scholarly journals Exact results for the roughness of a finite size random walk

2006 ◽  
Vol 370 (1) ◽  
pp. 127-131 ◽  
Author(s):  
V. Alfi ◽  
F. Coccetti ◽  
M. Marotta ◽  
A. Petri ◽  
L. Pietronero
Keyword(s):  
Random Walk ◽  
Finite Size ◽  
Exact Results

scholarly journals RANDOM WALKS ON DIRECTED NETWORKS: THE CASE OF PAGERANK

2007 ◽  
Vol 17 (07) ◽  
pp. 2343-2353 ◽  
Author(s):  
SANTO FORTUNATO ◽  
ALESSANDRO FLAMMINI
Keyword(s):  
Random Walk ◽  
Power Law ◽  
Degree Distribution ◽  
Directed Graphs ◽  
Damping Factor ◽  
Web Pages ◽  
Stationary Probability ◽  
Exact Results ◽  
Limit Values ◽  
Underlying Graph

PageRank, the prestige measure for Web pages used by Google, is the stationary probability of a peculiar random walk on directed graphs, which interpolates between a pure random walk and a process where all nodes have the same probability of being visited. We give some exact results on the distribution of PageRank in the cases in which the damping factor q approaches the two limit values 0 and 1. When q → 0 and for several classes of graphs the distribution is a power law with exponent 2, regardless of the in-degree distribution. When q → 1 it can always be derived from the in-degree distribution of the underlying graph, if the out-degree is the same for all nodes.

scholarly journals Some approximate results in renewal and dam theories

1971 ◽  
Vol 12 (4) ◽  
pp. 425-432 ◽  
Author(s):  
R. M. Phatarfod
Keyword(s):  
Random Walk ◽  
Sequential Analysis ◽  
Random Variables ◽  
Renewal Theory ◽  
Slight Modification ◽  
Exact Results ◽  
Fundamental Identity ◽  
Gambler’S Ruin ◽  
Gambler's Ruin Problem ◽  
Two Barriers

It is well known that Wald's Fundamental Identity (F.I.) in sequential analysis can be used to derive approximate (and, sometimes exact) results in most situations wherein we have essentially a random walk phenomenon. Bartlett [2] used it for the gambler's ruin problem and also for a simple renewal problem. Phatarfod [18] used it for a problem in dam theory. It is the purpose of this paper to show how a generalization of the Fundamental Identity to Markovian variables, (Phatarfod [19]) can be used to derive approximate results in some problems in dam and renewal theories where the random variables involved have Markovian dependence. The reason for considering both the theories together is that the models usually proposed for both the theories — input distribution for dam theory, and lifedistribution for renewal theory — are similar, and only a slight modification (to account for the ‘release rules’ in dam theory, plus the fact that we have two barriers) is necessary to derive results in dam theory from those of renewal theory.

Cayley Tree Random Walk Dynamics

2000 ◽  
Vol 651 ◽  
Author(s):  
Dimitrios Katsoulis ◽  
Panos Argyrakis ◽  
Alexander Pimenov ◽  
Lexei Vitukhnovsky
Keyword(s):  
Random Walk ◽  
Cayley Tree ◽  
Infinite System ◽  
Mean Square Displacement ◽  
Finite Size ◽  
Finite Size Effects ◽  
Cayley Trees ◽  
Large Trees ◽  
The Mean ◽  
Diffusion Properties

AbstractWe investigate diffusion on newly synthesized molecules with dendrimer structures. We model these structures with geometrical Cayley trees. We focus on diffusion properties, such as the excursion distance, the mean square displacement of the diffusing particles, and the area probed, as given by the walk parameter SN, the number of the distinct sites visited, on different coordination number, z, and different generation order g of a dendrimer structure. We simulate the trapping kinetics curves for randomly distributed traps on these structures, and compare the finite and the infinite system cases, and also with the cases of regular dimensionality lattices. For small dendrimer structures, SN approaches the overall number of the dendrimer nodes, while for large trees it grows linearly with time. The average displacement R also grows linearly with time. We find that the random walk on Cayley trees, due to the nature ot these structures, is indeed a type of a “biased” walk. Finally we find that the finite-size effects are particularly important in these structures.

scholarly journals Static and dynamic properties of selected stochastic processes on complex networks

2020 ◽  
Author(s):  
Jeremi Ochab
Keyword(s):  
Random Walk ◽  
Complex Networks ◽  
Random Walks ◽  
Dynamic Properties ◽  
Finite Size ◽  
Disease Spread ◽  
Maximal Entropy ◽  
Centrality Measures ◽  
Topological Features ◽  
The Subject

This thesis is concerned with the properties of a number of selected processes taking place on complex networks and the way they are affected by structure and evolution of the networks. What is meant here by 'complex networks' is the graph-theoretical representations and models of various empirical networks (e.g., the Internet network) which contain both random and deterministic structures, and are characterised among others by the small-world phenomenon, power-law vertex degree distributions, or modular and hierarchical structure. The mathematical models of the processes taking place on these networks include percolation and random walks we utilise.The results presented in the thesis are based on five thematically coherent papers. The subject of the first paper is calculating thresholds for epidemic outbreaks on dynamic networks, where the disease spread is modelled by percolation. In the paper, known analytical solutions for the epidemic thresholds were extended to a class of dynamically evolving networks; additionally, the effects of finite size of the network on the magnitude of the epidemic were studied numerically. The subject of the second and third paper is the static and dynamic properties of two diametrically opposed random walks on model highly symmetric deterministic graphs. Specifically, we analytically and numerically find the stationary states and relaxation times of the ordinary, diffusive random walk and the maximal-entropy random walk. The results provide insight into localisation of random walks or their trapping in isolated regions of networks. Finally, in the fourth and fifth paper, we examine the utility of random walks in detecting topological features of complex networks. In particular, we study properties of the centrality measures (roughly speaking, the ranking of vertices) based on random walks, as well as we conduct a systematic comparative study of random-walk based methods of detecting modular structure of networks.These studies thus aimed at specific problems in modelling and analysis of complex networks, including theoretical examination of the ways the behaviour of random processes intertwines with the structure of complex networks.

Improved Discrete Random Walk Stochastic Model for Simulating Particle Dispersion and Deposition in Inhomogeneous Turbulent Flows

2020 ◽  
Vol 142 (10) ◽  
Author(s):  
Amir A. Mofakham ◽  
Goodarz Ahmadi
Keyword(s):  
Random Walk ◽  
Turbulent Flows ◽  
Particle Deposition ◽  
Finite Size ◽  
Particle Dispersion ◽  
Concentration Profiles ◽  
Velocity Fluctuations ◽  
Point Particles ◽  
Deposition Velocities ◽  
Gradient Drift

Abstract The performance of different versions of the discrete random walk models in turbulent flows with nonuniform normal root-mean-square (RMS) velocity fluctuations and turbulence time scales were carefully investigated. The OpenFOAM v2−f low Reynolds number turbulence model was used for evaluating the fully developed streamwise velocity and the wall-normal RMS velocity fluctuations profiles in a turbulent channel flow. The results were then used in an in-house matlab particle tracking code, including the drag and Brownian forces, and the trajectories of randomly injected point-particles with diameters ranging from 10 nm to 30 μm were evaluated under the one-way coupling assumption. The distributions and deposition velocities of fluid-tracer and finite-size particles were evaluated using the conventional-discrete random walk (DRW) model, the modified-DRW model including the velocity gradient drift correction, and the new improved-DRW model including the velocity and time gradient drift terms. It was shown that the conventional-DRW model leads to superfluous migration of fluid-point particles toward the wall and erroneous particle deposition rate. The concentration profiles of tracer particles obtained by using the modified-DRW model still are not uniform. However, it was shown that the new improved-DRW model with the velocity and time scale drift corrections leads to uniform distributions for fluid-point particles and reasonable concentration profiles for finite-size heavy particles. In addition, good agreement was found between the estimated deposition velocities of different size particles by the new improved-DRW model with the available data.

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