Two opinions dynamics generated by inflexibles and non-contrarian and   contrarian floaters
  We assume a community whose members adopt one of two opinions $A$ or $B$. Each member appears as an inflexible, or as a non-contrarian or contrarian floater. An inflexible sticks to its opinion, whereas a floater may change into a floater of the alternative opinion. The occurrence of this change is governed by the local majority rule: members meet in groups of a fixed size, and a floater then changes its opinion provided it is a minority in the group. Subsequently, a non-contrarian floater keeps the opinion as adopted under the local majority rule, whereas a contrarian floater adopts the alternative opinion. Whereas the effects of on the one hand inflexibles and on the other hand non-contrarians and contrarians have previously been studied seperately, the current approach allows us to gain insight in the effect of their combined presence in a community. Given fixed proportions of inflexibles $(\alpha_{A}, \alpha_{B})$ for the two opinions, and fixed fractions of contrarians $(\gamma_{A}, \gamma_{B})$ among the $A$ and $B$ floaters, we derive the update equation $p_{t+1}$ for the overall support for opinion $A$ at time $t+1$, given $p_{t}$. The update equation is derived respectively for local group sizes 1, 2 and 3. The associated dynamics generated by repeated local updates is then determined to identify its asymptotic steady configuration. The full opinion flow diagram is thus obtained, showing conditions in terms of the parameters for each opinion to eventually win the competing dynamics. Various dynamical scenarios are thus exhibited, and it is derived that relatively small densities of inflexibles allow for more variation in the qualitative outcome of the dynamics than higher densities of inflexibles. 
Frans Jacobs and Serge Galam