The NFM separated-eigenvector conjecture under Dirichlet noise

Let $G=(V,W)$ be generated using the node-feature model (NFM), with simplex distribution given by a Dirichlet distribution with constant parameter $\bm{\alpha}$. For $j\in[k]$, define

$$V_j':= \{i\in[n]:\theta^i_j\geq t(k,\bm{\alpha})\},\qquad \Theta_j:=\Theta(V_j',:).$$

NFM separated-eigenvector conjecture. There exists a scalar $t(k,\bm{\alpha})\in(0.5,1/\sqrt{2})$ such that, for every $j\in[k]$, with probability not converging to $0$ as $n\to\infty$, the largest eigenvalue of

$$4.4\Theta_j\Theta_j^T-2.2E$$

is well separated from the remaining eigenvalues and its corresponding eigenvector is positive.

The conjecture formalizes the computational observation that strong-node subgraphs have a dominant positive spectral direction, which could support robust identification of cluster structure under asymmetric noise.