Unified view on random walk and Weisfeiler-Leman kernels, improving accuracy.
problem Improving graph kernel methods for better classification accuracy.
method Define and analyze walk-based node refinement methods, relate to Weisfeiler-Leman test, and introduce new walk-based kernels.
result Walk-based kernels are as expressive as Weisfeiler-Leman subtree kernel but support non-strict neighborhood comparison.
Tag2Vec learns tag representations in hybrid networks with semantic and hierarchical information.
problem Lack of semantic and hierarchical information in tag networks.
method Tag2Vec model that combines nodes and tags into hybrid networks, using parameterized random walks and hyperbolic Skip-gram model.
result Tag2Vec outperforms other models in learning rich semantic tag representations.
Study reveals neural scaling laws in random graphs and natural language models.
problem Understanding the origin of neural scaling laws in complex systems.
method Examined scaling laws in transformers trained on random walks and simplified natural language models.
result Neural scaling laws emerge in the absence of power law structure in data correlations.
Study large deviations in random walks on Lie groups.
problem Large deviations in sub-Riemannian random walks.
method Prove large deviation principle for random walks on stratified Lie groups.
result Proved a large deviation principle with a rate function adapted to sub-Riemannian geometry.
We review recent advances on the record statistics of strongly correlated time series, whose entries denote the positions of a random walk or a Lévy flight on a line. After a brief survey of the theory of records for independent and identically distributed random variables, we focus on random walks. During the last few…
This paper presents VEC-NBT, a variation on the unsupervised graph clustering technique VEC, which improves upon the performance of the original algorithm significantly for sparse graphs. VEC employs a novel application of the state-of-the-art word2vec model to embed a graph in Euclidean space via random walks on the n…
Quantum walks blend patterns into splines when averaged.
problem Understanding the asymptotic patterns of quantum random walks.
method Averaging over quantum coins using the Haar measure.
result Patterns blend into splines, showing a unified behavior.
Local limit theorem for random walks on hyperbolic groups with parabolic subgroups.
problem Analyzing the behavior of random walks on relatively hyperbolic groups.
method Study of convergent random walks with finite derivative of Green function at spectral radius.
result Proves a local limit theorem for the probability of returning to the origin.
Random walks on cell complexes link to Laplacians and Novikov-Shubin invariants.
problem Computing Novikov-Shubin invariants for complex cell structures.
method Construct random walks on cell complexes, relate to Laplacians, and use return probabilities.
result Novikov-Shubin invariants can be recovered from random walk return probabilities.
Study diffusions and random walks on hyperbolic spaces, focusing on their Martin boundaries.
problem Understanding diffusions and random walks on hyperbolic spaces.
method Analyzing specific diffusions and random walks on hyperbolic spaces, examining their Martin boundaries.
result Characterized the Martin boundaries of diffusions and random walks on hyperbolic spaces.
Random walks on metric spaces embed quasi-isometrically into the space.
problem Embedding random subgroups of metric spaces quasi-isometrically.
method Analyzing random walks and contracting elements in metric spaces.
result Random subgroups of isometry groups are quasi-isometrically embedded.
This work estimates edge weights of edge-reinforced random walks using observed data.
problem Statistical estimation of edge weights in edge-reinforced random walks.
method Proposes an estimator based on the generalized method of moments using the magic formula and hyperbolic Gaussian structure.
result Analyzes the sample complexity of the proposed estimator.
New proof shows rapid mixing for random walks on nilmanifolds.
problem Proving rapid mixing for random walks on nilmanifolds.
method Proved rapid mixing for almost all random walks generated by m translations on nilmanifolds under mild assumptions.
result For several classical classes of nilmanifolds, m=2 suffices for rapid mixing.
Random walks on hyperbolic spaces show linear growth in translation lengths.
problem Investigate the growth of translation lengths in random walks on hyperbolic spaces.
method Prove linear growth without moment conditions and apply to Teichmüller spaces.
result Linear growth of translation lengths in random walks on hyperbolic spaces.
Study random walks on groups with superlinear divergent geodesics.
problem Existence of superlinear divergent geodesics in groups.
method Developed theory of superlinear divergence and applied Gouëzel's pivoting technique.
result Established a central limit theorem for random walks on groups with superlinear divergent geodesics.
The paper develops a spectral theory for hypergraphs with edge-dependent vertex weights using random walks.
problem Lack of spectral theory for hypergraphs with edge-dependent vertex weights.
method Random walks on hypergraphs with edge-dependent vertex weights, deriving a random walk-based hypergraph Laplacian.
result Random walks on hypergraphs with edge-dependent vertex weights can capture higher-order relationships in data.
Study random walks on sub-Riemannian manifolds using retractions.
problem Modeling random walks on sub-Riemannian manifolds.
method Use retractions to approximate normal geodesics and study convergence to Brownian motion.
result Convergence of geodesic random walks defined with different connections.
A new method for embedding heterogeneous networks using spacey random walks.
problem Stationarity issues in meta-path guided random walks for HIN embedding.
method Heterogeneous personalized spacey random walk.
result Substantial improvement over existing network embedding algorithms.
A scalable framework preserves personalized higher-order network proximities.
problem Lack of expressive methods to preserve personalized higher-order network proximities.
method Incorporates random walk into a sound objective to preserve arbitrary higher-order proximities and introduces random walk with restart for personalized-weighted preservation.
result Consistently and substantially outperforms state-of-the-art methods on real-world networks.
The paper examines random walks on metric spaces and finds commensurable subgroups.
problem Determining commensurable subgroups via stationary measures in metric spaces.
method Analyzing random walks on isometry groups of metric spaces with non-singular stationary measures.
result Subgroups generated by random walks are commensurable under mild conditions.
Random walks on free groups reveal asymmetric expansion factors.
problem Understanding expansion factors in free groups.
method Random walks and BGIP on metric spaces.
result Generic outer automorphisms have different forward and backward expansion factors.
Survey on random walks on mapping class groups and their properties.
problem Understanding random walks on mapping class groups.
method Analyzing actions on Teichmüller spaces and curve complexes.
result Laws of large numbers and central limit theorems for random walks.
A new method improves NetGAN's random walk initialization for graph generation.
problem NetGAN's random walk initialization leads to inconsistent and variable results, especially with short walks.
method Estimating node importance based on random walk influence over its neighborhood.
result Significantly better accuracy, less variance, and fewer outliers achieved.
Deviation inequalities and limit laws for random walks on metric spaces.
problem Understanding random walks on metric spaces with contracting isometries.
method Adapting Gouëzel's pivotal time construction to establish deviation inequalities.
result Exponential bounds and limit laws for random walks on mapping class groups and CAT(0) spaces.
Random walks on Fuchsian Schottky groups have harmonic measures with lower dimension.
problem Understanding the dimensionality of harmonic measures for random walks.
method Analyzing finite range random walks on Fuchsian Schottky groups.
result Harmonic measures have dimension strictly less than the limit set's Hausdorff dimension.
UniNet efficiently learns network representations from large graphs.
problem Efficiently learning network representations from large graphs.
method Metropolis-Hastings sampling for efficient edge sampling and random walk model abstraction.
result UniNet outperforms existing NRL models on billion-edge networks.
The study of random walks on hyperbolic spaces and Teichmüller spaces, proving central limit theorems and geodesic tracking.
problem Analyzing random walks on hyperbolic and Teichmüller spaces.
method Proving central limit theorems and geodesic tracking using finite moments and logarithmic moments.
result Translation lengths of random isometries satisfy a central limit theorem if and only if the random walk has finite second moment.
Uniform drift estimates found for random walks on graph products.
problem Finding uniform lower bounds on drift for random walks on graph products.
method Extending Gouëzel's argument and introducing the combinatorial notion of piling.
result Uniform lower bounds on the drift for a family of random walks on graph products.
A random Heegaard splitting is a 3-manifold obtained by using a random walk of length n on the mapping class group as the gluing map between two handlebodies. We show that the joint distribution of random walks of length n and their inverses is asymptotically independent, and converges to the product of the harmonic an…
Geodesic walks converge to Brownian motion on Finsler manifolds.
problem Understanding random walks on Finsler manifolds.
method Analyzing convergence of geodesic random walks to diffusion processes.
result The Brownian motion on a Riemannian metric is a key result.
We extend some properties of random walks on hyperbolic groups to random walks on convergence groups. In particular we prove that if a convergence group G acts on a compact metrizable space M with the convergence property then we can provide G∪M with a compact topology such that random walks on G converge a…
For any pseudo-Anosov diffeomorphism on a closed orientable surface S of genus greater than one, it is known by the work of Bers and Thurston that the topological entropy agrees with the translation distance on the Teichmüller space with respect to the Teichmüller metric. In this paper, we consider random walks on th…
Quantum walk algorithm optimizes quantum state preparation for financial simulations.
problem Efficiently loading classical data into quantum states for quantum computers.
method Split-step quantum walks (SSQW) to design parameterized quantum circuits (PQC).
result SSQW facilitates generating desired probability amplitude distributions for quantum simulations.
Repelling random walks improve graph-based sampling efficiency.
problem Efficient graph-based sampling and statistical estimation.
method Induces correlations between trajectories of an ensemble of walkers on a graph, maintaining unbiasedness.
result Improves concentration of statistical estimators on graphs.
Abstract: Nonlinear random walk with distributionally robust transition probabilities.
problem Modeling nonlinear random walks with robust transition probabilities.
method Scaling limit and nonlinear semigroup approach.
result Explicit computation of the generator and corresponding PDE.
Random walks and polygons are used to model polymers. In this paper we consider the extension of writhe, self-linking number and linking number to open chains. We then study the average writhe, self-linking and linking number of random walks and polygons over the space of configurations as a function of their length. W…
Study large deviations and speed of random walks in hyperbolic spaces.
problem Understanding the speed of random walks in hyperbolic spaces.
method Large deviations analysis for random walks with a non-elementary semi-group.
result Established large deviations results for random walk distances.
The study analyzes convergence of random-walk embeddings in graph theory.
problem Understanding the convergence behavior of random-walk based vertex embeddings.
method Theoretical analysis of convergence in single and double limits of N and L. result Proved convergence of vertex embeddings under weak assumptions and derived concentration bounds.
Analyzes biased random walks and corrupted intervals in adversarial settings.
problem Learning thresholds and intervals in adversarial conditions.
method Analyzes biased random walks and corrupted intervals under adversarial design.
result Analyzes the expected behavior of biased random walks and corrupted intervals.
Random walks on mapping class groups identified with geodesic laminations.
problem Understanding random walks on mapping class groups.
method Electrification of curve graph, identifying Poisson boundary, using geodesic laminations.
result Random walk on mapping class group identified with geodesic laminations.
Study ratio-limit boundaries for random walks on hyperbolic groups.
problem Computing ratio-limit boundaries for relatively hyperbolic groups.
method Adapting Woess's strategy to non-hyperbolic groups and analyzing degenerate cases.
result Closure of minimal points in R-Martin boundary is the unique smallest invariant subspace in ratio-limit boundary. Fold maps associated to geodesic random walks on curved spaces.
problem Understanding the behavior of geodesic random walks on curved surfaces.
method Analyzing mappings from the unit tangent sphere to a manifold with non-positive curvature.
result For odd powers of the unit tangent sphere, these mappings are fold maps.
We show that the probability that a finitely supported random walk on a non-elementary subgroup of the the mapping class group gives a non-pseudo-Anosov element decays exponentially in the length of the random walk. More generally, we show that if R is a set of mapping class group elements with an upper bound on their …
Study examines large deviations in random walks on hyperbolic spaces.
problem Large deviations in random walks on Gromov-hyperbolic spaces.
method Established large deviations results for distance and translation length of random walks.
result Deduced a special case of a conjecture regarding spectral radii of random matrix products.
Random walk constructs Morse functions on surfaces.
problem Creating Morse functions on surfaces.
method Random walk method to construct Morse functions.
result Small set of Morse functions approximates any other function.
We consider a random walk on the mapping class group of a surface of finite type. We assume that the random walk is determined by a probability measure whose support is finite and generates a non-elementary subgroup H. We further assume that H is not consisting only of lifts with respect to any one covering. Then w…
We review statistical properties of models generated by the application of a (positive and negative order) fractional derivative operator to a standard random walk and show that the resulting stochastic walks display slowly-decaying autocorrelation functions. The relation between these correlated walks and the well-kno…
Random walk speed on Teichmüller space is a proper function.
problem Understanding the speed of random walks on Teichmüller space.
method Adaptation of Gouëzel's pivoting techniques to Teichmüller space.
result Speed of random walk is a proper function on Teichmüller space.