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We compare two versions of the nonlinear $q$-voter model: the original one,with annealed randomness, and the modified one, with quenched randomness. Inthe original model, each voter changes its opinion with a certain probability$\epsilon$ if the group of influence is not unanimous. In contrast, themodified version introduces two types of voters that act in a deterministic wayin case of disagreement in the influence group: the fraction $\epsilon$ ofvoters always change their current opinion, whereas the rest of them alwaysmaintain it. Although both concepts of randomness lead to the same averagenumber of opinion changes in the system on the microscopic level, they causequalitatively distinct results on the macroscopic level. We focus on themean-field description of these models. Our approach relies on the stabilityanalysis by the linearization technique developed within dynamical systemtheory. This approach allows us to derive complete, exact phase diagrams forboth models. The results obtained in this paper indicate that quenchedrandomness promotes continuous phase transitions to a greater extent, whereasannealed randomness favors discontinuous ones. The quenched model also createscombinations of continuous and discontinuous phase transitions unobserved inthe annealed model, in which the up-down symmetry may be spontaneously brokeninside or outside the hysteresis loop. The analytical results are confirmed byMonte Carlo simulations carried out on a complete graph.

Nonlinear $q$-voter model from the quenched perspective