Accuracy and precision of random walk with barrier model fitting: Simulations and applications in head and neck cancers.

Limit theorems for elephant random walks remembering the very recent past, with applications to the Takagi–van der Waerden class functions

Het2Gene: a phenotype-driven model for gene prioritization by heterogeneous graph embedding.

On elephant random walk with random memory

CycRW: Cycle-graph random walks for identifying high-impact spreaders in complex networks

Strong large deviation principles for pair empirical measures of random walks in the Mukherjee-Varadhan topology

Regulatory role of mobile fine particles in anomalous solute transport in porous media.

Drug combination therapy is pivotal for complex diseases, but identifying synergistic three-drug regimens remains challenging due to both combinatorial explosion and the opacity of existing computational models. To address this, we introduce HyperSynergyX, an explainable framework that integrates synergy prediction with mechanistic explanation. Its core predictive component, a Dual-Biased Random Walk on Hypergraphs (DBRWH), models higher-order interactions among drugs on a three drug hypergraph and identifies latent combination patterns via tensor decomposition. To enhance interpretability, we couple DBRWH with a knowledge-graph-enhanced retrieval augmented generation (KG-RAG) module that retrieves mechanistically relevant subgraphs and uses them to generate biologically grounded hypotheses for predicted synergies. On breast-cancer data, DBRWH achieves AUROC/AUPRC of 0.9593/0.9453 under 5-fold cross-validation, and on lung cancer data it achieves 0.9262/0.9481, outperforming strong deep learning and hypergraph baselines. By linking predictive performance with mechanistic interpretability, HyperSynergyX provides a robust and transparent tool to accelerate multi-drug discovery and support rational regimen design in precision oncology. The code is available at: https://github.com/wangqi27/HyperSynergyX.

Modeling discourse structure with 2D similarity-based random walks for improved understanding of online conversations.

Drought is a primary factor that adversely impacts rice yield and quality. Identifying drought-responsive genes is essential for developing drought-responsive cultivars. Recently, graph neural network methods based on embedding learning have shown considerable success in biological networks. However, challenges still remain in adequately capturing node attribute features, representing topological structures, and addressing class imbalance, which may constrain the model's predictive capability. To address these issues, we propose a distance-based prototypical graph neural network with path aggregation mechanism (DPGNNPAM) to mine drought-responsive genes in rice. First, we combine gene expression data and protein interaction networks in rice to construct graph-based datasets. Next, we utilize a random walk strategy to generate diverse walk paths and employ a recursive neural network-based path aggregator to encode node attributes along these paths. The prototypical network approach is subsequently employed during training to focus on global information and address the issue of sample imbalance. After that, the weighted similarity is computed by measuring the distance between the node embeddings and the class prototypes. Specifically, class prototypes are representative embeddings that capture the central characteristic of each class. Finally, we transform this value into a predictive probability using the softmax function. Experimental results demonstrate that DPGNNPAM outperforms traditional graph neural network algorithms in identifying drought-responsive genes in rice. Ultimately, we identify 17 candidate genes closely related to drought stress, 12 of which are confirmed in the literature as being involved in the plant's drought stress response.

Abstract Random walks in a finite Abelian group G are studied. They use Markov chains with doubly stochastic transition matrices, in a Birkhoff subpolytope B ( G ) associated with the group G . It is shown that all future probability vectors belong to a polytope which does not depend on the transition matrices, and which shrinks during time evolution. Various quantities are used to describe the probability vectors: the majorization preorder, Lorenz values and the Gini index, entropic quantities, and the total variation distance. The general results are applied to the additive group Z ( d ) , and to the Heisenberg–Weyl group H W ( d ) / Z ( d ) . A physical implementation of random walks in Z ( d ) that involves a sequence of non-selective projective measurements, is discussed. A physical implementation of random walks in the Heisenberg–Weyl group H W ( d ) / Z ( d ) using a sequence of non-selective positive operator-valued measure measurements with coherent states, is also presented.

With the increasing demands for autonomy and coverage efficiency in tasks such as security patrol and post-disaster exploration using mobile robots, achieving random, efficient, and complete coverage path planning has become a critical challenge. Traditional chaotic path planning methods, while capable of generating unpredictable trajectories, still have limitations in terms of randomness strength, traversal uniformity, and convergence coverage. To address this, this study proposes a complete-coverage random path planning method based on a novel four-dimensional fractal-fractional multi-scroll chaotic system. The main contributions of this research are as follows: First, by introducing additional state variables and fractal-fractional operators into the classical Chen system, a fractal-fractional chaotic system with a multi-scroll attractor structure is constructed. The output of this system is then mapped into robot angular velocity commands to achieve area coverage in unknown environments. Key findings include: the novel chaotic system possesses two positive Lyapunov exponents; Spectral Entropy (SE) and Complexity (CO) analyses indicate that when parameter B is fixed and the fractional order α increases, the dynamic complexity of the system significantly rises; in a 50 × 50 grid environment, the robot driven by this system achieved a coverage rate of 98.88% within 10,000 iterations, outperforming methods based on Lorenz, Chua systems, and random walks; ablation experiments further demonstrate that the combined effects of the fractal order β, fractional order α, and multi-scroll nonlinear terms are key to enhancing system complexity and coverage performance. The significance of this study lies in that it not only provides new ideas for constructing complex chaotic systems but also offers a reliable theoretical foundation and practical solution for mobile robots to perform efficient, random, and high-coverage autonomous inspection tasks in unknown regions.

Results from the mathematical literature on random walks reveal a closed-form analytical expression for the π-energy and bond number of graphene in the simplest tight-binding model and its Hartree-Fock Hubbard extension. Closed-form expressions follow for all π spectral moments of graphene. Bond numbers of carbon and boron nitride (BN) zigzag nanotubes are found as finite sums, with graphene and hexagonal boron nitride sheets as asymptotes.

Abstract We study the first-passage properties of a jump process with constant drift where jump amplitudes and inter-arrival times follow arbitrary light-tailed distributions with smooth densities. Using a mapping to an effective discrete-time random walk, we identify three regimes determined by the drift strength: survival (weak drift), absorption (strong drift), and critical. We derive explicit expressions for exponential decay rates in the survival and absorption regimes, and characterize algebraic decay at the critical point. We also obtain asymptotic behavior of the mean first-passage time, number of jumps, and their variances for processes starting either close to the origin or far from it.

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.

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.

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.

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.

Identifying influential nodes in complex networks is fundamental for understanding information diffusion, epidemic control, and network resilience. Conventional centrality measures often rely solely on local or global topological features while neglecting dynamic interactions, leading to limited accuracy. To address this issue, we propose an Improved Gravity Centrality based on Degree-Mixed Clustering Coefficient and Return Random Walk (DMCIGC) that integrates both structural and dynamic characteristics. DMCIGC constructs the degree-mixed clustering coefficient hybrid index based on merging node degree and clustering coefficient, and incorporates information entropy into an improved return random walk framework to capture dynamic distances. Experiments on multiple real-world networks datasets demonstrate that DMCIGC effectively detects nodes with dual influence, which are locally central and globally cohesive across the network. Comparative analyses show that DMCIGC consistently outperforms classical centrality measures in spreading ability, achieving the highest average Kendall rank correlation coefficient of 0.6940, which is 0.023 higher than the runner-up across the evaluated datasets. Overall, this study presents a unified and efficient framework for influential node identification, bridging local-global and static-dynamic perspectives in complex network analysis.