  A common trait involving the opinion dynamics in social networks is an anchor
on interacting network to characterize the opinion formation process among
participating social actors, such as information flow, cooperative and
antagonistic influence, etc. Nevertheless, interacting networks are generally
public for social groups, as well as other individuals who may be interested
in. This blocks a more precise interpretation of the opinion formation process
since social actors always have complex feeling, motivation and behavior, even
beliefs that are personally private. In this paper, we formulate a general
configuration on describing how individual's opinion evolves in a distinct
fashion. It consists of two functional networks: interacting network and
appraisal network. Interacting network inherits the operational properties as
DeGroot iterative opinion pooling scheme while appraisal network, forming a
belief system, quantifies certain cognitive orientation to interested
individuals' beliefs, over which the adhered attitudes may have the potential
to be antagonistic. We explicitly show that cooperative appraisal network
always leads to consensus in opinions. Antagonistic appraisal network, however,
causes opinion cluster. It is verified that antagonistic appraisal network
affords to guarantee consensus by imposing some extra restrictions. They hence
bridge a gap between the consensus and the clusters in opinion dynamics. We
further attain a gauge on the appraisal network by means of the random convex
optimization approach. Moreover, we extend our results to the case of mutually
interdependent issues.
