Properties Of Random Walks Research Articles

Nonequilibrium statistical mechanics of money/energy exchange models

Many-body dynamical models in which Boltzmann statistics can be derived directly from the underlying dynamical laws without invoking the fundamental postulates of statistical mechanics are scarce. Interestingly, one such model is found in econophysics and in chemistry classrooms: the money game, in which players exchange money randomly in a process that resembles elastic intermolecular collisions in a gas, giving rise to the Boltzmann distribution of money owned by each player. Although this model offers a pedagogical example that demonstrates the origins of Boltzmann statistics, such demonstrations usually rely on computer simulations. In fact, a proof of the exponential steady-state distribution in this model has only become available in recent years. Here, we study this random money/energy exchange model and its extensions using a simple mean-field-type approach that examines the properties of the one-dimensional random walk performed by one of its participants. We give a simple derivation of the Boltzmann steady-state distribution in this model. Breaking the time-reversal symmetry of the game by modifying its rules results in non-Boltzmann steady-state statistics. In particular, introducing ‘unfair’ exchange rules in which a poorer player is more likely to give money to a richer player than to receive money from that richer player, results in an analytically provable Pareto-type power-law distribution of the money in the limit where the number of players is infinite, with a finite fraction of players in the ‘ground state’ (i.e. with zero money). For a finite number of players, however, the game may give rise to a bimodal distribution of money and to bistable dynamics, in which a participant’s wealth jumps between poor and rich states. The latter corresponds to a scenario where the player accumulates nearly all the available money in the game. The time evolution of a player’s wealth in this case can be thought of as a ‘chemical reaction’, where a transition between ‘reactants’ (rich state) and ‘products’ (poor state) involves crossing a large free energy barrier. We thus analyze the trajectories generated from the game using ideas from the theory of transition paths and highlight non-Markovian effects in the barrier crossing dynamics.

A Random Journey Through the Math of Gambling

The laws of chance are often subtle and deceptive. This is why games of chance work. People are convinced that they obey seemingly intuitive laws, while the underlying mathematical structure reveals a different and more complex reality. This article is a brief and rigorous journey through the implications that the mathematical laws governing stochastic processes have on gambling. It addresses a specific process, the random walk, and analyze some instances of fair and unfair games by highlighting the fallacy of many of our intuitions and beliefs. The paper gradually moves from the analysis of the random walk properties to a comprehensive description of the ruin problem. The introduction of the idea of transient and persistent states concludes the discussion. Much emphasis is placed on concrete examples and on the numerical values, in particular of the involved probabilities, and the interpretation of the results is always more central than the demonstrative technical details, which are nevertheless available to the reader.

Some martingale properties of the simple random walk and its maximum process

Martingales related to simple random walks and their maximum processes are investigated, and characterizations of those martingales are obtained. As applications, derivation of the Kennedy martingale, proofs of the corresponding Doob inequalities, and a solution to the Skorokhod embedding problem are presented.

Cycles with many chords

AbstractHow many edges in an ‐vertex graph will force the existence of a cycle with as many chords as it has vertices? Almost 30 years ago, Chen, Erdős and Staton considered this question and showed that any ‐vertex graph with edges contains such a cycle. We significantly improve this old bound by showing that edges are enough to guarantee the existence of such a cycle. Our proof exploits a delicate interplay between certain properties of random walks in almost regular expanders. We argue that while the probability that a random walk of certain length in an almost regular expander is self‐avoiding is very small, one can still guarantee that it spans many edges (and that it can be closed into a cycle) with large enough probability to ensure that these two events happen simultaneously.

Liouville property and Poisson boundary of random walks with infinite entropy: what’s amiss?

Обсуждаются качественно новые свойства граничного поведения случайных блужданий на группах, возникающие в случае бесконечности энтропии переходной меры.

Random walks on mapping class groups

This survey is concerned with random walks on mapping class groups. We illustrate how the actions of mapping class groups on Teichmüller spaces or curve complexes reveal the nature of random walks and vice versa. Our emphasis is on the analogues of classical theorems such as laws of large numbers and central limit theorems and the properties of harmonic measures under optimal moment conditions. We also explain the geometric analogy between Gromov hyperbolic spaces and Teichmüller spaces that has been used to copy the properties of random walks from one to the other.

Isogeny graphs on superspecial abelian varieties: eigenvalues and connection to Bruhat–Tits buildings

Abstract We study for each fixed integer $g \ge 2$ , for all primes $\ell $ and p with $\ell \neq p$ , finite regular directed graphs associated with the set of equivalence classes of $\ell $ -marked principally polarized superspecial abelian varieties of dimension g in characteristic p, and show that the adjacency matrices have real eigenvalues with spectral gaps independent of p. This implies a rapid mixing property of natural random walks on the family of isogeny graphs beyond the elliptic curve case and suggests a potential construction of the Charles–Goren–Lauter-type cryptographic hash functions for abelian varieties. We give explicit lower bounds for the gaps in terms of the Kazhdan constant for the symplectic group when $g \ge 2$ . As a byproduct, we also show that the finite regular directed graphs constructed by Jordan and Zaytman also has the same property.

Poisson representation and Furstenberg entropy of hypergroups

We extend the theory of Poisson boundary, tail boundary and the associated entropy theory of groups to the class of discrete hypergroups. We establish a zero-entropy criterion for the Liouville property of random walks on discrete hypergroups and provide various other characterizations for it. Finally, we will solve the identification problem for the Poisson boundary of finite range random walks on permutation hypergroups associated with affine groups of homogenous trees. As a byproduct, we obtain the first examples of random walks on these hypergroups that have a countable infinite Poisson boundary.

Local nonlinear dimensionality reduction via preserving the geometric structure of data

Dimensionality reduction has many applications in data visualization and machine learning. Existing methods can be classified into global ones and local ones. The global methods usually learn the linear relationship in data, while the local ones learn the manifold intrinsic geometry structure, which has a significant impact on pattern recognition. However, most of existing local methods obtain an embedding with eigenvalue or singular value decomposition, where the computational complexities are very high in a large amount of high-dimensional data. In this paper, we propose a local nonlinear dimensionality reduction method named Vec2vec, which employs a neural network with only one hidden layer to reduce the computational complexity. We first build a neighborhood similarity graph from the input matrix, and then define the context of data points with the random walk properties in the graph. Finally, we train the neural network with the context of data points to learn the embedding of the matrix. We conduct extensive experiments of data classification and clustering on nine image and text datasets to evaluate the performance of our method. Experimental results show that Vec2vec is better than several state-of-the-art dimensionality reduction methods, except that it is equivalent to UMAP on data clustering tasks in the statistical hypothesis tests, but Vec2vec needs less computational time than UMAP in high-dimensional data. Furthermore, we propose a more lightweight method named Approximate Vec2vec (AVec2vec) with little performance degradation, which employs an approximate method to build the neighborhood similarity graph. AVec2vec is still better than some state-of-the-art local dimensionality reduction methods and competitive with UMAP on data classification and clustering tasks in the statistical hypothesis tests.

On the mean first passage time and mean cover time of a non-backtracking random walk on graphs

The non-backtracking random walk is known to have nicer properties than simple random walks. However, the effect of graph structure on the properties of the non-backtracking random walk has not been fully clarified. In this paper, we elucidate the mean first passage time of the non-backtracking random walk on two different types of graphs with different characteristics as well as the mean cover time on random regular graphs. Furthermore, we analyze the effects of the graph structure on the properties of the non-backtracking random walk using several numerical examples. Our findings include that the number of nodes in random regular graphs has a great impact on the mean first passage time of the non-backtracking random walk.

Ergodic Limits, Relaxations, and Geometric Properties of Random Walk Node Embeddings

Random walk-based node embedding algorithms learn vector representations of nodes by optimizing an objective function of embedding vectors and skip-bigram statistics computed from random walks on the network. They have demonstrated state-of-the-art performance in many supervised learning problems such as link prediction and node classification. Yet, their properties remain poorly understood. This paper studies properties of random walk based node embeddings in the unsupervised setting of discovering latent network block structure, i.e., learning node representations whose cluster structure in Euclidean space reflects node adjacencies within the network. We characterize the ergodic limits of the embedding objective, its generalization, and related convex relaxations to derive corresponding non-randomized versions of the node embedding objectives. We also characterize the optimal node embedding Grammians of the non-randomized objectives for the expected graph of a two-community Stochastic Block Model (SBM). We prove that the solution Grammian has rank 1 for a suitable nuclear norm relaxation of the non-randomized objective. Comprehensive experimental results on SBM random networks reveal that our non-randomized ergodic objectives yield node embeddings whose distribution is Gaussian-like, centered at the node embeddings of the expected network within each community, and concentrate in the linear degree-scaling regime as the number of nodes increases.

On improving the properties of random walk on graph using Q-learning

Studies have recently been conducted to model mobile agents on unknown graphs, such as random walks (RWs) on graphs, and to understand their mathematical properties. In this study, we investigate the extent to which the properties of RWs can be improved when mobile agents have access to very limited information. We propose Q-weighted random walk (QW-RW), in which an agent decides a next node by using Q-values learned by Q-learning, and examine its effectiveness. We find that in small scale-free graphs, QW-RW is 1.25 times faster than self-avoiding RW to cover 80% of the entire graph.

An Improved NSGA-II Algorithm Based on Adaptive Weighting and Searching Strategy

Non-dominated sorting genetic algorithm II is a classical multi-objective optimization algorithm but it suffers from poor diversity and the tendency to fall into a local optimum. In this paper, we propose an improved non-dominated sorting genetic algorithm, which aims to address the issues of poor global optimization ability and poor convergence ability. The improved NSGA-II algorithm not only uses Levy distribution for global search, which enables the algorithm to search a wider range, but also improves the local search capability by using the relatively concentrated search property of random walk. Moreover, an adaptive balance parameter is designed to adjust the respective contributions of the exploration and exploitation abilities, which lead to a faster search of the algorithm. It helps to expand the search area, which increases the diversity of the population and avoids getting trapped in a local optimum. The superiority of the improved NSGA-II algorithm is demonstrated through benchmark test functions and a practical application. It is shown that the improved strategy provides an effective improvement in the convergence and diversity of the traditional algorithm.

Statistical and transport properties of a one-dimensional random walk with periodically distributed trapping intervals

Statistical and transport properties of a one-dimensional random walk with periodically distributed trapping intervals

AVERAGE TRAPPING TIME ON THE 3-DIMENSIONAL 3-LEVEL SIERPINSKI GASKET NETWORK WITH A SET OF TRAP NODES

As a basic dynamic feature on complex networks, the property of random walk has received a lot of attention in recent years. In this paper, we first studied the analytical expression of the mean global first passage time (MGFPT) on the 3-dimensional 3-level Sierpinski gasket network. Based on the self-similar structure of the network, the correlation between the MGFPT and the average trapping time (ATT) is found, and then the analytical expression of the ATT is obtained. Finally, by establishing a joint network model, we further give the standard process of solving the analytical expression of the ATT when there is a set of trap nodes in the network. By illustrating examples and numerical simulations, it can be proved that when the trap node sets are different, the ATT will be quite different, but the super-linear relationship with the number of iterations will not be changed.

Random walks on convergence groups

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\cup M with a compact topology such that random walks on G converge almost surely to points in M . Furthermore, we prove that if G is finitely generated and the random walk has finite entropy and finite logarithmic moment with respect to the word metric, then M , with the corresponding hitting measure, can be seen as a model for the Poisson boundary of G .

Universal fluctuations and ergodicity of generalized diffusivity on critical percolation clusters

Despite a long history and a clear overall understanding of properties of random walks on an incipient infinite cluster in percolation, some important information on it seems to be missing in the literature. In the present work, we revisit the problem by performing massive numerical simulations for (sub)diffusion of particles on such clusters. Thus, we discuss the shape of the probability density function of particles’ displacements, and the way it converges to its long-time limiting scaling form. Moreover, we discuss the properties of the mean squared displacement (MSD) of a particle diffusing on the infinite cluster at criticality. This one is known not to be self-averaging. We show that the fluctuations of the MSD in different realizations of the cluster are universal, and discuss the properties of the distribution of these fluctuations. These strong fluctuations coexist with the ergodicity of subdiffusive behavior in the time domain. The dependence of the relative strength of fluctuations in time-averaged MSD on the total trajectory length (total simulation time) is characteristic for diffusion in a percolation system and can be used as an additional test to distinguish this process with disorder-induced memory from processes with otherwise similar behavior, like fractional Brownian motion with the same value of the Hurst exponent.

The random walk-based gravity model to identify influential nodes in complex networks

The identification of influential nodes in complex networks has been a topic of immense interest. In most cases, the local approach represented by degree centrality performs well but has limitations when dealing with the bridge nodes. In order to solve the problem of being trapped in the locality, researchers have proposed many useful methods. The gravity model is an emerging research direction among them. However, such models have to exhaust the shortest distance between all nodes, which renders them impractical and difficult to run over large graphs. In order to address this issue, we propose a random walk-based gravity model to identify influential spreaders. Our proposed model decreases the time complexity of calculating the shortest distance—a critical step in the conventional gravity models, from O(|V|2) to O(|V|*γ*lr(l-r)), and reduces space complexity of O(|V|2) to O(<K>2|V|), where <K>2≪|V| and γ*lr(l-r)≪|V|. Some random walk properties are also investigated to support our model. In order to demonstrate the feasibility of the proposed gravity centrality, we have verified its spreading ability and convergence speed under different random walk strategies. Experimental results indicate that our method performs far better than most gravity models.

Random Walks with Bounded First Moment on Finite-volume Spaces

Let G be a real Lie group, Lambda le G a lattice, and Omega =G/Lambda . We study the equidistribution properties of the left random walk on Omega induced by a probability measure mu on G. It is assumed that mu has a finite first moment, and that the Zariski closure of the group generated by the support of mu in the adjoint representation is semisimple without compact factors. We show that for every starting point xin Omega , the mu -walk with origin x has no escape of mass, and equidistributes in Cesàro averages toward some homogeneous measure. This extends several fundamental results due to Benoist-Quint and Eskin-Margulis for walks with finite exponential moment.

Self-Interacting Random Walks: Aging, Exploration, and First-Passage Times

Self-interacting random walks are endowed with long-range memory effects that emerge from the interaction of the random walker at time t with the territory that it has visited at earlier times t′<t. This class of non-Markovian random walks has applications in a broad range of examples, from insects to living cells, where a random walker locally modifies its environment—leaving behind footprints along its path and, in turn, responding to its own footprints. Because of their inherent non-Markovian nature, the exploration properties of self-interacting random walks have remained elusive. Here, we show that long-range memory effects can have deep consequences on the dynamics of generic self-interacting random walks; they can induce aging and nontrivial persistence and transience exponents, which we determine quantitatively, in both infinite and confined geometries. Based on this analysis, we quantify the search kinetics of self-interacting random walkers and show that the distribution of the first-passage time to a target site in a confined domain takes universal scaling forms in the large-domain size limit, which we characterize quantitatively. We argue that memory abilities induced by attractive self-interactions provide a decisive advantage for local space exploration, while repulsive self-interactions can significantly accelerate the global exploration of large domains.Received 13 July 2021Revised 8 November 2021Accepted 20 December 2021DOI:https://doi.org/10.1103/PhysRevX.12.011052Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasDiffusionFluctuations & noiseRandom walksStochastic processesStatistical PhysicsInterdisciplinary Physics