Distributed Algorithm for Making Scale-Free Network by Preferential Rewiring Without Growth

The digital payment system that uses cryptocurrency, such as Bitcoin, is a distributed ledger working on a peer-to-peer network. We present a method to make a scale-free network for such applications. Using some biased physical quantities that are observable in sites, we can make the scale-free network through processes of cooperating distributed sites. Each node only proposes connecting to the more attractive node among randomly known nodes. The candidate node that is found by each node agrees to set two-way links if the requesting node is more attractive than the old node that is already connected. Once they establish the new bidirectional relationship, they, respectively, remove the outgoing link to the less attractive node. We analytically calculate the connectivity distribution and show that the scaling exponent is 2.5. By Monte Carlo simulations, we confirm that a power law distribution of the scaling exponent 2.5 describes the degree distribution of the topology.