Joint finite-sample degree regime

Determine a joint asymptotic regime with degree threshold k=k(n) that quantifies the degree window in which the limiting directed and undirected degree formulas for range-renewal graphs approximate finite graphs, and explains the increasing far-tail fluctuations observed in the simulations.

Background

The paper establishes asymptotics by first letting the sequence length n tend to infinity for each fixed degree k and then analyzing the limiting degree laws as k tends to infinity. It explicitly makes no claim for a simultaneous limit in which k depends on n.

A joint regime k=k(n) would connect the limiting formulas to observable finite graphs and would characterize the range of degree thresholds for which the asymptotic exponent-two laws remain accurate, particularly where simulations show increasing fluctuations in the far tail.

References

Two natural extensions remain open. First, a joint regime $k=k(n)$ would quantify the degree window in which the limiting formula approximates a finite graph and would explain the increasing fluctuations in the far tail of Fig.~\ref{fig:tail}.

— Universal Exponent-Two Degree Laws in Range-Renewal Networks  (2609.05290 - Xie et al., 4 Sep 2026) in Section 6, Discussion and outlook