The NFM strong-node containment conjecture for ℓ₂-norm-diag

Let $G$ be generated according to the node-feature model (NFM), with simplex distribution given by a Dirichlet distribution with constant parameter. Let $V_1^{\mathrm{strong}},\dots,V_k^{\mathrm{strong}}$ denote the strong-node sets, and let $\ell_2$-norm-diag be the stated recovery procedure.

NFM strong-node containment conjecture. With probability not converging to $0$ as $n\to\infty$, $\ell_2$-norm-diag returns exactly $k$ disjoint clusters $V_1',\dots,V_k'$ such that, for every $j\in[k]$,

$$V_j^{\mathrm{strong}}\subseteq V_j'.$$

The conjecture is motivated by theoretical robustness bounds and experiments showing that the method can recover disjoint clusters containing all strong nodes, possibly together with fringe nodes. It remains open whether this behavior persists with nonvanishing probability asymptotically.