Google matrix, dynamical attractors, and Ulam networks.

@article{Shepelyansky2010GoogleMD,
  title={Google matrix, dynamical attractors, and Ulam networks.},
  author={Dima L. Shepelyansky and O. V. Zhirov},
  journal={Physical review. E, Statistical, nonlinear, and soft matter physics},
  year={2010},
  volume={81 3 Pt 2},
  pages={
          036213
        }
}
  • D. Shepelyansky, O. Zhirov
  • Published 26 May 2009
  • Mathematics
  • Physical review. E, Statistical, nonlinear, and soft matter physics
We study the properties of the Google matrix generated by a coarse-grained Perron-Frobenius operator of the Chirikov typical map with dissipation. The finite-size matrix approximant of this operator is constructed by the Ulam method. This method applied to the simple dynamical model generates directed Ulam networks with approximate scale-free scaling and characteristics being in certain features similar to those of the world wide web with approximate scale-free degree distributions as well as… 
Google matrix and Ulam networks of intermittency maps
  • L. Ermann, D. Shepelyansky
  • Mathematics, Computer Science
    Physical review. E, Statistical, nonlinear, and soft matter physics
  • 2010
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References

SHOWING 1-10 OF 94 REFERENCES
Delocalization transition for the Google matrix
TLDR
It is argued that, for networks with delocalized PageRank, the efficiency of information retrieval by Google-type search is strongly affected since the PageRank values have no clear hierarchical structure in this case.
PageRank of Scale-Free Growing Networks
TLDR
An analytical expression is found for the expected PageRank value in a scale-free growing network model as a function of the age of the growing network and theAge of a particular node and it is shown that PageRank follows closely a power law in the middle range of its values.
Using PageRank to Characterize Web Structure
TLDR
It is suggested that PageRank values on the web follow a power law, and generative models for the web graph are developed that explain this observation and moreover remain faithful to previously studied degree distributions.
Algorithms and Models for the Web-Graph, 5th International Workshop, WAW 2007, San Diego, CA, USA, December 11-12, 2007, Proceedings
TLDR
Bias Reduction in Traceroute Sampling - Towards a More Accurate Map of the Internet and Giant Component and Connectivity in Geographical Threshold Graphs.
Jordan Canonical Form of the Google Matrix: A Potential Contribution to the PageRank Computation
We consider the web hyperlink matrix used by Google for computing the PageRank whose form is given by A(c)=[cP +(1-c)E]T, where P is a row stochastic matrix, E is a row stochastic rank one matrix,
Distribution of PageRank Mass Among Principle Components of the Web
TLDR
A detailed study of the OUT component reveals the presence of "dead-ends" (small groups of pages linking only to each other) that receive an unfairly high ranking when c is close to one, and argues that this problem can be mitigated by choosing c as small as 1/2.
Algorithms and models for the web-graph : 6th international workshop, WAW 2009, Barcelona, Spain, February 12-13, 2009 : proceedings
TLDR
This work discusses Graph Models for Complex Networks, Characterization of Tail Dependence for In-Degree and PageRank, and Exploiting Positive and Negative Graded Relevance Assessments for Content Recommendation.
Diffusion and localization for the Chirikov typical map.
TLDR
The quantum dynamics is characterized by the effect of Chirikov localization (or dynamical localization), and it is found that the localization length depends in a subtle way on the two classical diffusion constants in the two time-scale regime.
Ruelle?Perron?Frobenius spectrum for Anosov maps
We extend a number of results from one-dimensional dynamics based on spectral properties of the Ruelle–Perron–Frobenius transfer operator to Anosov diffeomorphisms on compact manifolds. This allows
Simulating the Webgraph: a comparative analysis of models
TLDR
This work simulated several of these models and compared them against a 300-million-node sample of the Webgraph provided by the Stanford WebBase project, finding that the more random the model, the better the graph.
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