Tightness of upper bounds or existence of slower undirected graphs for db updating

Determine whether the general upper bound O(N^5 log N) on the expected absorption time for neutral death-birth updating on undirected graphs is tight, or construct undirected graph families whose expected absorption time asymptotically exceeds the Ω(N^4) lower bound achieved by the barbell graph family.

Background

Under db updating, the authors show a lower bound Ω(N4) via the barbell graph and a general upper bound of O(N5 log N) for any undirected graph. As with bd updating, there is a gap between known extremal constructions and the general upper bound.

Resolving whether the barbell is near-optimal or whether the theoretical upper bound can be tightened—or if undirected graphs exist with even slower expected absorption times—would settle the open question about maximal diversity maintenance under db updating.

References

For bd updating, there is a gap between the slowest undirected family of graphs we know (double star) and our theoretical upper bound for any undirected graph. Similarly, for db updating there is a gap between the slowest undirected family of graphs we know (barbell) and our theoretical upper bound for any undirected graph. In both cases, whether our analysis is not tight enough or there are even slower graph families that we have not found is unknown.

— Maintaining diversity in structured populations (2503.09841 - Brewster et al., 12 Mar 2025) in Table 1 caption, Results