Asymptotic convergence to the zero set in nonlinear preferential attachment models

Let $N$ be the number of types, let $m$ be the number of sampled vertices added at each step, and let $p_{\mathbf{u}}^i$ be the probability that a new vertex is assigned type $i$ when the sampled type-count vector is $\mathbf{u}$. Let $X_0^i$ be the initial number of vertices of type $i$, let $\mathbf{a}_n$ be the vector of type proportions at time $n$, and define

$$Z_{\mathbf{P}}:= \left\{ \mathbf{y} \in \Delta^N: \mathbf{P}\left(\mathbf{y}\right)=\mathbf{0} \right\},$$

where $\mathbf{P}$ is the vector field governing the nonlinear model on the probability simplex $\Delta^N$. Asymptotic zero-set conjecture. Assume that there exist $\mathbf{u}$ and $i \in \left[ N \right]$ such that $p_{\mathbf{u}}^i \neq \frac{u^i}{m}$, and that $X_0^i > 0$ for all $i \in \left[ N \right]$. Then $\mathbf{a}_n$ converges almost surely and the limit is a point in the zero set $Z_{\mathbf{P}}$. The complete theoretical analysis of nonlinear models with multiple types is out of reach, but partial results suggest that the asymptotic behavior should parallel the two-type case; the conjecture predicts convergence to an equilibrium of the associated vector field.