Pair approximation of the biased-independence q-voter model for innovation diffusion in organizational networks

Collective adaptation, including innovation adoption, pro-environmental change, and organizational change, emerges from the interplay between individual decisions and social influence. We study the biased-independence q-voter model, in which agents choose between adoption and non-adoption under conformity and independent choice. Independent choice is governed by an engagement parameter inspired by earlier models of eco-innovation diffusion. For engagement equal to 0.5, the model reduces to the standard q-voter model with independence; otherwise, the symmetry between the two options is broken. This asymmetry generates discontinuous phase transitions and irreversible hysteresis, reflecting path-dependent adoption dynamics. We first review variants of the asymmetric q-voter model and then analyze the model using mean-field approximation, pair approximation, and Monte Carlo simulations on artificial and empirical organizational networks. We derive, for the first time, a pair approximation for an asymmetric q-voter model and test how well it reproduces simulations on empirical networks. Pair approximation captures the dependence on network density and systematically improves upon mean-field approximation. However, while the improvement is substantial for random graphs, it is moderate for empirical networks. These results demonstrate the need to test analytical approximations directly in realistic settings, as such tests may show that, in some cases, a simple mean-field approach is sufficient.