Google matrix and Ulam networks of intermittency maps

We study the properties of the Google matrix of an Ulam network generated by intermittency maps. This network is created by the Ulam method which gives a matrix approximant for the Perron-Frobenius operator of dynamical map. The spectral properties of eigenvalues and eigenvectors of this matrix are analyzed. We show that the PageRank of the system is characterized by a power law decay with the exponent beta dependent on map parameters and the Google damping factor alpha . Under certain conditions the PageRank is completely delocalized so that the Google search in such a situation becomes inefficient.

[1]  Y. Pomeau,et al.  Intermittent transition to turbulence in dissipative dynamical systems , 1980 .

[2]  Konstantin Avrachenkov,et al.  PageRank of Scale-Free Growing Networks , 2006, Internet Math..

[3]  Eli Upfal,et al.  Using PageRank to Characterize Web Structure , 2002, Internet Math..

[4]  Sergey Brin,et al.  The Anatomy of a Large-Scale Hypertextual Web Search Engine , 1998, Comput. Networks.

[5]  Tien-Yien Li.  Finite approximation for the Frobenius-Perron operator. A solution to Ulam's conjecture , 1976 .

[6]  T. Geisel,et al.  Anomalous diffusion in intermittent chaotic systems , 1984 .

[7]  Jiu Ding,et al.  Finite approximations of Frobenius-Perron operators. A solution of Ulam's conjecture to multi-dimensional transformations , 1996 .

[8]  Konstantin Avrachenkov,et al.  Algorithms and models for the web-graph : 6th international workshop, WAW 2009, Barcelona, Spain, February 12-13, 2009 : proceedings , 2009 .

[9]  A. Lichtenberg,et al.  Regular and Chaotic Dynamics , 1992 .

[10]  Pikovsky.  Statistical properties of dynamically generated anomalous diffusion. , 1991, Physical review. A, Atomic, molecular, and optical physics.

[11]  Dima Shepelyansky,et al.  Delocalization transition for the Google matrix , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[12]  D L Shepelyansky,et al.  Google matrix, dynamical attractors, and Ulam networks. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[13]  T. Morrison,et al.  Dynamical Systems , 2021, Nature.

[14]  Eric A Sobie,et al.  An Introduction to Dynamical Systems , 2011, Science Signaling.

[15]  Mark Holland.  Slowly mixing systems and intermittency maps , 2004, Ergodic Theory and Dynamical Systems.

[16]  Kaufmann,et al.  Eigenvalue spectrum of the Frobenius-Perron operator near intermittency. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[17]  Geisel,et al.  Accelerated diffusion in Josephson junctions and related chaotic systems. , 1985, Physical review letters.

[18]  Rua Murray,et al.  Ulam's method for some non-uniformly expanding maps , 2009 .

[19]  Debora Donato,et al.  Large scale properties of the Webgraph , 2004 .

[20]  Gerhard Keller,et al.  Ruelle?Perron?Frobenius spectrum for Anosov maps , 2002 .

[21]  Instability statistics and mixing rates. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[22]  Werner R. W. Scheinhardt,et al.  In-Degree and PageRank: Why Do They Follow Similar Power Laws? , 2007, Internet Math..

[23]  Anthony Bonato,et al.  Algorithms and Models for the Web-Graph, 5th International Workshop, WAW 2007, San Diego, CA, USA, December 11-12, 2007, Proceedings , 2007, WAW.

[24]  George Osipenko.  Dynamical systems, graphs, and algorithms , 2007 .

[25]  Mw Hirsch,et al.  Chaos In Dynamical Systems , 2016 .

[26]  G. Froyland,et al.  Rigorous numerical approximation of Ruelle–Perron–Frobenius operators and topological pressure of expanding maps , 2008 .

[27]  Gary Froyland,et al.  Extracting Dynamical Behavior via Markov Models , 2001 .

[28]  Efficient computation of topological entropy, pressure, conformal measures, and equilibrium states in one dimension. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[29]  James Hendler,et al.  Google’s PageRank and Beyond: The Science of Search Engine Rankings , 2007 .

[30]  Amy Nicole Langville,et al.  Google's PageRank and beyond - the science of search engine rankings , 2006 .

[31]  Tatsuya Hagino,et al.  Proceedings of the 14th international conference on World Wide Web , 2005 .

[32]  S. Ulam.  A collection of mathematical problems , 1960 .

[33]  C. Caramanis.  What is ergodic theory , 1963 .