3,552 research outputs found
In-Degree and PageRank of Web pages: Why do they follow similar power laws?
The PageRank is a popularity measure designed by Google to rank Web pages.
Experiments confirm that the PageRank obeys a `power law' with the same
exponent as the In-Degree. This paper presents a novel mathematical model that
explains this phenomenon. The relation between the PageRank and In-Degree is
modelled through a stochastic equation, which is inspired by the original
definition of the PageRank, and is analogous to the well-known distributional
identity for the busy period in the M/G/1 queue. Further, we employ the theory
of regular variation and Tauberian theorems to analytically prove that the tail
behavior of the PageRank and the In-Degree differ only by a multiplicative
factor, for which we derive a closed-form expression. Our analytical results
are in good agreement with experimental data.Comment: 20 pages, 3 figures; typos added; reference adde
In-Degree and PageRank of web pages: why do they follow similar power laws?
PageRank is a popularity measure designed by Google to rank Web pages. Experiments confirm that PageRank values obey a power law with the same exponent as In-Degree values. This paper presents a novel mathematical model that explains this phenomenon. The relation between PageRank and In-Degree is modelled through a stochastic equation, which is inspired by the original definition of PageRank, and is analogous to the well-known distributional identity for the busy period in the queue. Further, we employ the theory of regular variation and Tauberian theorems to analytically prove that the tail distributions of PageRank and In-Degree differ only by a multiple factor, for which we derive a closed-form expression. Our analytical results are in good agreement with experimental data
Investigating the Impact of the Blogsphere: Using PageRank to Determine the Distribution of Attention
Much has been written in recent years about the blogosphere and its impact on political, educational and scientific debates. Lately the issue has received significant attention from the industry. As the blogosphere continues to grow, even doubling its size every six months, this paper investigates its apparent impact on the overall Web itself. We use the popular Google PageRank algorithm which employs a model of Web used to measure the distribution of user attention across sites in the blogosphere. The paper is based on an analysis of the PageRank distribution for 8.8 million blogs in 2005 and 2006. This paper addresses the following key questions: How is PageRank distributed across the blogosphere? Does it indicate the existence of measurable, visible effects of blogs on the overall mediasphere? Can we compare the distribution of attention to blogs as characterised by the PageRank with the situation for other forms of Web content? Has there been a growth in the impact of the blogosphere on the Web over the two years analysed here? Finally, it will also be necessary to examine the limitations of a PageRank-centred approach
PageRank model of opinion formation on Ulam networks
We consider a PageRank model of opinion formation on Ulam networks, generated
by the intermittency map and the typical Chirikov map. The Ulam networks
generated by these maps have certain similarities with such scale-free networks
as the World Wide Web (WWW), showing an algebraic decay of the PageRank
probability. We find that the opinion formation process on Ulam networks have
certain similarities but also distinct features comparing to the WWW. We
attribute these distinctions to internal differences in network structure of
the Ulam and WWW networks. We also analyze the process of opinion formation in
the frame of generalized Sznajd model which protects opinion of small
communities.Comment: 7 pages, 6 figures. Updated version for publicatio
Google matrix analysis of directed networks
In past ten years, modern societies developed enormous communication and
social networks. Their classification and information retrieval processing
become a formidable task for the society. Due to the rapid growth of World Wide
Web, social and communication networks, new mathematical methods have been
invented to characterize the properties of these networks on a more detailed
and precise level. Various search engines are essentially using such methods.
It is highly important to develop new tools to classify and rank enormous
amount of network information in a way adapted to internal network structures
and characteristics. This review describes the Google matrix analysis of
directed complex networks demonstrating its efficiency on various examples
including World Wide Web, Wikipedia, software architecture, world trade, social
and citation networks, brain neural networks, DNA sequences and Ulam networks.
The analytical and numerical matrix methods used in this analysis originate
from the fields of Markov chains, quantum chaos and Random Matrix theory.Comment: 56 pages, 58 figures. Missed link added in network example of Fig3
A framework for evaluating statistical dependencies and rank correlations in power law graphs
We analyze dependencies in power law graph data (Web sample, Wikipedia sample and a preferential attachment graph) using statistical inference for multivariate regular variation. To the best of our knowledge, this is the first attempt to apply the well developed theory of regular variation to graph data. The new insights this yields are striking: the three above-mentioned data sets are shown to have a totally different dependence structure between different graph parameters, such as in-degree and PageRank. Based on the proposed methodology, we suggest a new measure for rank correlations. Unlike most known methods, this measure is especially sensitive to rank permutations for topranked nodes. Using this method, we demonstrate that the PageRank ranking is not sensitive to moderate changes in the damping factor
PageRank in scale-free random graphs
We analyze the distribution of PageRank on a directed configuration model and
show that as the size of the graph grows to infinity it can be closely
approximated by the PageRank of the root node of an appropriately constructed
tree. This tree approximation is in turn related to the solution of a linear
stochastic fixed point equation that has been thoroughly studied in the recent
literature
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