Linear response theory for Google matrix

We develop the linear response theory for the Google matrix PageRank algorithm with respect to a general weak perturbation and a numerical efficient and accurate algorithm, called LIRGOMAX algorithm, to compute the linear response of the PageRank with respect to this perturbation. We illustrate its efficiency on the example of the English Wikipedia network with more than 5 millions of articles (nodes). For a group of initial nodes (or simply a pair of nodes) this algorithm allows to identify the effective pathway between initial nodes thus selecting a particular subset of nodes which are most sensitive to the weak perturbation applied to them (injection or pumping at one node and absorption of probability at another node). The further application of the reduced Google matrix algorithm (REGOMAX) allows to determine the effective interactions between the nodes of this subset. General linear response theory already found numerous applications in various areas of science including statistical and mesoscopic physics. Based on these grounds we argue that the developed LIRGOMAX algorithm will find broad applications in the analysis of complex directed networks.

[1]  Dima Shepelyansky,et al.  Wikipedia mining of hidden links between political leaders , 2016, The European Physical Journal B.

[2]  Alexei Chepelianskii,et al.  Towards physical laws for software architecture , 2010, ArXiv.

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

[4]  Hui Cao,et al.  Lasing in random media , 2003 .

[5]  K. M. Frahm,et al.  Reduced Google matrix , 2016 .

[6]  Dima Shepelyansky,et al.  Google matrix analysis of directed networks , 2014, ArXiv.

[7]  Andrei Zinovyev,et al.  Inferring hidden causal relations between pathway members using reduced Google matrix of directed biological networks , 2017 .

[8]  Dima Shepelyansky,et al.  Influence of petroleum and gas trade on EU economies from the reduced Google matrix analysis of UN COMTRADE data , 2019, The European Physical Journal B.

[9]  R. Kubo.  Statistical-Mechanical Theory of Irreversible Processes : I. General Theory and Simple Applications to Magnetic and Conduction Problems , 1957 .

[10]  Baranger,et al.  Electrical linear-response theory in an arbitrary magnetic field: A new Fermi-surface formation. , 1989, Physical review. B, Condensed matter.

[11]  D. Ruelle.  A review of linear response theory for general differentiable dynamical systems , 2009, 0901.0484.

[12]  Peter Hänggi,et al.  Stochastic processes: Time evolution, symmetries and linear response , 1982 .

[13]  Kohn,et al.  Current-Dependent Exchange-Correlation Potential for Dynamical Linear Response Theory. , 1996, Physical review letters.

[14]  T. Guhr,et al.  RANDOM-MATRIX THEORIES IN QUANTUM PHYSICS : COMMON CONCEPTS , 1997, cond-mat/9707301.

[15]  Dima Shepelyansky,et al.  World influence and interactions of universities from Wikipedia networks , 2018, The European Physical Journal B.