Opinion formation models with extreme switches and disorder: critical behaviour and dynamics

Abstract: In a three state kinetic exchange opinion formation model, the effect of extreme switches was considered in a paper. In the present work, we study the same model with disorder. Here disorder implies that negative interactions may occur with a probability $p$. In absence of extreme switches, the known critical point is at $p_c =1/4$ in the mean field model. With a nonzero value of $q$ that denotes the probability of such switches, the critical point is found to occur at $ p = \frac{1-q}{4}$ where the order parameter vanishes with a universal value of the exponent $\beta =1/2$. Stability analysis of initially ordered states near the phase boundary reveals the exponential growth/decay of the order parameter in the ordered/disordered phase with a timescale diverging with exponent $1$. The fully ordered state also relaxes exponentially to its equilibrium value with a similar behaviour of the associated timescale. Exactly at the critical points, the order parameter shows a power law decay with time with exponent $1/2$. Although the critical behaviour remains mean field like, the system behaves more like a two state model as $q \to 1$. At $q=1$ the model behaves like a binary voter model with random flipping occurring with probability $p$.