Ultrametric identities in glassy models of Natural Evolution

TL;DR

This paper investigates ultrametric identities in simple evolutionary spin-glass models, revealing new identities in certain models and comparing them with biological genome data, thus advancing understanding of evolutionary landscape structures.

Contribution

It introduces a new class of ultrametric identities in flat landscape models and analyzes their validity across different evolutionary models, contrasting with traditional identities.

Findings

New ultrametric identities found in specific models

Classic identities do not hold in some models

Preliminary genome data aligns better with new identities

Abstract

Spin-glasses constitute a well-grounded framework for evolutionary models. Of particular interest for (some of) these models is the lack of self-averaging of their order parameters (e.g. the Hamming distance between the genomes of two individuals), even in asymptotic limits, much as like the behavior of the overlap between the configurations of two replica in mean-field spin-glasses. In the latter, this lack of self-averaging is related to peculiar fluctuations of the overlap, known as Ghirlanda-Guerra identities and Aizenman-Contucci polynomials, that cover a pivotal role in describing the ultrametric structure of the spin-glass landscape. As for evolutionary models, such identities may therefore be related to a taxonomic classification of individuals, yet a full investigation on their validity is missing. In this paper, we study ultrametric identities in simple cases where solely random mutations take place, while selective pressure is absent, namely in {\em flat landscape} models. In particular, we study three paradigmatic models in this setting: the {\em one parent model} (which, by construction, is ultrametric at the level of single individuals), the {\em homogeneous population model} (which is replica symmetric), and the {\em species formation model} (where a broken-replica scenario emerges at the level of species). We find analytical and numerical evidence that in the first and in the third model nor the Ghirlanda-Guerra neither the Aizenman-Contucci constraints hold, rather a new class of ultrametric identities is satisfied; in the second model all these constraints hold trivially. Very preliminary results on a real biological human genome derived by {\em The 1000 Genome Project Consortium} and on two artificial human genomes (generated by two different types neural networks) seem in better agreement with these new identities rather than the classic ones.