The 'Butterfly effect' in Cayley graphs, and its relevance for evolutionary genomics

TL;DR

This paper explores how small changes in permutation sequences affect outcomes in Cayley graphs, with applications to evolutionary genomics and insights into the sensitivity of genomic transformations.

Contribution

It introduces a framework for analyzing the sensitivity of permutation sequences in Cayley graphs, linking group theory to genomic evolution and statistical implications.

Findings

Sensitivity to permutation sequence changes varies across different groups.

Small perturbations can lead to significant differences in outcomes.

Implications for understanding evolutionary processes in genomics.

Abstract

Suppose a finite set $X$ is repeatedly transformed by a sequence of permutations of a certain type acting on an initial element $x$ to produce a final state $y$. We investigate how 'different' the resulting state $y'$ to $y$ can be if a slight change is made to the sequence, either by deleting one permutation, or replacing it with another. Here the 'difference' between $y$ and $y'$ might be measured by the minimum number of permutations of the permitted type required to transform $y$ to $y'$, or by some other metric. We discuss this first in the general setting of sensitivity to perturbation of walks in Cayley graphs of groups with a specified set of generators. We then investigate some permutation groups and generators arising in computational genomics, and the statistical implications of the findings.