We consider a coupled system mimicking opinion formation under the influence of a group of $q$ neighbors ($q$-lobby) that consists of an Ising part governed by temperature-like parameter $T$ and a voter dynamics parametrized by noise probability $p$ (independence of choice). Using rigorous analytical calculations backed by extensive Monte Carlo simulations, we examine the interplay between these two quantities. Based on the theory of phase transitions, we derive the relation between $T$ and $p$ at the critical line dividing the ordered and disordered phases, which takes a very simple and generic form $T(p-a)=b$ in the high temperature limit. For specific lobby sizes, we show where the temperature and noise are balanced, and we hint that for large $q$, the temperature-like dynamics prevails.

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• Phase transitions
• Multilayer & multiplex networks
• Social dynamics
• Mean field theory
• Monte Carlo methods