  Trust and distrust are common in the opinion interactions among agents in
social networks, and they are described by the edges with positive and negative
weights in the signed digraph, respectively. It has been shown in social
psychology that although the opinions of most agents (followers) tend to
prevail, sometimes one agent (leader) with a firm stand and strong influence
can impact or even overthrow the preferences of followers. This paper aims to
analyze how the leader influences the formation of followers' opinions in
signed social networks. In addition, this paper considers an asynchronous
evolution mechanism of trust/distrust level based on opinion difference, in
which the trust/distrust level between neighboring agents is portrayed as a
nonlinear weight function of their opinion difference, and each agent interacts
with the neighbors to update the trust/distrust level and opinion at the times
determined by its own will. Based on the related properties of sub-stochastic
and super-stochastic matrices, the inequality conditions about positive and
negative weights to achieve opinion consensus and polarization are established.
Some numerical simulations based on two well-known networks called the ``12
Angry Men" network and the Karate Club network are provided to verify the
correctness of the theoretical results.
