Polarisation conjecture for the co-evolutionary opinion model

Let $u\in[0,1]$, let $\vec{v}(t,u)=(f_+(t,u),f_-(t,u))^\top$ be the vector of opinion densities, and let $\pi^g_+,\pi^g_-,\pi^r_+,\pi^r_-\in[0,1]$ and $q\in\mathbb N$ be the model parameters. Consider choices of parameters not covered by Theorem (i). Polarisation conjecture. For every $u\in[0,1]$, the following statements hold. If $\pi^g_+=\pi^r_-$ and $\pi^r_+=\pi^g_-$, then for every $q\in\mathbb N$ there are unique limiting densities $f_+$ and $f_-$ corresponding either to consensus as in --, or to different levels of polarisation depending on $q$, with

$$\int_0^1 \operatorname{d}u\,f_+(\infty,u)=\int_0^1 \operatorname{d}u\,f_-(\infty,u)=\dfrac{1}{2},$$

and the density of edges connecting vertices with different opinions is a non-zero monotone decreasing function of $q$. In the limit $q\to\infty$, only consensus occurs as in --, or strong polarisation is admissible, namely

$$\lim_{q\to\infty}\lim_{t\to\infty}\vec{v}(t,u)=\dfrac{1}{2}\begin{pmatrix}\delta_1(u)\\delta_0(u)\end{pmatrix},$$

and the density of edges connecting vertices with different opinions is zero. Moreover, if $\pi^g_+,\pi^r_-,\pi^r_+,\pi^g_-\in(0,1)$, then the density of edges connecting vertices with the same opinion also tends to zero as $q\to\infty$, but more slowly than the density of disagreeing edges. The conjecture is presented as a proposed completion of the large-time limiting picture in the co-evolutionary setting; the preceding theorems establish only the parameter regimes covered by them, leaving these complementary regimes unresolved.