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 alte
rnative 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.
We propose a novel approach framed in terms of information theory and entropy to tackle the issue of conspiracy theories propagation. We start with the report of an event (such as 9/11 terroristic attack) represented as a series of individual strings
of information denoted respectively by two-state variable Ei=+/-1, i=1,..., N. Assigning Ei value to all strings, the initial order parameter and entropy are determined. Conspiracy theorists comment on the report, focusing repeatedly on several strings Ek and changing their meaning (from -1 to +1). The reading of the event is turned fuzzy with an increased entropy value. Beyond some threshold value of entropy, chosen by simplicity to its maximum value, meaning N/2 variables with Ei=1, doubt prevails in the reading of the event and the chance is created that an alternative theory might prevail. Therefore, the evolution of the associated entropy is a way to measure the degree of penetration of a conspiracy theory. Our general framework relies on online content made voluntarily available by crowds of people, in response to some news or blog articles published by official news agencies. We apply different aggregation levels (comment, person, discussion thread) and discuss the associated patterns of entropy change.
