# The fluid limit of a random graph model for a shared ledger

## TL;DR

This paper analyzes the growth of the tangle, a shared ledger protocol used in IOTA, modeling it as a random directed acyclic graph and deriving a fluid limit to understand its behavior at high transaction rates.

## Contribution

It introduces a fluid model for the tangle, proves convergence of the stochastic process to this model, and establishes its global stability using martingale techniques.

## Key findings

- Derived a delay differential equation for the fluid model.
- Proved convergence in probability of the tangle process to the fluid model.
- Established global stability of the fluid model.

## 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.