RANDOM WALKS ON DIRECTED NETWORKS: THE CASE OF PAGERANK

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.

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Step-by-Step Random Walk Network with Power-Law Clique-Degree Distribution

Chinese Physics Letters ◽

10.1088/0256-307x/25/7/106 ◽

2008 ◽

Vol 25 (7) ◽

pp. 2718-2720 ◽

Cited By ~ 11

Author(s):

Yang Han-Xin ◽

Wang Bing-Hong ◽

Liu Jian-Guo ◽

Han Xiao-Pu ◽

Zhou Tao

Keyword(s):

Random Walk ◽

Power Law ◽

Degree Distribution

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The structure of co-publications multilayer network

Computational Social Networks ◽

10.1186/s40649-021-00089-w ◽

2021 ◽

Vol 8 (1) ◽

Author(s):

Ghislain Romaric Meleu ◽

Paulin Yonta Melatagia

Keyword(s):

Power Law ◽

Degree Distribution ◽

Preferential Attachment ◽

Small World ◽

Generation Model ◽

Small World Network ◽

Power Law Distribution ◽

Multilayer Networks ◽

Multilayer Network ◽

Scientific Papers

AbstractUsing the headers of scientific papers, we have built multilayer networks of entities involved in research namely: authors, laboratories, and institutions. We have analyzed some properties of such networks built from data extracted from the HAL archives and found that the network at each layer is a small-world network with power law distribution. In order to simulate such co-publication network, we propose a multilayer network generation model based on the formation of cliques at each layer and the affiliation of each new node to the higher layers. The clique is built from new and existing nodes selected using preferential attachment. We also show that, the degree distribution of generated layers follows a power law. From the simulations of our model, we show that the generated multilayer networks reproduce the studied properties of co-publication networks.

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Zero forcing number of graphs with a power law degree distribution

Physical Review E ◽

10.1103/physreve.103.022301 ◽

2021 ◽

Vol 103 (2) ◽

Author(s):

Alexei Vazquez

Keyword(s):

Power Law ◽

Degree Distribution ◽

Zero Forcing ◽

Zero Forcing Number ◽

Forcing Number ◽

Law Degree

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A Simulation of the Structure of the World-Wide Web

Sociological Research Online ◽

10.5153/sro.684 ◽

2002 ◽

Vol 7 (1) ◽

pp. 9-25 ◽

Cited By ~ 2

Author(s):

Moses Boudourides ◽

Gerasimos Antypas

Keyword(s):

World Wide Web ◽

Power Law ◽

Web Sites ◽

World Wide ◽

The Internet ◽

Web Pages ◽

Small Worlds ◽

Web Page ◽

Simple Simulation ◽

The World

In this paper we are presenting a simple simulation of the Internet World-Wide Web, where one observes the appearance of web pages belonging to different web sites, covering a number of different thematic topics and possessing links to other web pages. The goal of our simulation is to reproduce the form of the observed World-Wide Web and of its growth, using a small number of simple assumptions. In our simulation, existing web pages may generate new ones as follows: First, each web page is equipped with a topic concerning its contents. Second, links between web pages are established according to common topics. Next, new web pages may be randomly generated and subsequently they might be equipped with a topic and be assigned to web sites. By repeated iterations of these rules, our simulation appears to exhibit the observed structure of the World-Wide Web and, in particular, a power law type of growth. In order to visualise the network of web pages, we have followed N. Gilbert's (1997) methodology of scientometric simulation, assuming that web pages can be represented by points in the plane. Furthermore, the simulated graph is found to possess the property of small worlds, as it is the case with a large number of other complex networks.

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Diamond aggregation

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s030500411000006x ◽

2010 ◽

Vol 149 (2) ◽

pp. 351-372

Author(s):

WOUTER KAGER ◽

LIONEL LEVINE

Keyword(s):

Random Walk ◽

Growth Model ◽

Power Law ◽

Simple Random Walk ◽

Internal Diffusion ◽

Diffusion Limited Aggregation ◽

Directional Bias ◽

Model Based ◽

Diffusion Limited ◽

Logarithmic Fluctuations

AbstractInternal diffusion-limited aggregation is a growth model based on random walk in ℤd. We study how the shape of the aggregate depends on the law of the underlying walk, focusing on a family of walks in ℤ2 for which the limiting shape is a diamond. Certain of these walks—those with a directional bias toward the origin—have at most logarithmic fluctuations around the limiting shape. This contrasts with the simple random walk, where the limiting shape is a disk and the best known bound on the fluctuations, due to Lawler, is a power law. Our walks enjoy a uniform layering property which simplifies many of the proofs.

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Speed-detachment tradeoff and its effect on track bound transport of single motor protein

10.1101/482604 ◽

2018 ◽

Author(s):

Mohd Suhail Rizvi

Keyword(s):

Experimental Data ◽

Random Walk ◽

Power Law ◽

Average Distance ◽

Motor Protein ◽

Fixed Duration ◽

Biological Cells ◽

Single Motor ◽

Stochastic Nature ◽

Time Required

AbstractThe transportation of the cargoes in biological cells is primarily driven by the motor proteins on filamentous protein tracks. The stochastic nature of the motion of motor protein often leads to its spontaneous detachment from the track. Using the available experimental data, we demonstrate a tradeoff between the speed of the motor and its rate of spontaneous detachment from the track. Further, it is also shown that this speed-detachment relation follows a power law where its exponent dictates the nature of the motor protein processivity. We utilize this information to study the motion of motor protein on track using a random-walk model. We obtain the average distance travelled in fixed duration and average time required for covering a given distance by the motor protein. These analyses reveal non-monotonic dependence of the motor protein speed on its transport and, therefore, optimal motor speeds can be identified for the time and distance controlled conditions.

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Models And Mining Of On-Line Social Networks

10.32920/ryerson.14655462 ◽

2021 ◽

Author(s):

Yanhua Tian

Keyword(s):

Social Networks ◽

Power Law ◽

Degree Distribution ◽

Small World ◽

Spectral Expansion ◽

On Line ◽

Simulation Results ◽

Small World Property ◽

P Type ◽

Law Degree

Power law degree distribution, the small world property, and bad spectral expansion are three of the most important properties of On-line Social Networks (OSNs). We sampled YouTube and Wikipedia to investigate OSNs. Our simulation and computational results support the conclusion that OSNs follow a power law degree distribution, have the small world property, and bad spectral expansion. We calculated the diameters and spectral gaps of OSNs samples, and compared these to graphs generated by the GEO-P model. Our simulation results support the Logarithmic Dimension Hypothesis, which conjectures that the dimension of OSNs is m = [log N]. We introduced six GEO-P type models. We ran simulations of these GEO-P-type models, and compared the simulated graphs with real OSN data. Our simulation results suggest that, except for the GEO-P (GnpDeg) model, all our models generate graphs with power law degree distributions, the small world property, and bad spectral expansion.

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