The fluid limit of a random graph model for a shared ledger
Christopher King
TL;DR
This work analyzes a stochastic growth model for the tangle, a DAG-based shared ledger, and derives a fluid limit described by a pair of delay differential equations $\frac{da}{dt}=1-2\frac{a}{b}$, $\frac{db}{dt}=1-2\frac{a(t-h)}{b(t-h)}$. It proves that the rescaled tip process $B^{(\\lambda)}(t)$ converges in probability to the deterministic fluid solution as the arrival rate $\\lambda\\to\\infty$, with fluctuations of order $\\lambda^{-1/2}$, using martingale methods. It further shows that the fluid solution converges exponentially to the steady-state corresponding to $b(t)\to 2h$, implying $L(t)\approx 2\\lambda h$ for large $\\lambda$ and long times. The results provide a rigorous, tractable approximation for tip dynamics in high-rate regimes and offer a foundation for analyzing stability and extensions of the tangle protocol and related shared-ledger models.
Abstract
A shared ledger is a record of transactions that can be updated by any member of a group of users. The notion of independent and consistent record-keeping in a shared ledger is important for blockchain and more generally for distributed ledger technologies. In this paper we analyze the growth of a model for the tangle, which is the shared ledger protocol used as the basis for the IOTA cryptocurrency. The model is a random directed acyclic graph, and its growth is described by a non-Markovian stochastic process. We derive a delay differential equation for the fluid model which describes the tangle at high arrival rate. We prove convergence in probability of the tangle process to the fluid model, and also prove global stability of the fluid model. The convergence proof relies on martingale techniques.