Bernoulli

  • Bernoulli
  • Volume 22, Number 2 (2016), 653-680.

The number of accessible paths in the hypercube

Julien Berestycki, Éric Brunet, and Zhan Shi

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Motivated by an evolutionary biology question, we study the following problem: we consider the hypercube $\{0,1\}^{L}$ where each node carries an independent random variable uniformly distributed on $[0,1]$, except $(1,1,\ldots,1)$ which carries the value $1$ and $(0,0,\ldots,0)$ which carries the value $x\in[0,1]$. We study the number $\Theta$ of paths from vertex $(0,0,\ldots,0)$ to the opposite vertex $(1,1,\ldots,1)$ along which the values on the nodes form an increasing sequence. We show that if the value on $(0,0,\ldots,0)$ is set to $x=X/L$ then $\Theta/L$ converges in law as $L\to\infty$ to $\mathrm{e}^{-X}$ times the product of two standard independent exponential variables.

As a first step in the analysis, we study the same question when the graph is that of a tree where the root has arity $L$, each node at level 1 has arity $L-1$, …, and the nodes at level $L-1$ have only one offspring which are the leaves of the tree (all the leaves are assigned the value 1, the root the value $x\in[0,1]$).

Article information

Source
Bernoulli Volume 22, Number 2 (2016), 653-680.

Dates
Received: November 2013
First available in Project Euclid: 9 November 2015

Permanent link to this document
https://projecteuclid.org/euclid.bj/1447077757

Digital Object Identifier
doi:10.3150/14-BEJ641

Mathematical Reviews number (MathSciNet)
MR3449796

Zentralblatt MATH identifier
1341.60103

Keywords
branching processes evolutionary biology percolation trees

Citation

Berestycki, Julien; Brunet, Éric; Shi, Zhan. The number of accessible paths in the hypercube. Bernoulli 22 (2016), no. 2, 653--680. doi:10.3150/14-BEJ641. https://projecteuclid.org/euclid.bj/1447077757


Export citation

References

  • [1] Aldous, D. and Steele, J.M. (2004). The objective method: Probabilistic combinatorial optimization and local weak convergence. In Probability on Discrete Structures. Encyclopaedia Math. Sci. 110 1–72. Berlin: Springer.
  • [2] Altenberg, L. (1997). NK fitness landscapes. In Handbook of Evolutionary Computation (T. Bäck, D.B. Fogel and Z. Michalewicz, eds.) B2.7:5–B2.7:10. New York: Oxford Univ. Press.
  • [3] Berestycki, J., Brunet, E. and Shi, Z. (2014). Accessibility percolation with backsteps. Preprint. Available at arXiv:1401.6894.
  • [4] Carneiro, M. and Hartl, D.L. (2010). Adaptive landscapes and protein evolution. Proc. Natl. Acad. Sci. USA 107, Suppl 1 1747–1751.
  • [5] Chen, X. (2014). Increasing paths on $N$-ary trees. Preprint. Available at arXiv:1403.0843.
  • [6] Franke, J., Klözer, A., de Visser, J.A.G.M. and Krug, J. (2011). Evolutionary accessibility of mutational pathways. PLoS Comput. Biol. 7 e1002134, 9.
  • [7] Gillespie, J.H. (1983). A simple stochastic gene substitution model. Theor. Popul. Biol. 23 202–215.
  • [8] Hegarty, P. and Martinsson, A. (2014). On the existence of accessible paths in various models of fitness landscapes. Ann. Appl. Probab. 24 1375–1395.
  • [9] Kauffman, S. and Levin, S. (1987). Towards a general theory of adaptive walks on rugged landscapes. J. Theoret. Biol. 128 11–45.
  • [10] Kingman, J.F.C. (1978). A simple model for the balance between selection and mutation. J. Appl. Probab. 15 1–12.
  • [11] Klozner, A. (2008). NK fitness landscapes. Diplomarbeit Universität zu Köln.
  • [12] Lalley, S.P. and Sellke, T. (1987). A conditional limit theorem for the frontier of a branching Brownian motion. Ann. Probab. 15 1052–1061.
  • [13] Nowak, S. and Krug, J. (2013). Accessibility percolation on $n$-trees. Europhys. Lett. 101 66004.
  • [14] On-line Encyclopedia of Integer Sequences. Available at http://oeis.org/A003319.
  • [15] Roberts, M.I. and Zhao, L.Z. (2013). Increasing paths in regular trees. Electron. Commun. Probab. 18 1–10.
  • [16] Weinreich, D.M., Delaney, N.F., DePristo, M.A. and Hartl, D.M. (2006). Darwinian evolution can follow only very few mutational paths to fitter proteins. Science 312 111–114.
  • [17] Weinreich, D.M., Watson, R.A. and Chao, L. (2005). Perspective: Sign epistasis and genetic constraints on evolutionary trajectories. Evolution 59 1165–1174.