Stability of shares in the Proof of Stake Protocol -- Concentration and Phase Transitions

TL;DR

This paper analyzes the stability and concentration phenomena of shares in Proof of Stake cryptocurrencies, revealing phase transitions and the impact of reward schemes on investor behavior and network centralization.

Contribution

It introduces a mathematical framework distinguishing investor behaviors and characterizes phase transition thresholds in PoS protocols, including a dynamic model for growing investor populations.

Findings

Large investors' shares are stable

Medium and small investors' shares are volatile or shrink to zero

Chaotic centralization occurs with geometric rewards

Abstract

This paper is concerned with the stability of shares in a cryptocurrency where the new coins are issued according to the Proof of Stake protocol. We identify large, medium and small investors under various rewarding schemes, and show that the limiting behaviors of these investors are different -- for large investors their shares are stable, while for medium to small investors their shares may be volatile or even shrink to zero. For instance, with a geometric reward there is chaotic centralization, where all the shares will eventually concentrate on one investor in a random manner. This leads to the phase transition phenomenon, and the thresholds for stability are characterized. In response to the increasing activities in blockchain networks, we also propose and analyze a dynamical population model for the PoS protocol, which allows the number of investors to grow over the time. Numerical experiments are provided to corroborate our theory.