Hodge Decomposition of Bitcoin Money Flow

Abstract

How money flows among users of Bitcoin is an interesting question in order to understand the dynamics on the complex network of Bitcoin transactions, and also how the transactions are related to the price in the exchange market indirectly. We employ the data of blockchain in the Bitcoin from 2013 to 2018 (compiled by a Hungary research group), utilize a simple algorithm to partially identify anonymous users from addresses, and construct snapshots of temporarily changing network with the users as nodes and the transactions as directed links. In order to understand how users are located in the entire flow, in particular upstream and downstream, we use the so-called Hodge decomposition (or Helmholtz-Hodge-Kodaira decomposition). In addition, we examine the so-called “bow-tie” structure of the binary network disregarding flow to find how the users in the upstream/downstream peripheries (the so-called IN/OUT) are located away from the core of strongly connected component (SCC). We compare the Hodge potential of each node with such a location in the bow-tie structure, and also with the net demand/supply of each node measured from the money flow, to find a significant correlation among the potential, the topological position, and the net demand/supply. We also decompose the flow of each link into potential flow and circular flow to find that circulation of the flow is quite significant. We shall discuss about possible implication of these findings.