Linear response theory for Google matrix

Abstract: 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.