  Dynamic social opinion models have been widely studied on undirected
networks, and most of them are based on spin interaction models that produce a
consensus. In reality, however, many networks such as Twitter and the World
Wide Web are directed and are composed of both unidirectional and bidirectional
links. Moreover, from choosing a coffee brand to deciding who to vote for in an
election, two or more competing opinions often coexist. In response to this
ubiquity of directed networks and the coexistence of two or more opinions in
decision-making situations, we study a non-consensus opinion model introduced
by Shao et al. \cite{shao2009dynamic} on directed networks. We define
directionality $\xi$ as the percentage of unidirectional links in a network,
and we use the linear correlation coefficient $\rho$ between the indegree and
outdegree of a node to quantify the relation between the indegree and
outdegree. We introduce two degree-preserving rewiring approaches which allow
us to construct directed networks that can have a broad range of possible
combinations of directionality $\xi$ and linear correlation coefficient $\rho$
and to study how $\xi$ and $\rho$ impact opinion competitions. We find that, as
the directionality $\xi$ or the indegree and outdegree correlation $\rho$
increases, the majority opinion becomes more dominant and the minority
opinion's ability to survive is lowered.
