A Small World Network of Prime Numbers

Abstract

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 {\em one} decomposition with a probability $|p - q|^α$. Through computation of average shortest distance and clustering coefficient, we conclude that for $α> -1.8$ the network is of small world type and for $α< -1.8$ it is of regular type. We also present a theoretical justification for such behaviour.