  Opinion dynamics on social networks have been received considerable
attentions in recent years. Nevertheless, just a few works have theoretically
analyzed the condition in which a certain opinion can spread in the whole
structured population. In this paper, we propose an evolutionary game approach
for a binary opinion model to explore the conditions for an opinion's
spreading. Inspired by real-life observations, we assume that an agent's choice
to select an opinion is not random, but is based on a score rooted both from
public knowledge and the interactions with neighbors. By means of coalescing
random walks, we obtain a condition in which opinion $A$ can be favored to
spread on social networks in the weak selection limit. We find that the
successfully spreading condition of opinion $A$ is closely related to the basic
scores of binary opinions, the feedback scores on opinion interactions, and the
structural parameters including the edge weights, the weighted degrees of
vertices, and the average degree of the network. In particular, when
individuals adjust their opinions based solely on the public information, the
vitality of opinion $A$ depends exclusively on the difference of basic scores
of $A$ and $B$. When there are no negative (positive) feedback interactions
between connected individuals, we find that the success of opinion $A$ depends
on the ratio of the obtained positive (negative) feedback scores of competing
opinions. To complete our study, we perform computer simulations on
fully-connected, small-world, and scale-free networks, respectively, which
support and confirm our theoretical findings.
