Topological properties of integer networks

Inspired by Pythagoras's belief that numbers represent the reality, we study the topological properties of networks of composite numbers, in which the vertices represent the numbers and two vertices are connected if and only if there exists a divisibility relation between them. The network has a fairly large clustering coefficient C approximate to 0.34, which is insensitive to the size of the network. The average distance between two nodes is shown to have an upper bound that is independent of the size of the network, in contrast to the behavior in small-world and ultra-small-world networks. The out-degree distribution is shown to follow a power-law behavior of the form k(-2). In addition, these networks possess hierarchical structure as C(k)similar to k(-1) in accord with the observations of many real-life networks. (c) 2005 Elsevier B.V. All rights reserved.