  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.
