Tokunaga self-similarity arises naturally from time invariance

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.