Families and clustering in a natural numbers network

We develop a network in which the natural numbers are the vertices. The decomposition of natural numbers by prime numbers is used to establish the connections. We perform data collapse and show that the degree distribution of these networks scales linearly with the number of vertices. We explore the families of vertices in connection with prime numbers decomposition. We compare the average distance of the network and the clustering coefficient with the distance and clustering coefficient of the corresponding random graph. In case we set connections among vertices each time the numbers share a common prime number the network has properties similar to a random graph. If the criterion for establishing links becomes more selective, only prime numbers greater than p(l) are used to establish links, where the network has high clustering coefficient.