction. This approach enables a more accurate construction of the exponentiated Markov matrix (cell neighborhood graph), surpassing methods like MAGIC. sc-PHENIX significantly mitigates over-smoothing, as validated through various scRNA-seq datasets, demonstrating improved cell phenotype representation. Applied to a multicellular tumor spheroid dataset, sc-PHENIX identified known extreme phenotype states, showcasing its effectiveness. sc-PHENIX is open-source and available for use and modification. [ABSTRACT FROM AUTHOR]

We propose and illustrate a general framework in which spatial autocorrelation is measured by the Frobenius product of two kernels, a feature kernel and a spatial kernel. The resulting autocorrelation index δ$$ \delta $$ generalizes Moran's index in the weighted, multivariate setting, where regions, differing in importance, are characterized by multivariate features. Spatial kernels can traditionally be obtained from a matrix of spatial weights, or directly from geographical distances. In the former case, the Markov transition matrix defined by row‐normalized spatial weights must be made compatible with the regional weights, as well as reversible. Equivalently, space is specified by a symmetric exchange matrix containing the joint probabilities to select a pair of regions. Four original weight‐compatible constructions, based upon the binary adjacency matrix, are presented and analyzed. Weighted multidimensional scaling on kernels yields a low‐dimensional visualization of both the feature and the spatial configurations. The expected values of the first four moments of δ$$ \delta $$ under the null hypothesis of absence of spatial autocorrelation can be exactly computed under a new approach, invariant orthogonal integration, thus permitting to test the significance of δ$$ \delta $$ beyond the normal approximation, which only involves its expectation and expected variance. Various illustrations are provided, investigating the spatial autocorrelation of political and social features among French departments. [ABSTRACT FROM AUTHOR]

We consider invertible, row diagonally dominant real matrices and give inequalities on their minors and diagonal entries of their inverses. A very special case is that all diagonal entries of an inverse, of a row stochastic, row diagonally dominant and invertible matrix, are at least 1, with strict inequality at least when the dominance is strict. This was conjectured in international trade theory in economics and motivated the present work (though much more is proven). Some of the results generalize previously known facts for M -matrices. [ABSTRACT FROM AUTHOR]

We consider a face of the polytope of doubly stochastic matrices, whose non-zero entries coincide with that of \[ V_{l,m,n}=\left[\begin{matrix} 0_{l,l} & 0_{l,m} & J_{l,n} \\ 0_{m,l} & I_{m} & J_{m,n}\\ J_{n,l} & J_{n,m} & J_{n,n}\\ \end{matrix}\right]. \] V l , m , n = [ 0 l , l 0 l , m J l , n 0 m , l I m J m , n J n , l J n , m J n , n ]. Here, $ 0_{r,s} $ 0 r , s is the $ r\times s $ r × s zero matrix, $ J_{u,v} $ J u , v denotes the $ u\times v $ u × v matrix all of whose entries are 1 and $ I_m $ I m is the identity matrix of order m. We determine the minimum permanent and minimizing matrices on this face of the polytope of doubly stochastic matrices. This research contributes towards solution of two problems from Minc's well-known lists of unsolved problems on permanents. [ABSTRACT FROM AUTHOR]

For a non-empty set I, the sub-defect of an $ I\times I $ I × I doubly substochastic matrix $ A=[a_{ij}]_{i,j \in I}, $ A = [ a ij ] i , j ∈ I , denoted by $ \mathrm {sd}(A), $ sd (A) , is the smallest cardinal number α for which there is a set J with $ \mathrm {card}(J) =\alpha, $ card (J) = α , $ I\cap J =\emptyset, $ I ∩ J = ∅ , and there exists a doubly stochastic matrix $ D=[d_{ij}]_{i,j\in I\cup J} $ D = [ d ij ] i , j ∈ I ∪ J which contains A as a sub-matrix. In this paper, we show the set of all finite sub-defect matrices is closed under multiplication. We also show that the inequality $ \max \{\mathrm {sd}(A), \mathrm {sd}(B)\} \leq \mathrm {sd}(AB) \leq \min \{n, \mathrm {sd}(A) + \mathrm {sd}(B)\} $ max { sd (A) , sd (B) } ≤ sd (AB) ≤ min { n , sd (A) + sd (B) } which is obtained by Lei Cao et al. remains valid for all finite sub-defect matrices. [ABSTRACT FROM AUTHOR]

The need to replace nonrenewable sources is a global challenge, especially for emerging countries with financial constraints to invest in energy transition. In Brazil, electrical generation is predominantly renewable, since the hydraulic source is responsible for more than half of the domestic electricity supply. Other alternative sources, such as biomass, solar, and wind, complement the renewable portion of the Brazilian electrical matrix. However, periods of drought and climate disturbances threaten the stability of the electrical system and increase dependence on thermal generation. Based on the global movement to intensify the use of renewable sources, which are volatile and influenced by climate variations, the objective of this research is to predict electricity generation using the Heston model. This model, originally applied to predict option prices, assumes nonconstant volatility and has the ability to estimate the average value including the stochastic volatility process. The database is made up of monthly values of the volume of electrical energy generation from biomass, coal, fuel oil, hydraulic, industrial waste, natural gas, nuclear, solar, and wind sources. Compared to the Black‐Scholes model, the chosen method proved to be superior in predicting the generation from coal, hydropower, and solar energy, according to the MAE, MAPE, and RMSE statistics. We conclude that a financial model can be effectively applied to the energy sector and the Heston model can be a useful instrument to assist in the planning and management of the national electrical system. One suggestion for the Brazilian case is to encourage the use of alternative energies and low‐carbon sources, such as solar, wind, and natural gas, providing greater energy security and a more reliable electrical system. [ABSTRACT FROM AUTHOR]

A formulation is presented for the assessment of the CO2 generated by ships in operation and their evolution with time, conditional on the current legislation using Markov chains. Any potential deep repair or retrofitting of the ship propulsion system or enhancement of route operational characteristics during the service life are not accounted for. The Markov transition matrix is defined based on the ship operations and CO2 history of A, B, C, D, and E carbon intensity indicator (CII) rates. The transition between different CII rate states in the survey data is used to estimate the probability of transition of the analysed ships between different CII grates. Distinct transition matrices employing the progressively tightened legislation of CII are employed and analysed. In addition, the transition matrices can be fed into risk-based models that take the CII rates as input for defining the most appropriate ship energy efficiency management plan. [ABSTRACT FROM AUTHOR]

Purpose: This work ensures the higher performability of this complex system, which consists of five different subsystems, i.e. shearing machine, V-cutting machine, center hole punch, edge cutting burr and drilling machine. These subsystems are placed in combinations of both series and parallel arrangement. The concerned plant management must be aware of the failures that have the greatest/least impact on the system's performance. Design/methodology/approach: Performability analysis has been done for the Shearing, Punch and V- Cutting (SPVC) line system by using a probabilistic approach (i.e. Markov method). This system was further divided into five subsystems, and single-order differential equations are derived using the transition diagram. MATLAB software was used to determine the performability of the system for various combinations of repair and failure rates. Findings: In this research work, performability analysis was done using different combinations of repair and failure rates for these subsystems. Further, a decision matrix (DM) has been developed that indicates that edge cutting burr is the most critical subsystem, which requires the top level of maintenance priorities among the various subsystems. This matrix will facilitate policymaking related to various maintenance activities for the respective system. Originality/value: In this research work, a mathematical modeling based on a single differential equation using a transition diagram has been developed for the SPVC line system. The novelty of this work is to consider interaction among different subsystem, which generates more realistic situation during modeling. The purposed DM helps make future maintenance planning, which reduces maintenance costs and enhances system's performability. [ABSTRACT FROM AUTHOR]

This study investigates distributed convex optimisation that accounts for the inequality constraints over unbalanced directed graphs. In distributed optimisation, agents exchange information on a network to obtain an optimal solution when they only know their own cost function. We propose a distributed primal-dual subgradient method based on a row stochastic weight matrix that is associated with a communication network. In the proposed method, the normalised left eigenvector of the weight matrix is estimated by a consensus algorithm. Then, the subgradient of the Lagrange function is scaled by the estimated values of the left eigenvector to compensate for the imbalance of the information flow in the network. We show that the pair of estimations for the primal and dual problems converge to an optimal primal-dual solution. We also show the relation between the convergence rate of the proposed algorithm and the step-size rule. A numerical example confirms the validity of the proposed method. [ABSTRACT FROM AUTHOR]

We explore the application of reduced density matrix-based approaches to molecules interacting with the lattice vibrations of metals, an interaction responsible for the temperature dependence of many of the fundamental steps of catalysis. We avoid the use of simple models for the bath and instead use density functional theory to compute all molecule–phonon interactions and the properties of the lattice phonons, for methane scattering from Ir(111). We find that while the large metal mass leads to long bath correlation times, these are not significantly longer than the time over which the reduced density matrix changes due to interactions with the bath. We show that the neglect of memory is reasonable and the use of the Redfield equation is justified. We also show how the commonly used rotating wave approximation is far too severe for this scattering problem. A less restrictive approximation that is nearly exact for our system gives an equation of motion in the Lindblad form. As a result, the Monte Carlo wave packet methods can be used to describe gas–phonon scattering, guaranteeing positivity, and with all couplings derived from first-principles. [ABSTRACT FROM AUTHOR]

Agricultural ecological efficiency is an important tool with which to measure the coordination of the sustainable development of agricultural economies and ecological environments. In this paper, a super-efficiency slacks-based measures model was used to measure the agricultural ecological efficiency in Hebei Province. The characteristics of spatial and temporal evolution patterns were explored using a spatial Markov transfer matrix. The results showed that (i) based on measurements, the agricultural ecological efficiency in Hebei Province showed regional differences in four regions (eastern, northern, central and southern Hebei) and 141 counties; (ii) from the perspective of evolutionary characteristics of agricultural ecological efficiency, the overall development of in Hebei Province was good, with more concentrated spatial distribution and more obvious direction, while the type of transfer of agricultural ecological efficiency in Hebei Province showed strong stability that was significantly affected by geographical neighborhood conditions and the club convergence phenomenon; (iii) from the perspective of the long-term evolutionary trend of agricultural ecological efficiency, the areas adjacent to counties with low efficiency had limited potential for improvement, and the areas adjacent to counties with high grade had great potential. However, it was difficult to achieve large-scale improvement in agricultural ecological efficiency in Hebei Province, whether the impact of geospatial backgrounds was considered or not. [ABSTRACT FROM AUTHOR]

The paper introduces and explores the notions of ergodicity index and ergodicity exponent of a set of stochastic matrices. For the ergodicity exponent a sharp upper bound is obtained. A particular case of this bound is the well-known Paz bound. Also a connection between the ergodicity index and the Protasov–Voynov imprimitivity index is established. [ABSTRACT FROM AUTHOR]

In this paper we investigate the impact of lensing magnification on the analysis of Euclid's spectroscopic survey using the multipoles of the two-point correlation function for galaxy clustering. We determine the impact of lensing magnification on cosmological constraints as well as the expected shift in the best-fit parameters if magnification is ignored. We considered two cosmological analyses: (i) a full-shape analysis based on the Λ cold dark matter (CDM) model and its extension w0waCDM and (ii) a model-independent analysis that measures the growth rate of structure in each redshift bin. We adopted two complementary approaches in our forecast: the Fisher matrix formalism and the Markov chain Monte Carlo method. The fiducial values of the local count slope (or magnification bias), which regulates the amplitude of the lensing magnification, have been estimated from the Euclid Flagship simulations. We used linear perturbation theory and modelled the two-point correlation function with the public code coffe. For a ΛCDM model, we find that the estimation of cosmological parameters is biased at the level of 0.4–0.7 standard deviations, while for a w0waCDM dynamical dark energy model, lensing magnification has a somewhat smaller impact, with shifts below 0.5 standard deviations. For a model-independent analysis aimed at measuring the growth rate of structure, we find that the estimation of the growth rate is biased by up to 1.2 standard deviations in the highest redshift bin. As a result, lensing magnification cannot be neglected in the spectroscopic survey, especially if we want to determine the growth factor, one of the most promising ways to test general relativity with Euclid. We also find that, by including lensing magnification with a simple template, this shift can be almost entirely eliminated with minimal computational overhead. [ABSTRACT FROM AUTHOR]

Markov Decision Process Tayloring for Approximation Design Optimal control problems are difficult to solve for problems on large state spaces, calling for the development of approximate solution methods. In "A Low-rank Approximation for MDPs via Moment Coupling," Zhang and Gurvich introduce a novel framework to approximate Markov decision processes (MDPs) that stands on two pillars: (i) state aggregation, as the algorithmic infrastructure, and (ii) central-limit-theorem-type approximations, as the mathematical underpinning. The theoretical guarantees are grounded in the approximation of the Bellman equation by a partial differential equation (PDE) where, in the spirit of the central limit theorem, the transition matrix of the controlled Markov chain is reduced to its local first and second moments. Instead of solving the PDE, the algorithm introduced in the paper constructs a "sister"' (controlled) Markov chain whose two local transition moments are approximately identical with those of the focal chain. Because of this moment matching, the original chain and its sister are coupled through the PDE, facilitating optimality guarantees. Embedded into standard soft aggregation, moment matching provides a disciplined mechanism to tune the aggregation and disaggregation probabilities. We introduce a framework to approximate Markov decision processes (MDPs) that stands on two pillars: (i) state aggregation, as the algorithmic infrastructure, and (ii) central-limit-theorem-type approximations, as the mathematical underpinning of optimality guarantees. The theory is grounded in recent work by Braverman et al. [Braverman A, Gurvich I, Huang J (2020) On the Taylor expansion of value functions. Oper. Res. 68(2):631–65] that relates the solution of the Bellman equation to that of a partial differential equation (PDE) where, in the spirit of the central limit theorem, the transition matrix is reduced to its local first and second moments. Solving the PDE is not required by our method. Instead, we construct a "sister" (controlled) Markov chain whose two local transition moments are approximately identical with those of the focal chain. Because of this moment matching, the original chain and its sister are coupled through the PDE, a coupling that facilitates optimality guarantees. Embedded into standard soft aggregation algorithms, moment matching provides a disciplined mechanism to tune the aggregation and disaggregation probabilities. Computational gains arise from the reduction of the effective state space from N to N 1 2 + ϵ is as one might intuitively expect from approximations grounded in the central limit theorem. Funding: This work was supported by the National Science Foundation [Grant CMMI-1662294]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/opre.2022.2392. [ABSTRACT FROM AUTHOR]

We study a semismooth Newton-type method for the nearest doubly stochastic matrix problem where the nonsingularity of the Jacobian can fail. The optimality conditions for this problem are formulated as a system of strongly semismooth functions. We show that the nonsingularity of the Jacobian does not hold for this system. By exploiting the problem structure, we construct a modified two step semismooth Newton method that guarantees a nonsingular Jacobian matrix at each iteration, and that converges to the nearest doubly stochastic matrix quadratically. Funding: This work was supported by Canadian Network for Research and Innovation in Machining Technology. The research of H. Hu, H. Im, and H. Wolkowocz was supported by The Natural Sciences and Engineering Research Council of Canada. The research of X. Li was supported by the National Natural Science Foundation of China [No. 11601183] and Natural Science Foundation for Young Scientist of Jilin Province [No. 20180520212JH]. [ABSTRACT FROM AUTHOR]

Aims: This study aimed to evaluate whether voluntary and mandatory prescription drug monitoring program (PDMP) use in Victoria, Australia, had an impact on prescribing behaviour, focusing on individual patients' prescribed opioid doses and transition to prescribing of nonmonitored medications. Methods: This was a retrospective cross‐sectional study using routinely collected primary healthcare data. A 90‐day moving average prescribed opioid dose in oral morphine equivalents was used to estimate opioid dosage. A Markov transition matrix was used to describe how patients prescribed medications transitioned between opioid dose groups and other nonopioid treatment options during 3 transition periods: transition between 2 control periods prior to PDMP implementation (T1 to T2); during the voluntary PDMP implementation (T2 to T3); and during mandatory PDMP implementation (T3 to T4). Results: Among patients prescribed opioids in our study, we noted an increased probability of transitioning to not being prescribed opioids during the mandatory PDMP period (T3 to T4). This increase was attributed mainly to the ceasing of low‐dose opioid prescribing. Membership in an opioid dose group remained relatively stable for most patients who were prescribed high opioid doses. For those who were only prescribed nonmonitored medications initially, the probability of being prescribed opioids increased during the mandatory PDMP when compared to other transition periods. Conclusion: The introduction of PDMP mandates appeared to have an impact on the prescribing for patients who were prescribed low‐dose opioids, while its impact on individuals prescribed higher opioid doses was comparatively limited. [ABSTRACT FROM AUTHOR]

In this paper, we study the flocking behavior of a locally interacting multi‐agent system with a unit speed constraint. Using the spectral gap of a connected stochastic matrix, together with an elaborate estimation on perturbations of a linearized system, we obtain a sufficient condition imposed only on initial data and model parameters to guarantee flocking and further show that the system achieves a consensus at an exponential rate. [ABSTRACT FROM AUTHOR]

This paper focuses on optimizing the green multimodal transportation path under the condition of transit time subjected to random distribution with dynamic changes. The study utilizes the transition matrix of a homogeneous Markov chain and the theory of time-space network to simulate the transit time subjected to random distribution with dynamic changes. A time-space network path optimization model is established and solved using a commercial solver. The solutions are compared under different transit time scenarios and carbon tax rates. It is observed that the reliability of the same solution decreases in a transportation environment with higher randomness. Additionally, new solutions that did not appear in a single situation are generated under dynamic changes in transit time. As the carbon tax rate increases, the transportation strategy gradually shifts away from road transportation. Furthermore, the time-space network model may yield better solutions compared to the physical network model. The choice of different types of distributions to simulate transit times also influences the solution results. [ABSTRACT FROM AUTHOR]