  The issue of opinion sharing and formation has received considerable
attention in the academic literature, and a few models have been proposed to
study this problem. However, existing models are limited to the interactions
among nearest neighbors, ignoring those second, third, and higher-order
neighbors, despite the fact that higher-order interactions occur frequently in
real social networks. In this paper, we develop a new model for opinion
dynamics by incorporating long-range interactions based on higher-order random
walks. We prove that the model converges to a fixed opinion vector, which may
differ greatly from those models without higher-order interactions. Since
direct computation of the equilibrium opinion is computationally expensive,
which involves the operations of huge-scale matrix multiplication and
inversion, we design a theoretically convergence-guaranteed estimation
algorithm that approximates the equilibrium opinion vector nearly linearly in
both space and time with respect to the number of edges in the graph. We
conduct extensive experiments on various social networks, demonstrating that
the new algorithm is both highly efficient and effective.
