Degree Distributions in Recursive Trees with Fitnesses

TL;DR

This paper analyzes a recursive tree model with fitness-based attachment, deriving limiting degree and weight distributions, and explores phenomena like condensation and degenerate distributions, extending understanding of complex network growth.

Contribution

It provides explicit formulas for degree and weight distributions in recursive trees with fitness, including cases with condensation and degenerate limits, using branching process theory.

Findings

Derived almost sure limiting distributions for degrees and weights.

Proved condensation phenomena where edges concentrate on high-weight vertices.

Established stochastic convergence of degree distribution under strong law assumptions.

Abstract

We study a general model of recursive trees where vertices are equipped with independent weights and at each time-step a vertex is sampled with probability proportional to its fitness function (a function of its weight and degree) and connects to $\ell$ new-coming vertices. Under a certain technical assumption, applying the theory of Crump-Mode-Jagers branching processes, we derive formulas for the almost sure limiting distribution of the proportion of vertices with a given degree and weight, and proportion of edges with endpoint having a certain weight. As an application of this theorem, we prove rigorously observations of Bianconi related to the evolving Cayley tree in [$\mathit{Phys. \, Rev. \, E} \; \mathbf{66}, \text{ 036116 (2002)}$]. We also study the process in depth when the technical condition can fail in the particular case when the fitness function is affine, a model we call "generalised preferential attachment with fitness". We show that this model can exhibit condensation where a positive proportion of edges accumulate around vertices with maximal weight, or, more drastically, have a degenerate limiting degree distribution where the entire proportion of edges accumulate around these vertices. Finally, we prove stochastic convergence for the degree distribution under a different assumption of a strong law of large numbers for the partition function associated with the process.