Exploring activity-driven network with biased walks

2017 ◽  
Vol 28 (09) ◽  
pp. 1750111
Author(s):  
Yan Wang ◽  
Ding Juan Wu ◽  
Fang Lv ◽  
Meng Long Su
Keyword(s):  
Random Walk ◽  
Diffusion Process ◽  
Random Walks ◽  
Stationary Distribution ◽  
Significant Role ◽  
Search Strategy ◽  
Transition Probability ◽  
Edge Weighting ◽  
Coverage Function ◽  
Analytical Expressions

We investigate the concurrent dynamics of biased random walks and the activity-driven network, where the preferential transition probability is in terms of the edge-weighting parameter. We also obtain the analytical expressions for stationary distribution and the coverage function in directed and undirected networks, all of which depend on the weight parameter. Appropriately adjusting this parameter, more effective search strategy can be obtained when compared with the unbiased random walk, whether in directed or undirected networks. Since network weights play a significant role in the diffusion process.

Transition probability matrices for correlated random walks

1980 ◽  
Vol 17 (01) ◽  
pp. 253-258 ◽  
Author(s):  
R. B. Nain ◽  
Kanwar Sen
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Difference Equations ◽  
Generating Functions ◽  
Transition Probability ◽  
Correlated Random Walk ◽  
One Dimensional ◽  
Correlated Random Walks ◽  
Transition Probability Matrices ◽  
Probability Matrices

For correlated random walks a method of transition probability matrices as an alternative to the much-used methods of probability generating functions and difference equations has been investigated in this paper. To illustrate the use of transition probability matrices for computing the various probabilities for correlated random walks, the transition probability matrices for restricted/unrestricted one-dimensional correlated random walk have been defined and used to obtain some of the probabilities.

scholarly journals Stochastic LU factorizations, Darboux transformations and urn models

2018 ◽  
Vol 55 (3) ◽  
pp. 862-886 ◽  
Author(s):  
F. Alberto Grünbaum ◽  
Manuel D. de la Iglesia
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Free Parameter ◽  
Transition Probabilities ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Urn Models ◽  
Triangular Matrices ◽  
New Family ◽  
Step Transition

Abstract We consider upper‒lower (UL) (and lower‒upper (LU)) factorizations of the one-step transition probability matrix of a random walk with the state space of nonnegative integers, with the condition that both upper and lower triangular matrices in the factorization are also stochastic matrices. We provide conditions on the free parameter of the UL factorization in terms of certain continued fractions such that this stochastic factorization is possible. By inverting the order of the factors (also known as a Darboux transformation) we obtain a new family of random walks where it is possible to state the spectral measures in terms of a Geronimus transformation. We repeat this for the LU factorization but without a free parameter. Finally, we apply our results in two examples; the random walk with constant transition probabilities, and the random walk generated by the Jacobi orthogonal polynomials. In both situations we obtain urn models associated with all the random walks in question.

STATIONARY PROCESSES THAT LOOK LIKE RANDOM WALKS— THE BOUNDED RANDOM WALK PROCESS IN DISCRETE AND CONTINUOUS TIME

2002 ◽  
Vol 18 (1) ◽  
pp. 99-118 ◽  
Author(s):  
João Nicolau
Keyword(s):  
Time Series ◽  
Random Walk ◽  
Diffusion Process ◽  
Random Walks ◽  
Interest Rates ◽  
Continuous Time ◽  
Financial Time Series ◽  
Finite Limit ◽  
Closed Expression ◽  
Continuous Time Process

Several economic and financial time series are bounded by an upper and lower finite limit (e.g., interest rates). It is not possible to say that these time series are random walks because random walks are limitless with probability one (as time goes to infinity). Yet, some of these time series behave just like random walks. In this paper we propose a new approach that takes into account these ideas. We propose a discrete-time and a continuous-time process (diffusion process) that generate bounded random walks. These paths are almost indistinguishable from random walks, although they are stochastically bounded by an upper and lower finite limit. We derive for both cases the ergodic conditions, and for the diffusion process we present a closed expression for the stationary distribution. This approach suggests that many time series with random walk behavior can in fact be stationarity processes.

Mean first return time for random walks on weighted networks

2015 ◽  
Vol 26 (06) ◽  
pp. 1550068 ◽  
Author(s):  
Xing-Li Jing ◽  
Xiang Ling ◽  
Jiancheng Long ◽  
Qing Shi ◽  
Mao-Bin Hu
Keyword(s):  
Random Walk ◽  
Complex Networks ◽  
Random Walks ◽  
Stationary Distribution ◽  
Real World ◽  
Return Time ◽  
Edge Weight ◽  
Weighted Networks ◽  
Strength Based ◽  
First Return Time

Random walks on complex networks are of great importance to understand various types of phenomena in real world. In this paper, two types of biased random walks on nonassortative weighted networks are studied: edge-weight-based random walks and node-strength-based random walks, both of which are extended from the normal random walk model. Exact expressions for stationary distribution and mean first return time (MFRT) are derived and examined by simulation. The results will be helpful for understanding the influences of weights on the behavior of random walks.

Diffusion Approximation for Random Walks on Anisotropic Lattices

1998 ◽  
Vol 35 (01) ◽  
pp. 206-212
Author(s):  
Lajos Horváth
Keyword(s):  
Random Walk ◽  
Diffusion Process ◽  
Random Walks ◽  
Diffusion Approximation ◽  
Horizontal Position ◽  
Two Dimensional ◽  
Anisotropic Lattices ◽  
Anisotropic Lattice

We show that the horizontal position of a random walk on a two-dimensional anisotropic lattice converges weakly to a diffusion process.

Transition probability matrices for correlated random walks

1980 ◽  
Vol 17 (1) ◽  
pp. 253-258 ◽  
Author(s):  
R. B. Nain ◽  
Kanwar Sen
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Difference Equations ◽  
Generating Functions ◽  
Transition Probability ◽  
Correlated Random Walk ◽  
One Dimensional ◽  
Correlated Random Walks ◽  
Transition Probability Matrices ◽  
Probability Matrices

For correlated random walks a method of transition probability matrices as an alternative to the much-used methods of probability generating functions and difference equations has been investigated in this paper. To illustrate the use of transition probability matrices for computing the various probabilities for correlated random walks, the transition probability matrices for restricted/unrestricted one-dimensional correlated random walk have been defined and used to obtain some of the probabilities.

scholarly journals Characterizing the Relationship Between Unitary Quantum Walks and Non-Homogeneous Random Walks

2021 ◽  
Author(s):  
Matheus Guedes de Andrade ◽  
Franklin De Lima Marquezino ◽  
Daniel Ratton Figueiredo
Keyword(s):  
Random Walk ◽  
Probability Distribution ◽  
Random Walks ◽  
Transition Probability ◽  
Stochastic Matrix ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Time Dependent ◽  
Matrix Sequence ◽  
The Relationship

Quantum walks on graphs are ubiquitous in quantum computing finding a myriad of applications. Likewise, random walks on graphs are a fundamental building block for a large number of algorithms with diverse applications. While the relationship between quantum and random walks has been recently discussed in specific scenarios, this work establishes a formal equivalence between the two processes on arbitrary finite graphs and general conditions for shift and coin operators. It requires empowering random walks with time heterogeneity, where the transition probability of the walker is non-uniform and time dependent. The equivalence is obtained by equating the probability of measuring the quantum walk on a given node of the graph and the probability that the random walk is at that same node, for all nodes and time steps. The first result establishes procedure for a stochastic matrix sequence to induce a random walk that yields the exact same vertex probability distribution sequence of any given quantum walk, including the scenario with multiple interfering walkers. The second result establishes a similar procedure in the opposite direction. Given any random walk, a time-dependent quantum walk with the exact same vertex probability distribution is constructed. Interestingly, the matrices constructed by the first procedure allows for a different simulation approach for quantum walks where node samples respect neighbor locality and convergence is guaranteed by the law of large numbers, enabling efficient (polynomial-time) sampling of quantum graph trajectories. Furthermore, the complexity of constructing this sequence of matrices is discussed in the general case.

scholarly journals Correlated random walks with stay

1988 ◽  
Vol 1 (3) ◽  
pp. 197-222
Author(s):  
Ram Lal ◽  
U. Narayan Bhat
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Random Walks ◽  
Equilibrium Solution ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Finite State ◽  
Direction Of Movement ◽  
One Step ◽  
Bivariate Markov Chain

A random walk describes the movement of a particle in discrete time, with the direction and the distance traversed in one step being governed by a probability distribution. In a correlated random walk (CRW) the movement follows a Markov chain and induces correlation in the state of the walk at various epochs. Then, the walk can be modelled as a bivariate Markov chain with the location of the particle and the direction of movement as the two variables. In such random walks, normally, the particle is not allowed to stay at one location from one step to the next. In this paper we derive explicit results for the following characteristics of the CRW when it is allowed to stay at the same location, directly from its transition probability matrix: (i) equilibrium solution and the fast passage probabilities for the CRW restricted on one side, and (ii) equilibrium solution and first passage characteristics for the CRW restricted on bath sides (i.e., with finite state space).

Diffusion Approximation for Random Walks on Anisotropic Lattices

1998 ◽  
Vol 35 (1) ◽  
pp. 206-212 ◽  
Author(s):  
Lajos Horváth
Keyword(s):  
Random Walk ◽  
Diffusion Process ◽  
Random Walks ◽  
Diffusion Approximation ◽  
Horizontal Position ◽  
Two Dimensional ◽  
Anisotropic Lattices ◽  
Anisotropic Lattice

We show that the horizontal position of a random walk on a two-dimensional anisotropic lattice converges weakly to a diffusion process.

scholarly journals Current Trends in Random Walks on Random Lattices

Mathematics ◽  
2021 ◽  
Vol 9 (10) ◽  
pp. 1148
Author(s):  
Jewgeni H. Dshalalow ◽  
Ryan T. White
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Real Time ◽  
Real Space ◽  
Historical Background ◽  
Escape Time ◽  
The Real ◽  
Current Trends ◽  
One Step ◽  
Time Position

In a classical random walk model, a walker moves through a deterministic d-dimensional integer lattice in one step at a time, without drifting in any direction. In a more advanced setting, a walker randomly moves over a randomly configured (non equidistant) lattice jumping a random number of steps. In some further variants, there is a limited access walker’s moves. That is, the walker’s movements are not available in real time. Instead, the observations are limited to some random epochs resulting in a delayed information about the real-time position of the walker, its escape time, and location outside a bounded subset of the real space. In this case we target the virtual first passage (or escape) time. Thus, unlike standard random walk problems, rather than crossing the boundary, we deal with the walker’s escape location arbitrarily distant from the boundary. In this paper, we give a short historical background on random walk, discuss various directions in the development of random walk theory, and survey most of our results obtained in the last 25–30 years, including the very recent ones dated 2020–21. Among different applications of such random walks, we discuss stock markets, stochastic networks, games, and queueing.

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