Accessibility Percolation with Crossing Valleys on n-ary Trees
- 20 Downloads
Abstract
In this paper, we study a variation of the accessibility percolation model. This work 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 \(\root k \of {h/(ek)}\); and tends to 0, if n grows significantly slower than \(\root k \of {h/(ek)}\).
Keywords
Percolation Dynamics of evolution Fitness landscapeMathematics Subject Classification
60K35 60C05 92D15Notes
Acknowledgements
The authors are thankful to Ricardo Restrepo and Daya K. Nagar for helpful discussions. Thanks are due to the anonymous referees for their careful reading, criticism and suggestions which helped us to considerably improve the paper. Research Partially supported by FORDECYT 265667 (Mexico). Research Partially supported by Universidad de Antioquia (Colombia).
References
- 1.Berestycki, J., Brunet, E., Shi, Z.: The number of accessible paths in the hypercube. Bernoulli 22(2), 653–680 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
- 2.Berestycki, J., Brunet, É., Shi, Z.: Accessibility percolation with backsteps. ALEA Lat. Am. J. Probab. Math. Stat 14, 45–62 (2017)MathSciNetzbMATHGoogle Scholar
- 3.Franke, J., Klözer, A., de Visser, J.A.G.M., Krug, J.: Evolutionary accessibility of mutational pathways. PLOS Comput. Biol. 7(8), 1–9 (2011)MathSciNetCrossRefGoogle Scholar
- 4.Gillespie, J.H.: Molecular evolution over the mutational landscape. Evolution 38(5), 1116–1129 (1984)CrossRefGoogle Scholar
- 5.Hegarty, P., Martinsson, A.: On the existence of accessible paths in various models of fitness landscapes. Ann. Appl. Probab. 24(4), 1375–1395 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
- 6.Li, L.: Phase transition for accessibility percolation on hypercubes. J. Theor. Probab. 31, 2072–2111 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
- 7.Martinsson, A.: Accessibility percolation and first-passage site percolation on the unoriented binary hypercube. arXiv:1501.02206 (2015)Google Scholar
- 8.Nowak, S., Krug, J.: Accessibility percolation on n-trees. EPL (Europhys. Lett.) 101(6), 66004 (2013)ADSCrossRefGoogle Scholar
- 9.Richter, H., Engelbrecht, A.: Recent advances in the theory and application of fitness landscapes. Springer, New York (2014)CrossRefGoogle Scholar
- 10.Roberts, M., Zhao, L.: Increasing paths in regular trees. Electron. Commun. Probab. 18(87), 1–10 (2013)MathSciNetzbMATHGoogle Scholar
- 11.Stadler, P.F., Stephens, C.R.: Landscapes and effective fitness. Comment Theor. Biol. 8((4–5)), 389–431 (2003)CrossRefGoogle Scholar
- 12.Weinreich, D.M., Chao, L.: Rapid evolutionary escape by large populations from local fitness peaks is likely in nature. Evolution 59(6), 1175–1182 (2005)CrossRefGoogle Scholar
- 13.Weinreich, D.M., Watson, R.A., Chao, L.: Perspective:sign epistasis and genetic constraint on evolutionary trajectories. Evolution 59, 1165–1174 (2005)Google Scholar
- 14.Weinreich, D.M., Delaney, N.F., DePristo, M.A., Hartl, D.L.: Darwinian evolution can follow only very few mutational paths to fitter proteins. Science 312(5770), 111–114 (2006)ADSCrossRefGoogle Scholar
- 15.Weissman, D.W., Desai, M.M., Fisher, D.S., Feldman, M.W.: The rate at which asexual populations cross fitness valleys. Theor. Popul. Biol. 75, 286–300 (2009)CrossRefzbMATHGoogle Scholar
- 16.Wright, S.: The roles of mutation, inbreeding, crossbreeding, and selection in evolution. Proc. Sixth Int. Congress Genet. 1, 356–366 (1932)Google Scholar