  We study a two states opinion formation model driven by PageRank node
influence and report an extensive numerical study on how PageRank affects
collective opinion formations in large-scale empirical directed networks. In
our model the opinion of a node can be updated by the sum of its neighbor
nodes' opinions weighted by the node influence of the neighbor nodes at each
step. We consider PageRank probability and its sublinear power as node
influence measures and investigate evolution of opinion under various
conditions. First, we observe that all networks reach steady state opinion
after a certain relaxation time. This time scale is decreasing with the
heterogeneity of node influence in the networks. Second, we find that our model
shows consensus and non-consensus behavior in steady state depending on types
of networks: Web graph, citation network of physics articles, and LiveJournal
social network show non-consensus behavior while Wikipedia article network
shows consensus behavior. Third, we find that a more heterogeneous influence
distribution leads to a more uniform opinion state in the cases of Web graph,
Wikipedia, and Livejournal. However, the opposite behavior is observed in the
citation network. Finally we identify that a small number of influential nodes
can impose their own opinion on significant fraction of other nodes in all
considered networks. Our study shows that the effects of heterogeneity of node
influence on opinion formation can be significant and suggests further
investigations on the interplay between node influence and collective opinion
in networks.
