Ising-PageRank model of opinion formation on social networks

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.