Families and clustering in a natural numbers network

We develop a network in which the natural numbers are the vertices. We use the decomposition of natural numbers by prime numbers to establish the connections. We perform data collapse and show that the degree distribution of these networks scale linearly with the number of vertices. 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 is not a small-world type. If the criterium for establishing links becomes more selective, only prime numbers greater than $p_l$ are used to establish links, the network shows small-world effect, it means, it has high clustering coefficient and low distance.