Almost sure one-endedness of a random graph model of distributed ledgers

TL;DR

This paper proves that the directed acyclic graph model of the IOTA distributed ledger almost surely develops a single infinite end, ensuring consensus and structural stability over time.

Contribution

It introduces a stochastic model for the IOTA ledger and proves its almost sure one-endedness, a key property for consensus in distributed ledgers.

Findings

Number of leaves in the DAG is bounded infinitely often.

Certain events occur infinitely often in the DAG process.

The IOTA DAG is almost surely one-ended as time approaches infinity.

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). One essential property of a properly functioning shared ledger is that all network members holding a copy of the ledger agree on a sequence of information added to the ledger, which is referred to as consensus and is known to be related to a structural property of DAG called one-endedness. In this paper, we consider a model of distributed ledger with sequential stochastic arrivals that mimic attachment rules from the IOTA cryptocurrency. We first prove that the number of leaves in the random DAG is bounded by a constant infinitely often through the identification of a suitable martingale, and then prove that a sequence of specific events happens infinitely often. Combining those results we establish that, as time goes to infinity, the IOTA DAG is almost surely one-ended.