Competing effects of social balance and influence

The theory of social balance is one of the key drivers of social dynamics. We study a model of social interactions in which the dynamics of social balance is competing with external influence. In this model, each node in a social network is in one of the three possible states - leftist, rightist, centrist. Only a link between two unequal extremist nodes is considered unfriendly and a triangle is balanced if it contains even number of unfriendly links. Thus triangles formed by a centrist, a leftist and a rightist are unbalanced. In this model, at each time step with probability $p$, we pick a random node and convert it into a centrist while with probability $(1-p)$, a randomly picked triangle is checked for balance and if needed, it is balanced by updating the state of one of the nodes in the triangle. We find that there exists a critical value $p_c$ such that for $p