q-neighbor Ising model on a polarized network

In this paper, we examine the interplay between the lobby size $q$ in the $q$-neighbor Ising model of opinion formation (Phys. Rev. E 92, 052105) and the level of overlap $v$ of two fully connected graphs. Results suggest that for each lobby size $q \ge 3$, a specific level of overlap $v^*$ exists, which destroys initially polarized clusters of opinions. By performing Monte-Carlo simulations, backed by an analytical approach, we show that the dependence of the $v^*$ on the lobby size $q$ is far from trivial in the absence of temperature, showing consecutive maximum and minimum, that additionally depends on the parity of $q$. The temperature is, in general, a destructive factor; its increase leads to the collapse of polarized clusters for smaller values of $v$ and additionally brings a substantial decrease in the level of polarization. However, we show that this behavior is counter-intuitively inverted for specific lobby sizes and temperature ranges.