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

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.