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Non-conservative kinetic exchange model of opinion dynamics with randomness and bounded confidence

The concept of a bounded confidence level is incorporated in a nonconservative kinetic exchange model of opinion dynamics model where opinions have continuous values $\in [-1,1]$. The characteristics of the unrestricted model, which has one parameter $λ$ representing conviction, undergo drastic changes with the introduction of bounded confidence parametrised by $δ$. Three distinct regions are identified in the phase diagram in the $δ-λ$ plane and the evidences of a first order phase transition for $δ\geq 0.3$ are presented. A neutral state with all opinions equal to zero occurs for $λ\leq λ_{c_1} \simeq 2/3$, independent of $δ$, while for $λ_{c_1} \leq λ\leq λ_{c_2}(δ)$, an ordered region is seen to exist where opinions of only one sign prevail. At $λ_{c_2}(δ)$, a transition to a disordered state is observed, where individual opinions of both signs coexist and move closer to the extreme values ($\pm 1$) as $λ$ is increased. For confidence level $δ< 0.3$, the ordered phase exists for a narrow range of $λ$ only. The line $δ= 0$ is apparently a line of discontinuity and this limit is discussed in some detail.