Accessibility Percolation with Crossing Valleys on n-ary Trees

Abstract

In this paper we study a variation of the accessibility percolation model, this is also motivated by evolutionary biology and evolutionary computation. Consider a tree whose vertices are labeled with random numbers. We study the probability of having a monotone subsequence of a path from the root to a leaf, where any k consecutive vertices in the path contain at least one vertex of the subsequence. An n-ary tree, with height h, is a tree whose vertices at distance at most h − 1 to the root have n children. For the case of n-ary trees, we prove that, as h tends to infinity the probability of having such subsequence: tends to 1, if n grows significantly faster than k h/(ek) ; and tends to 0, if n grows significantly slower than k h/(ek) . Date: June 28, 2018. 2010 Mathematics Subject Classification. 60K35, 60C05, 92D15.

Percolação Acessível

“…Recentemente, um problema relacionado foi analisado no hipercubo H N = {0, 1} N , com N ∈ N, para mais confira os artigos [4,6]. Sugerimos consultar tambémos artigos [8,9,10,11,12] para questões relacionadas. No artigo [3], a principal questão estudadaé a existência de um caminho acessível infinito naárvore esfericamente simétrica.…”