Tokunaga self-similarity arises naturally from time invariance

TL;DR

This paper provides a theoretical foundation for the Tokunaga self-similarity condition in trees by linking it to time invariance in a geometric branching process, explaining its empirical success.

Contribution

It establishes a formal connection between time invariance in a geometric branching process and the Tokunaga self-similarity condition, offering a theoretical justification.

Findings

Tokunaga condition is equivalent to time invariance in the process.

The process exhibits symmetries similar to critical binary Galton-Watson trees.

Reproduces properties of Galton-Watson process at specific parameters.

Abstract

The Tokunaga condition is an algebraic rule that provides a detailed description of the branching structure in a self-similar tree. Despite a solid empirical validation and practical convenience, the Tokunaga condition lacks a theoretical justification. Such a justification is suggested in this work. We define a geometric branching processes $\mathcal{G}(s)$ that generates self-similar rooted trees. The main result establishes the equivalence between the invariance of $\mathcal{G}(s)$ with respect to a time shift and a one-parametric version of the Tokunaga condition. In the parameter region where the process satisfies the Tokunaga condition (and hence is time invariant), $\mathcal{G}(s)$ enjoys many of the symmetries observed in a critical binary Galton-Watson branching process and reproduce the latter for a particular parameter value.