In this paper, we propose a Lagrange multiplier (LM)-type unit root test for functional time series. The key novelty lies not in introducing a new LM principle but in establishing the asymptotic validity of such a test under the functional random walk null hypothesis without relying on functional principal component analysis (FPCA) or finite-dimensional unit root subspace assumptions. We derive the limit distribution of our proposed test statistics under the null hypothesis of a random walk and its asymptotic behavior of alternative hypotheses of trend stationary, weakly dependent stationary, and autoregressive stationary models. Specifically, we establish the theoretical consistency of the test under all aforementioned alternative hypotheses. Simulation studies corroborate these theoretical findings and demonstrate the desirable finite-sample performance of the proposed functional unit root test. The proposed test is also applied to real data of intraday stock price curves, and the test results are plausible.

Random walks on spaces with hyperbolic properties tend to sublinearly track geodesic rays which point in certain hyperbolic-like directions. Qing-Rafi-Tiozzo recently introduced the sublinearly Morse boundary and proved that this boundary is a quasi-isometry invariant which captures this notion of generic direction in a broad context. In this article, we develop the geometric foundations of sublinear Morseness in the mapping class group and Teichmüller space. We completely characterize sublinear Morseness in terms of the hierarchical structures of these spaces, and use this to prove that their sublinearly Morse boundaries admit continuous equivariant injections into the boundary of the curve graph. It was already known that the Gromov boundary of the curve graph is a Poisson model for sufficiently nice random walks of the mapping class group on itself and on Teichmüller space. As corollary, we prove that the corresponding hitting measure is fully supported on the image of the sublinearly Morse boundary, which was previously unknown. Our techniques include developing tools for modeling the hulls of median rays in hierarchically hyperbolic spaces via CAT(0) cube complexes. Part of this analysis involves establishing direct connections between the geometry of the curve graph and the combinatorics of hyperplanes in the approximating cube complexes.

DIFFUSION OF QUANTUM STATES GENERATED BY CLASSIC RANDOM WALK

Hepatic cirrhosis is a progressive chronic liver disease driven by sustained inflammation, cell death, and tissue remodeling, and effective disease-modifying options remain limited. Here, we applied a multiscale interactome framework to prioritize candidate herbs and active compounds for hepatic cirrhosis. Herb-compound associations were collected from the OASIS database and mapped to experimentally supported compound-target interactions (DrugBank/TTD/STITCH), while cirrhosis-related proteins were curated from DisGeNET. Using a biased random-walk algorithm, we generated disease and herb/compound diffusion profiles on the multiscale network and ranked candidates by profile similarity and target overlap. Among the top-ranked herbs, Magnoliae Cortex, Notoginseng Radix et Rhizoma, Polygoni Cuspidati Rhizoma et Radix, and Capsici Fructus were supported by prior literature, whereas several high-ranking herbs lacked curated evidence and were highlighted as underexplored candidates, including Saposhnikoviae Radix and Fritillariae Cirrhosae Bulbus. Enrichment analyses indicated convergence on inflammatory and innate-immune pathways (TNF, Toll-like receptor, NF-κB) and apoptosis-related processes, with additional signals involving HIF-1 and PI3K-Akt pathways. Disease-focused subnetworks suggested mechanistic hypotheses for evidence-lacking compounds, including bergapten, oleic acid, and octadecanoic acid. Overall, we systematically prioritize herbal candidates and provides a mechanistic basis for follow-up validation in hepatic cirrhosis.

Abstract We identify the size of the largest connected component in a subcritical inhomogeneous random graph with a kernel of preferential attachment type. The component is polynomial in the graph size with an explicitly given exponent, which is strictly larger than the exponent for the largest degree in the graph. This is in stark contrast to the behaviour of inhomogeneous random graphs with a kernel of rank one. Our proof uses local approximation by branching random walks going well beyond the weak local limit and novel results on subcritical killed branching random walks.

Abstract A mixed equation in a group G is given by a non-trivial element w ( x ) of the free product $$G *\mathbb {Z}$$ G ∗ Z , and a solution is some $$g\in G$$ g ∈ G such that w ( g ) is the identity. For G acylindrically hyperbolic with trivial finite radical (e.g. torsion-free), we show that any mixed equation of length n has a non-solution of length comparable to $$\log (n)$$ log ( n ) , which is the best possible bound. Similarly, we show that there is a common non-solution of length O ( n ) to all mixed equations of length n , again the best possible bound. In fact, in both cases, we show that a random walk of appropriate length yields a non-solution with positive probability.

Abstract We investigate the spectrum of the normalized Laplacian L for finite graphs over non-Archimedean ordered fields. We prove a Cheeger’s inequality for the first non-zero eigenvalue. Further, we show that, on the contrary to the real case, for the transition operator P = I − L and for any non-bipartite non-complete graph over a non-Archimedean field there is always a subspace of functions for which an analogue of random walk, i.e. P m f , does not converge. Moreover, we show that the strong Cheeger estimate α 1 ≼ 1 − h 2 for the second largest eigenvalue of P is crucial for the investigation of convergence of P m f to the equilibrium over non-Archimedean ordered fields. We provide examples of convergence and non-convergence for graphs over the Levi-Civita field.

An improved gravity centrality based on degree-mixed clustering coefficient and return random walk for identifying influential nodes in complex networks

Knowledge graph completion via link prediction is critical for intelligent equipment maintenance systems to support accurate fault diagnosis and maintenance decision making. However, existing approaches struggle to simultaneously capture local structural dependencies and perform effective multi-hop reasoning due to limited receptive fields or inefficient path exploration mechanisms. Traditional path-based methods implicitly assume path symmetry, treating all reasoning chains equally without considering their task-specific relevance. To address this issue, we propose a Graph Attention Network (GAT)-guided semantic path reasoning framework that breaks this symmetry through attention-driven asymmetric weighting, integrating local structural encoding with global multi-hop inference. The key innovation lies in a target-guided biased path sampling strategy, which transforms GAT attention weights into probabilistic transition biases, enabling adaptive exploration of high-quality semantic paths relevant to specific prediction targets. GATs learn importance-aware local representations, which guide biased random walks to efficiently sample task-relevant reasoning paths. The sampled paths are encoded and aggregated to form global semantic context representations, which are then fused with local embeddings through a gating mechanism for final link prediction. Experimental evaluations on FB15k-237, WN18RR, and a real-world equipment maintenance knowledge graph demonstrate that the proposed method consistently outperforms state-of-the-art baselines, achieving an MRR of 0.614 on the maintenance dataset and 0.485 on WN18RR. Further analysis shows that the learned path attention weights provide interpretable asymmetric reasoning evidence, enhancing transparency for safety-critical maintenance applications.

Sensitivity analysis characterises input-output relationships for mathematical models, and has been widely applied to deterministic models across many applications in the life sciences. In contrast, sensitivity analysis for stochastic models has received less attention, with most previous work focusing on well-mixed, non-spatial problems. For explicit spatio-temporal stochastic models, such as random walk models (RWMs), sensitivity analysis has received far less attention. Here we present a new type of sensitivity analysis, called parameter-wise prediction, for two types of biologically-motivated and computationally expensive RWMs. To overcome the limitations of directly analysing stochastic simulations, we employ continuum-limit partial differential equation (PDE) descriptions as surrogate models, and we link these efficient surrogate descriptions to the RWMs using a range of biophysically-motivated measurement error models. Our approach is likelihood-based, which means that we also consider likelihood-based parameter estimation and identifiability analysis along with parameter sensitivity. The new approach is presented for two important classes of lattice-based RWM including a classical model where crowding effects are neglected, and an exclusion process model that explicitly incorporates crowding. Our workflow illustrates how different process models can be combined with different measurement error models to reveal how each parameter impacts the outcome of the expensive stochastic simulation. Open-access software to replicate all results is available on GitHub (Liu, 2025).

As circular non-coding RNA (circRNA) is closely associated with various human diseases, identifying disease-related circRNAs can provide a deeper understanding of the mechanisms underlying disease pathogenesis. Advanced circRNA-disease association prediction methods mainly focus on graph learning techniques such as graph convolutional networks. However, these methods do not fully encode the multiscale neighbor topologies of each node, and the dependencies among the pairwise attributes. We propose a multi-scale neighbor topology-guided transformer with Kolmogorov-Arnold network (KAN) enhanced feature learning for circRNA and disease association prediction, termed MKCD. First, MKCD incorporates an adaptive multiscale neighbor topology embedding construction strategy (AMNE), which generates neighbor topologies covering varying scopes of neighbors by random walks. Second, we design a dynamic multi-scale neighbor topology-guided transformer (DMTT) that leverages the multi-scale neighbor topologies to guide the learning of relationships among circRNA, miRNA, and disease nodes. The multi-scale neighbor topology is dynamically evolved, providing adaptive guidance to the transformer's learning process. Third, we establish a feature-gated network (FGN) to evaluate the importance of topological features and the original node attributes. Finally, we propose an adaptive joint convolutional neural networks and KAN learning strategy (ACK) to learn the global and local dependencies of pairwise features. Comprehensive comparison experiments show that MKCD outperforms six state-of-the-art methods, improving AUC and AUPR by at least 14.1% and 7.6%, respectively. Ablation experiments further validate the effectiveness of AMNE, DMTT, FGN and ACK innovations. Case studies on three diseases further validate the application value of our method in discovering reliable circRNA candidates for the diseases.

Integrated computational analysis for Escherichia coli prevalence, genetic evolution, and antimicrobial resistance evolution: Implications for public health and environmental sustainability in Asia.

Outlier detection is an effective technique for identifying abnormal samples in complex data. Random walks effectively detect outliers by analyzing graph transition patterns. However, existing methods only consider local transitions and fail to capture complex structural patterns in complex data. Granular-ball computing based outlier detection methods have better robustness and efficiency. Nevertheless, these methods use a coarse-granularity representation of granular-balls and ignore the large number of sample information within the granular-balls. To address the above issues, this paper develops a bi-level granular-ball based second-order biased random walk outlier detection method. First, a bi-level granular-ball knowledge representation method is proposed to address the information distortion inherent in granular-ball computing-based methods. Then, a granular-ball anomaly membership evaluation metric is introduced, which leverages second-order biased random walk, to endow granular-balls with coarse-granularity anomaly degrees. Subsequently, a bi-level granular-ball anomaly classifier is designed to map coarse-granularity granular-ball anomaly degrees to fine-granularity sample-level anomaly degrees. Finally, a distilled outlier factor is defined, which selects optimal attribute sequences through granular-ball construction on attributes, for outlier detection. At the same time, a corresponding outlier detection algorithm is proposed. Experiments on datasets are conducted to compare the proposed algorithm with six other algorithms. The experimental results show that the algorithm has better performance and a certain degree of robustness.

Accurate estimation and forecasts for neonatal mortality rates (NMRs) in low- and middle-income countries is an urgent problem. Much of child mortality is preventable, and understanding temporal trends is of great interest when evaluating past performance and planning future policy or programming. In countries without robust vital registration, we rely on modeled estimates based on survey data to understand trends. A toolkit of compelling temporal models exists, but these methods have not been comprehensively evaluated for their application for the estimation of the NMR in low- and middle-income countries using household survey data. Using Demographic and Health Surveys (DHS) and Multiple Indicator Cluster Surveys (MICS) data from 41 countries in sub-Saharan Africa, we estimate and forecast the national-level NMR for 1970-2030 separately with random walk, auto-regressive, penalized spline, natural spline, and logit-linear latent temporal models. We examine the statistical behavior of these temporal models with both an out-of-sample analysis using the DHS and MICS data and a simulation study. We find that the second-order random walk and the penalized spline have the least bias, and short-term forecasts from the penalized spline tend to have narrower intervals with better out-of-sample performance. From the analysis of the NMR in sub-Saharan Africa, we estimate that 6 or fewer of the 41 countries included are on track to achieve the Sustainable Development Goals target of 12 neonatal deaths per 1000 live births by 2030.

Time-dependent diffusion MRI enables quantification of tumor microstructural parameters useful for diagnosis and prognosis. Nevertheless, current model fitting approaches exhibit suboptimal bias-variance trade-offs; specifically, nonlinear least squares fitting (NLLS) demonstrated low bias but high variance, whereas supervised deep learning methods trained with mean squared error loss (MSE-Net) yielded low variance but elevated bias. This study investigates these bias-variance characteristics and proposes a method to control fitting bias and variance. Random walk with barrier model was used as a representative biophysical model. NLLS and MSE-Net were reformulated within the Bayesian framework to elucidate their bias-variance behaviors. We introduced B2V-Net, a supervised learning approach using a loss function with adjustable bias-variance weighting, to control bias-variance trade-off. B2V-Net was evaluated and compared against NLLS and MSE-Net numerically across a wide range of parameters and noise levels, as well as in vivo in patients with head and neck cancer. Flat posterior distributions that were not centered at ground truth parameters explained the bias-variance behaviors of NLLS and MSE-Net. B2V-Net controlled the bias-variance trade-off, achieving a 56% reduction in standard deviation relative to NLLS and an 18% reduction in bias compared to MSE-Net. In vivo parameter maps from B2V-Net demonstrated a balance between smoothness and accuracy. We demonstrated and explained the low bias-high variance of NLLS and the low variance-high bias of MSE-Net. The proposed B2V-Net can balance bias and variance. Our work provided insights and methods to guide the design of customized loss functions tailored to specific clinical imaging needs.

We derive explicit estimates for the asymptotics of the first-passage function for random walks on free groups supported on the powers of standard generators, and use them to prove the singularity of the hitting measure for a similarly defined class of random walks on Fuchsian groups.

A large number of complex systems can be represented as directed networks, where the connections (edge weights) between nodes (system states) are such that the flux of the quantity of interest (e.g., energy or information) from node i to j is not equal to that from j to i. The network Laplacian underpins a suite of techniques for studying diffusion and random walks on networks, but many such methods have been developed for undirected graphs. In this paper, we propose a method for characterizing the relative degree of directedness of a network. Our approach is based on the pseudospectrum of the Laplacian and examines the difference in the nature of the "containing pseudospectral frontier" between the Laplacian itself and a variant formed by removing its non-normal structure. The distance between the containing and comparator frontiers then characterizes relative directedness as a function of the shape of the underlying pseudospectral surface. We demonstrate how this metric's behavior is consistent with that expected for a cyclic permutation adjacency matrix and is appropriate for a network where directedness is controlled systematically. We also extend an existing definition of network directedness and then apply our method in detail to a network capturing the Lagrangian dynamics of a turbulent flow. Two local maxima for our metric arise in distinct regions of the complex plane, but the eigenvalues driving these maxima are dominated by nodes from the network that are both very similar to each other and highly atypical in terms of the physics they embody. These nodes are also very different in nature from those highlighted by a stochastic differential equationmodel for these dynamics that successfully captures many key features of turbulence. This highlights the sensitivity of our approach and its potential utility in model validation, in addition to network characterization.

Abstract As spherical shell mantle convection models become increasingly commonplace, understanding how plates are generated has raised the issue of how to recognize whether rigid plates are present in model output. Tectonocists have long recognized that intraplate regions are not rigid without exception. Specifically, lithospheric deformation, widely evident on the continents, also occurs within oceanic regions and additionally results in tectonic plate boundaries having varying widths. These, non‐rigid, diffuse regions comprise roughly 15% of the terrestrial surface and identification of their analogs in models is an important step in recognizing progress on the goal of modeling plate generation. We describe a new plate detection tool, platerecipy , that utilizes the Random Walker segmentation algorithm to identify candidate plates in both mantle convection model output and global geophysical data sets. The method produces a set of probabilities for each surface data point that can be used to both assess confidence in the association of each location with a distinct rigid plate, and identify diffuse regions across the surface. Verification of the rigidity of each region identified as a distinct plate can be obtained by inverting the associated data for the candidate plate's Euler vector. We demonstrate the method's sensitivity to the three controlling parameters used by platerecipy 's algorithm and how the method can be used to determine the Euler vectors of plates identified in a mantle convection model. We also present promising results found by inverting for the Euler vectors of the Earth's major plates through applying platerecipy to a global strain‐rate field.

Abstract We investigate the use of discrete-time quantum walks to sample from an almost-uniform distribution, in the absence of any external source of randomness. Integers are encoded on the vertices of a cycle graph, and a quantum walker evolves for a fixed number of steps before its position is measured and recorded. The walker is then reset to the measured site, and the procedure is iterated to generate the sequence of outcomes. We show that when the number of nodes of the graph is odd, i.e. the condition of the ergodic theorem for classical random walks on finite groups are satisfied, the marginal distributions converges to the uniform distribution. Although correlations between successive outcomes are unavoidable, they can be significantly reduced by a suitable choice of the evolution time.

We investigate the bond percolation model on transient weighted graphs G induced by the excursion sets of the Gaussian free field on the corresponding metric graph. We assume that balls in G have polynomial volume growth with growth exponent α and that the Green’s function for the random walk on G exhibits a power law decay with exponent ν, in the regime 1≤ν≤α2. In particular, this includes the cases of G=Z3 for which ν=1, and G=Z4 for which ν=α2=2. For all such graphs, we determine the leading-order asymptotic behavior for the critical one-arm probability, which we prove decays with distance R, like R−ν2+o(1). Our results are, in fact, more precise and yield logarithmic corrections when ν>1 as well as corrections of order loglogR when ν=1. We further obtain very sharp upper bounds on truncated two-point functions close to criticality, which are new when ν>1 and essentially optimal when ν=1. This extends previous results from (Invent. Math. 232 (2023) 229–299; Ann. Probab. 48 (2020) 1411–1435).