Hamiltonian Cycle on the Set of Genotypes and Non-Ergodic Quadratic Stochastic Operators

TL;DR

This paper investigates the conditions under which Volterra quadratic stochastic operators on genotypes are non-ergodic, linking non-ergodicity to the existence of Hamiltonian cycles or specific vertices, with proofs for cases where m=2,3,4.

Contribution

It establishes new criteria for non-ergodicity of Volterra quadratic stochastic operators based on Hamiltonian cycles and vertex properties, extending understanding for low-dimensional cases.

Findings

Non-ergodicity linked to Hamiltonian cycles.

Vertices of the simplex can be sources causing non-ergodicity.

Results proven explicitly for dimensions 2, 3, and 4.

Abstract

On the set of genotypes $\Phi=\{1,...,m\}$ we introduce a binary relation generated by Volterra quadratic stochastic operator $V$ on $(m-1)$ dimensional simplex $S^{m-1}$ and prove that the operator $V$ be non-ergodic if either there exists a Hamiltonian cycle or one of the vertices $M_i=(\delta_{1i},\delta_{2i},...,\delta_{mi})$ of the simplex $S^{m-1}$ is a source and restriction of $V$ to the invariant face $F_i=\{x\in S^{m-1}: x_i=0\}$ is non-ergodic. In this paper we prove this result for $m=2,3,4.$