Primordial-degree distribution for RRSC(m) with m≥2

Determine the probability distribution P(δ_1,N) of the degree of the primordial vertex for every RRSC(m) model with m≥2, including its finite-size and asymptotic forms.

Background

For RRSC(2), the evolution of the primordial degree depends on the jointly evolving edge count S_1, making the process non-Markovian in the degree variable alone. This prevents the direct solution available for the recursive random tree.

The paper derives only the average growth exponent and a conjectural scaling description, while explicitly leaving the full degree distribution undetermined for all m≥2.

References

Although we do not know the probability distribution $P(\delta_1,N)$ for the RRSC(m) models with $m\geq 2$, it is possible to establish the growth law of the average degree $\delta_1(N)=\langle \delta_1\rangle$ of the primordial vertex.

— Random Recursive Simplicial Complexes  (2608.26547 - Krapivsky et al., 27 Aug 2026) in Section 4, paragraph following equation (delta-change)

Despite of the partial exact results partial, we have not been able to guess an exact general solution of Eq.~P-delta:HSC.

— Random Recursive Simplicial Complexes  (2608.26547 - Krapivsky et al., 27 Aug 2026) in Appendix, Section “Homogeneous simplicial complexes”