Conformity in numbers—Does criticality in social responses exist?

Abstract

Within this paper we explore the idea of a critical value representing the proportion of majority members within a group that affects dramatic changes in influence targets’ conformity. We consider the threshold q-voter model when the responses of the Willis-Nail model, a well-established two-dimensional model of social response, are used as a foundation. Specifically, we study a generalized threshold q-voter model when all basic types of social response described by Willis-Nail model are considered, i.e. conformity, anticonformity, independence, and uniformity/congruence. These responses occur in our model with complementary probabilities. We introduce independently two thresholds: one needed for conformity, as well as a second one for anticonformity. In the case of conformity, at least r individuals among q neighbors have to share the same opinion in order to persuade a voter to follow majority’s opinion, whereas in the case of anticonformity, at least w individuals among q neighbors have to share the same opinion in order to influence voters to take an opinion that goes against that of their own reference group. We solve the model on a complete graph and show that the threshold for conformity significantly influences the results. For example, there is a critical threshold for conformity above which the system behaves as in the case of unanimity, i.e. displays continuous and discontinuous phase transitions. On the other hand, the threshold for anticonformity is almost irrelevant. We discuss our results from the perspective of theories of social psychology, as well as the philosophy of agent-based modeling.

Homogeneous Symmetrical Threshold Model with Nonconformity: Independence versus Anticonformity

“…Discontinuous phase transitions do not appear even for highly connected networks. This result is consistent with the result obtained for the generalized threshold q-voter model [21,28], as described above.…”

“…Another possibility to obtain a discontinuous phase transition has been recently suggested within the generalized threshold q-voter model, in which two types of nonconformity were introduced simultaneously [28]. It has been shown that discontinuous phase transition could be obtained even without independence if only the threshold for anticonformity would be smaller than for conformity.…”

“…in lack of unanimity, spin at site i can flip to the opposite direction with probability ǫ. In most of later modifications ǫ = 0 [21,26,28,36,[40][41][42] and here we also refer to this case. 5.…”

“…The q-voter model with independence was studied on the complete graph [21,28,43], as well as on various complex networks [36,37], whereas the q-voter model with anticonformity only on the complete graph. It has been shown, that within Model A only continuous phase transitions are possible for all q ≥ 2, whereas within Model I both types of phase transitions appear: for 2 ≤ q ≤ 5 there is continuous phase transition and for q > 5 transition is discontinuous.…”

“…It has been also shown that in the case of the threshold q-voter model there is a critical threshold r * = r * (q) that decreases with q, above which discontinuous phase transitions are possible [21,28]. In the most general case, when independence and anticonformity are introduced simultaneously, it occurs that r * = r * (q) is monotonically decreasing function of q and r * is always greater than 0.5, even for very large q [28]. Although, the analytical form of r * = r * (q) has not been found, it was predicted that for q → N the critical value r * → 0.5.…”