Scaling of non-dominant subtrees in explosive attachment models

Determine the scaling required for the number of large subtrees in explosive superlinear preferential attachment models whose attachment functions satisfy \sum_j1/f(j)<\infty but f(n)=o(n^\alpha) for every \alpha>1, and establish whether their maximum degree still satisfies \Delta(T_n)=n-o(n).

Background

For attachment functions of order n\alpha with \alpha>1, the paper proves precise sublinear fluctuation bounds for the maximum degree and consequently for the largest common subtree. The authors note that other explosive models can have slower-than-polynomial attachment functions, for which the relevant subtree-count proposition fails and the number of sufficiently large descendant subtrees diverges.

The remark explicitly identifies both the appropriate scaling and the persistence of near-total condensation as unresolved.

References

It would be interesting to understand what kind of scaling would be required, and if the result $\Delta(T_n)=n-o(n)$, as in Theorem~\ref{thrm:supermax}, still~holds.

— On the largest common subtree of uniform attachment trees  (2609.30098 - Bäumler et al., 24 Sep 2026) in Remark following Proposition 1.21 in Section 6.2, “Superlinear preferential attachment”