# The power of choice in growing trees

- 89 Downloads
- 18 Citations

## Abstract.

The “power of choice” has been shown to radically alter the behavior of a number of randomized algorithms. Here we explore the effects of choice on models of random tree growth. In our models each new node has k randomly chosen contacts, where k > 1 is a constant. It then attaches to whichever one of these contacts is most desirable in some sense, such as its distance from the root or its degree. Even when the new node has just two choices, i.e., when k = 2, the resulting tree can be very different from a random graph or tree. For instance, if the new node attaches to the contact which is closest to the root of the tree, the distribution of depths changes from Poisson to a traveling wave solution. If the new node attaches to the contact with the smallest degree, the degree distribution is closer to uniform than in a random graph, so that with high probability there are no nodes in the tree with degree greater than O(log log N). Finally, if the new node attaches to the contact with the largest degree, we find that the degree distribution is a power law with exponent -1 up to degrees roughly equal to k, with an exponential cutoff beyond that; thus, in this case, we need k ≫ 1 to see a power law over a wide range of degrees.

## PACS.

89.75.Hc Networks and genealogical trees 02.50.Ey Stochastic processes 05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion## Preview

Unable to display preview. Download preview PDF.

## References

- Y. Azar, A.Z. Broder, A.R. Karlin, E. Upfal,
*Balanced allocations*, in*Proc. 26th ACM Symp. Theory of Computing*(1994) pp. 593–602 Google Scholar - Y. Azar, A.Z. Broder, A.R. Karlin, E. Upfal, SIAM J. Comp.
**29**, 180 (1999) zbMATHCrossRefMathSciNetGoogle Scholar - M. Adler, S. Chakarabarti, M. Mitzenmacher, L. Rasmussen, Rand. Struct. Alg.
**13**, 159 (1998) zbMATHCrossRefGoogle Scholar - M. Mitzenmacher, E. Upfal,
*Probability and Computing: Randomized Algorithms and Probabilistic Analysis*(Cambridge University Press, New York, 2005) Google Scholar - R.T. Smythe, H. Mahmoud, Theory Probab. Math. Statist.
**51**, 1 (1995) MathSciNetGoogle Scholar - M. Drmota, B. Gittenberger, Random Struct. Alg.
**10**, 421 (1997); M. Drmota, H.-K. Hwang, Adv. Appl. Probab.**37**, 321 (2005) zbMATHCrossRefMathSciNetGoogle Scholar - P.L. Krapivsky, S. Redner, Phys. Rev. Lett.
**89**, 258703 (2002); e-print arXiv:cond-mat/0207370 CrossRefADSGoogle Scholar - S. Janson, Alea Lat. Am. J. Probab. Math. Stat.
**1**, 347 (2006); e-print arXiv:math/0509471 zbMATHADSMathSciNetGoogle Scholar - We always assume that the initial tree is a single node which is the root Google Scholar
- A.L. Barabási, R. Albert, Science
**286**, 509–512 (1999) CrossRefMathSciNetGoogle Scholar - P.L. Krapivsky, S. Redner, Phys. Rev. E
**63**, 066123 (2001); e-print arXiv:cond-mat/0011094 CrossRefADSGoogle Scholar - Equation \(Q_j(N+1)=Q_j(N)+\frac{1}{N}\,Q_{j-1}(N)\) appears in many contexts, e.g. for networks with copying: P.L. Krapivsky, S. Redner, Phys. Rev. E
**71**, 036118 (2005); e-print arXiv:cond-mat/0410379 CrossRefGoogle Scholar - R.L. Graham, D.E. Knuth, O. Patashnik,
*Concrete Mathematics: A Foundation for Computer Science*(Reading, Mass.: Addison-Wesley, 1989) Google Scholar - E. Ben-Naim, P.L. Krapivsky, S.N. Majumdar, Phys. Rev. E
**64**, 035101 (2001); e-print arXiv:cond-mat/0105 CrossRefADSGoogle Scholar - W. van Saarloos, Phys. Rep.
**386**, 29 (2003) zbMATHCrossRefADSGoogle Scholar - S.N. Majumdar, P.L. Krapivsky, Physica A
**318**, 161 (2003); e-print arXiv:cond-mat/0205581 zbMATHCrossRefADSMathSciNetGoogle Scholar - R. van Zon, H. van Beijeren, Ch. Dellago, Phys. Rev. Lett.
**80**, 2035 (1998) CrossRefADSGoogle Scholar - P.L. Krapivsky, S.N. Majumdar, Phys. Rev. Lett.
**85**, 5492 (2000) CrossRefADSGoogle Scholar - M. Mitzenmacher, IEEE Trans. Paral. Dist. Sys.
**12**, 1094 (2001) CrossRefGoogle Scholar