Asymptotic behaviors of random graph models of distributed ledgers

Abstract

Blockchain and other decentralized databases, known as distributed ledgers, are designed to store information online where all trusted network members can update the data with transparency. The dynamics of ledger's development can be mathematically represented by a directed acyclic graph (DAG). In the first part of the thesis, we propose a random DAG model with sequential stochastic arrivals that mimic attachment rules from the IOTA cryptocurrency and study its asymptotic behavior as time goes to infinity. Our analysis establishes that the DAG is almost surely one-ended which is a crucial indicator of security of a decentralized database. In the second part of the paper, we study a modified DAG model and analyze its property as the arrival rate goes to infinity and the inter arrival time goes to zero. We establish that the number of leaves in the DAG and various random variables characterizing the vertices in the DAG can be approximated by its fluid limit, represented as delayed partial differential equations. Furthermore, we establish the stable state of this fluid limit and validate our findings through simulations.--Author's abstract