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 $|p-q{|}^{\alpha }$. 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.

6 pages, 10 figures

References in corpus (6)

• Statistical mechanics of complex networks
• The structure and function of complex networks
• Assortative mixing in networks
• Evolution of networks
• Small-world properties of the Indian Railway network
• Families and clustering in a natural numbers network

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• Structure Properties of Koch Networks Based on Networks Dynamical Systems