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

Part of: Combinatorial probability Graph theory

Published online by Cambridge University Press:  17 March 2021

Christopher King
Christopher King*
Affiliation:
Northeastern University
*
*Postal address: Department of Mathematics, Northeastern University, Boston, MA 02115. Email address: c.king@northeastern.edu
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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 a stochastic model for the shared ledger known as the tangle, which was devised 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 first prove ergodicity of the stochastic process, and then 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.

Keywords

Directed acyclic graphrandom growth modelfluid limitdelay differential equation

MSC classification

Primary: 60C05: Combinatorial probability
Secondary: 05C80: Random graphs
Type
Original Article
Information
Advances in Applied Probability , Volume 53 , Issue 1 , March 2021 , pp. 81 - 106
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

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