The NFM positive-semidefinite subgraph 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

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

NFM positive-semidefinite subgraph conjecture. There exists a scalar $t(k,\bm{\alpha})\in(0.5,1/\sqrt{2})$ such that, for every $j\in[k]$, $G[V_j']$ satisfies the hypothesis of Theorem 2 with probability not converging to $0$ as $n\to\infty$.

Here $G[V_j']$ is the subgraph induced by the nodes whose $j$th feature is at least the threshold. The conjecture is motivated by experiments suggesting that the relevant Laplacians remain positive semidefinite with nonvanishing probability, potentially enabling recovery on suitably selected strong-node subgraphs.