  We propose a new Ising-PageRank model of opinion formation on a social
network by introducing an Ising- or spin-like structure of the corresponding
Google matrix. Each elector or node of the network has two components
corresponding to a red or blue opinion in the society. Also each elector
propagates either the red or the blue opinion on the network so that the links
between electors are described by two by two matrices favoring one or the other
of the two opinions. An elector votes for red or blue depending on the
dominance of its red or blue PageRank vector components. We determine the
dependence of the final society vote on the fraction of nodes with red (or
blue) influence allowing to determine the transition for the election outcome
border between the red or blue option. We show that this transition border is
significantly affected by the opinion of society elite electors composed of the
top PageRank, CheiRank or 2DRank nodes of the network even if the elite
fraction is very small. The analytical and numerical studies are preformed for
the networks of English Wikipedia 2017 and Oxford University 2006.
