  As an important factor governing opinion dynamics, stubbornness strongly
affects various aspects of opinion formation. However, a systematically
theoretical study about the influences of heterogeneous stubbornness on opinion
dynamics is still lacking. In this paper, we study a popular opinion model in
the presence of inhomogeneous stubbornness. We show analytically that
heterogeneous stubbornness has a great impact on convergence time, expressed
opinion of every node, and the overall expressed opinion. We provide an
explanation of the expressed opinion in terms of stubbornness-dependent
spanning diverging forests. We propose quantitative indicators to quantify some
social concepts, including conflict, disagreement, and polarization by
incorporating heterogeneous stubbornness, and develop a nearly linear time
algorithm to approximate these quantities, which has a proved theoretical
guarantee for the error of each quantity. To demonstrate the performance of our
algorithm, we perform extensive experiments on a large set of real networks,
which indicate that our algorithm is both efficient and effective, scalable to
large networks with millions of nodes.
