Analyzing percolation of networks inspired by the 3x+1 problem

Zhi-Ming Gu, Chen-Ping Zhu, Tao Zhou and Ming Zhao

Physica A: Statistical Mechanics and its Applications, 2012, vol. 391, issue 20, 4986-4994

Abstract: In this paper, we investigate percolation in a sort of networks inspired by the observation of a Collatz graph (CG) which is the network version of the famous 3x+1 problem in mathematics. The CG consists of positive integers that are connected based on the iteration relations. Actually, we never mean to solve the 3x+1 problem exactly but we observe it from the viewpoint of statistical physics. We focus on the so-called reduced Collatz graph (RCG) that is a subgraph with all odd numbers since even numbers can be iterated into odd ones through 3x+1 rules. Considering boundary conditions, we obtain a special degree distribution of RCG for finite size set of odd integers, and treat the infinite case as its limit. With the percolation criterion through the approach of a generating function, we determine the critical condition for the network ensemble of RCG. Furthermore, we generalize the graph model with RCG-type degree distributions beyond the 3x+1 problem, get the generic criterion of percolation and phase diagram for an ensemble of positive-integer networks with RCG as its extreme case.

Keywords: The 3x+1 problem; Collatz graph; Network; Percolation (search for similar items in EconPapers)
Date: 2012
References: View complete reference list from CitEc
Citations:

Downloads: (external link)
http://www.sciencedirect.com/science/article/pii/S0378437112001793
Full text for ScienceDirect subscribers only. Journal offers the option of making the article available online on Science direct for a fee of $3,000

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:eee:phsmap:v:391:y:2012:i:20:p:4986-4994

DOI: 10.1016/j.physa.2012.02.032

Access Statistics for this article

Physica A: Statistical Mechanics and its Applications is currently edited by K. A. Dawson, J. O. Indekeu, H.E. Stanley and C. Tsallis

More articles in Physica A: Statistical Mechanics and its Applications from Elsevier
Bibliographic data for series maintained by Catherine Liu ().