A small world network of prime numbers

According to Goldbach conjecture, any even number can be broken up as the sum of two prime numbers: n = p + q. We construct a network where each node is a prime number and corresponding to every even number n, we put a link between the component primes p and q. In most cases, an even number can be broken up in many ways, and then we chose one decomposition with a probability vertical bar p-q vertical bar(alpha). Through computation of average shortest distance and clustering coefficient, we conclude that for alpha > - 1.8 the network is of small world type and for alpha < - 1.8 it is of regular type. We also present a theoretical justification for such behaviour. (c) 2005 Published by Elsevier B.V.