Later-position marginals and sampling complexity of the conditional discovery-order law

Analyze the later-position marginals of the conditional distribution on growth orders of a fixed first-entrance tree and determine the computational cost of sampling from this distribution.

Background

The paper identifies the order weights as the conditional distribution of the discovery order given a fixed first-entrance tree. Although the order-mass identity normalizes these weights, the discussion states that the behavior of positions later in the order and the computational effort required to sample an order have not been analyzed.

The paper also notes that the weighted counting result for Plackett–Luce-type order weights motivates analogous investigations for other matrix choices, but the explicitly unresolved aspects are the later-position marginals and sampling cost for the conditional law established in Theorem 3.2.

References

Several questions remain. First, the conditional law in \cref{thm:orderlaw} gives a distribution on the growth orders of a fixed tree. Its later-position marginals and the cost of sampling from it remain to be analyzed.

— The Order Mass Identity of Rooted Forests, with a Direct Proof of the Extended Aldous--Broder Theorem  (2610.13063 - Zheng, 9 Oct 2026) in Section Discussion

Designing samplers with complete support and stable importance weights, and determining their computational cost on graphs used in clustering and signal processing, remain problems for future work.

— The Order Mass Identity of Rooted Forests, with a Direct Proof of the Extended Aldous--Broder Theorem  (2610.13063 - Zheng, 9 Oct 2026) in Section Discussion