  We investigate opinion dynamics in multi-agent networks when a bias toward
one of two possible opinions exists; for example, reflecting a status quo vs a
superior alternative. Starting with all agents sharing an initial opinion
representing the status quo, the system evolves in steps. In each step, one
agent selected uniformly at random adopts the superior opinion with some
probability $\alpha$, and with probability $1 - \alpha$ it follows an
underlying update rule to revise its opinion on the basis of those held by its
neighbors. We analyze convergence of the resulting process under two well-known
update rules, namely majority and voter. The framework we propose exhibits a
rich structure, with a non-obvious interplay between topology and underlying
update rule. For example, for the voter rule we show that the speed of
convergence bears no significant dependence on the underlying topology, whereas
the picture changes completely under the majority rule, where network density
negatively affects convergence. We believe that the model we propose is at the
same time simple, rich, and modular, affording mathematical characterization of
the interplay between bias, underlying opinion dynamics, and social structure
in a unified setting.
