Transition Probability Of Random Walk Research Articles

Delayed random walk on deterministic weighted scale-free small-world network with a deep trap

How to effectively control the trapping process in complex systems is of great importance in the study of trapping problem. Recently, the approach of delayed random walk has been introduced into several deterministic network models to steer trapping process. However, exploring delayed random walk on pseudo-fractal web with the co-evolution of topology and weight has remained out of reach. In this paper, we employ delayed random walk to guide trapping process on a salient deterministic weighted scale-free small-world network with the co-evolution of topology and weight. In greater detail, we first place a deep trap at one of initial nodes of the network. Then, a tunable parameter [Formula: see text] is introduced to modulate the transition probability of random walk and dominate the trapping process. Subsequently, trapping efficiency is used as readout of trapping process and average trapping time is employed to measure trapping efficiency. Finally, the closed form solution of average trapping time (ATT) is deduced analytically, which agrees with corresponding numerical solution. The analytical solution of ATT shows that the delayed parameter [Formula: see text] only modifies the prefactor of ATT, and keeps the leading scaling unchanged. In other words, ATT grows sublinearly with network size, whatever values [Formula: see text] takes. In summary, the work may serves as one piece of clues for modulating trapping process toward desired efficiency on more general deterministic networks.

Average trapping time on weighted directed Koch network

Numerous recent studies have focused on random walks on undirected binary scale-free networks. However, random walks with a given target node on weighted directed networks remain less understood. In this paper, we first introduce directed weighted Koch networks, in which any pair of nodes is linked by two edges with opposite directions, and weights of edges are controlled by a parameter θ . Then, to evaluate the transportation efficiency of random walk, we derive an exact solution for the average trapping time (ATT), which agrees well with the corresponding numerical solution. We show that leading behaviour of ATT is function of the weight parameter θ and that the ATT can grow sub-linearly, linearly and super-linearly with varying θ . Finally, we introduce a delay parameter p to modify the transition probability of random walk, and provide a closed-form solution for ATT, which still coincides with numerical solution. We show that in the closed-form solution, the delay parameter p can change the coefficient of ATT, but cannot change the leading behavior. We also show that desired ATT or trapping efficiency can be obtained by setting appropriate weight parameter and delay parameter simultaneously. Thereby, this work advance the understanding of random walks on directed weighted scale-free networks.

A serendipity-biased Deepwalk for collaborators recommendation.

Scientific collaboration has become a common behaviour in academia. Various recommendation strategies have been designed to provide relevant collaborators for the target scholars. However, scholars are no longer satisfied with the acquainted collaborator recommendations, which may narrow their horizons. Serendipity in the recommender system has attracted increasing attention from researchers in recent years. Serendipity traditionally denotes the faculty of making surprising discoveries. The unexpected and valuable scientific discoveries in science such as X-rays and penicillin may be attributed to serendipity. In this paper, we design a novel recommender system to provide serendipitous scientific collaborators, which learns the serendipity-biased vector representation of each node in the co-author network. We first introduce the definition of serendipitous collaborators from three components of serendipity: relevance, unexpectedness, and value, respectively. Then we improve the transition probability of random walk in DeepWalk, and propose a serendipity-biased DeepWalk, called Seren2vec. The walker jumps to the next neighbor node with the proportional probability of edge weight in the co-author network. Meanwhile, the edge weight is determined by the three indices in definition. Finally, Top-N serendipitous collaborators are generated based on the cosine similarity between scholar vectors. We conducted extensive experiments on the DBLP data set to validate our recommendation performance, and the evaluations from serendipity-based metrics show that Seren2vec outperforms other baseline methods without much loss of recommendation accuracy.

Electrical networks with prescribed current and applications to random walks on graphs

In this paper we study Current Density Impedance Imaging (CDII) on Electrical Networks. The inverse problem is to determine the conductivity matrix of an electrical network from the prescribed knowledge of the magnitude of the induced current along the edges coupled with the imposed voltage or injected current on the boundary nodes. This problem leads to a weighted $ l^1 $ minimization problem for the corresponding voltage potential. We also investigate the problem of determining the transition probabilities of random walks on graphs from the prescribed expected net number of times the walker passes along the edges of the graph. Convergent numerical algorithms for solving such problems are also presented. Our results can be utilized in the design of electrical networks when certain current flow on the network is desired as well as the design of random walk models on graphs when the expected net number of the times the walker passes along the edges is prescribed. We also show that a mass preserving flow $ J = (J_{ij}) $ on a network can be uniquely recovered from the knowledge of $ |J| = (|J_{ij}|) $ and the flux of the flow on the boundary nodes, where $ J_{ij} $ is the flow from node $ i $ to node $ j $ and $ J_{ij} = -J_{ji} $, and discuss its potential application in cryptography.

Local time of Lévy random walks: A path integral approach.

The local time of a stochastic process quantifies the amount of time that sample trajectories x(τ) spend in the vicinity of an arbitrary point x. For a generic Hamiltonian, we employ the phase-space path-integral representation of random walk transition probabilities in order to quantify the properties of the local time. For time-independent systems, the resolvent of the Hamiltonian operator proves to be a central tool for this purpose. In particular, we focus on the local times of Lévy random walks (Lévy flights), which correspond to fractional diffusion equations.

Updatable, Accurate, Diverse, and Scalable Recommendations for Interactive Applications

Recommender systems form the backbone of many interactive systems. They incorporate user feedback to personalize the user experience typically via personalized recommendation lists. As users interact with a system, an increasing amount of data about a user’s preferences becomes available, which can be leveraged for improving the systems’ performance. Incorporating these new data into the underlying recommendation model is, however, not always straightforward. Many models used by recommender systems are computationally expensive and, therefore, have to perform offline computations to compile the recommendation lists. For interactive applications, it is desirable to be able to update the computed values as soon as new user interaction data is available: updating recommendations in interactive time using new feedback data leads to better accuracy and increases the attraction of the system to the users. Additionally, there is a growing consensus that accuracy alone is not enough and user satisfaction is also dependent on diverse recommendations. In this work, we tackle this problem of updating personalized recommendation lists for interactive applications in order to provide both accurate and diverse recommendations. To that end, we explore algorithms that exploit random walks as a sampling technique to obtain diverse recommendations without compromising on efficiency and accuracy. Specifically, we present a novel graph vertex ranking recommendation algorithm called RP 3 β that reranks items based on three-hop random walk transition probabilities. We show empirically that RP 3 β provides accurate recommendations with high long-tail item frequency at the top of the recommendation list. We also present approximate versions of RP 3 β and the two most accurate previously published vertex ranking algorithms based on random walk transition probabilities and show that these approximations converge with an increasing number of samples. To obtain interactively updatable recommendations, we additionally show how our algorithm can be extended for online updates at interactive speeds. The underlying random walk sampling technique makes it possible to perform the updates without having to recompute the values for the entire dataset. In an empirical evaluation with three real-world datasets, we show that RP 3 β provides highly accurate and diverse recommendations that can easily be updated with newly gathered information at interactive speeds (≪ 100 ms ).

Harnack inequalities and local central limit theorem for the polynomial lower tail random conductance model

We prove upper bounds on the transition probabilities of random walks with i.i.d. random conductances with a polynomial lower tail near 0. We consider both constant and variable speed models. Our estimates are sharp. As a consequence, we derive local central limit theorems, parabolic Harnack inequalities and Gaussian bounds for the heat kernel. Some of the arguments are robust and applicable for random walks on general graphs. Such results are stated under a general setting.

Equidistribution of random walks on spheres

We study the equidistribution on spheres of the n-step transition probabilities of random walks on graphs. We give sufficient conditions for this property being satisfied and for the weaker property of asymptotical equidistribution. We analyze the asymptotical behaviour of the Green function of the simple random walk on ℤ2 and we provide a class of random walks on Cayley graphs of groups, whose transition probabilities are not even asymptotically equidistributed.

Almost-sure central limit theorem for a Markov model of random walk in dynamical random environment

We consider a model of random walk on ℤν, ν≥2, in a dynamical random environment described by a field ξ={ξ t (x): (t,x)∈ℤν+1}. The random walk transition probabilities are taken as P(X t +1= y|X t = x,ξ t =η) =P 0( y−x)+ c(y−x;η(x)). We assume that the variables {ξ t (x):(t,x) ∈ℤν+1} are i.i.d., that both P 0(u) and c(u;s) are finite range in u, and that the random term c(u;·) is small and with zero average. We prove that the C.L.T. holds almost-surely, with the same parameters as for P 0, for all ν≥2. For ν≥3 there is a finite random (i.e., dependent on ξ) correction to the average of X t , and there is a corresponding random correction of order to the C.L.T.. For ν≥5 there is a finite random correction to the covariance matrix of X t and a corresponding correction of order to the C.L.T.. Proofs are based on some new L p estimates for a class of functionals of the field.