In distributed ledger technologies (DLTs) with a directed acyclic graph (DAG) data structure, a block-issuing node can decide where to append new blocks and, consequently, how the DAG grows. This DAG data structure is typically decomposed into two pools of blocks, dependent on whether another block already references them. The unreferenced blocks are called the tips. Due to network delay, nodes can perceive the set of tips differently, giving rise to local tip pools. We present a new mathematical model to analyse the stability of the different local perceptions of the tip pools and allow heterogeneous and random network delay in the underlying peer-to-peer communication layer. Under natural assumptions, we prove that the number of tips is ergodic, converges to a stationary distribution, and provide quantitative bounds on the tip pool sizes. We conclude our study with agent-based simulations to illustrate the convergence of the tip pool sizes and the pool sizes' dependence on the communication delay and degree of centralization.

翻译：在采用有向无环图数据结构的分布式账本技术中，区块发布节点可自主决定新块的附加位置，从而影响有向无环图的生长方式。该数据结构通常将区块划分为两个池：依据是否已被其他区块引用，未引用的区块被称为尖端。由于网络延迟，各节点对尖端集合的感知存在差异，由此形成局部尖端池。我们提出一种新的数学模型，用于分析尖端池局部感知差异的稳定性，并允许底层点对点通信层存在异构且随机的网络延迟。在自然假设条件下，我们证明了尖端数量的遍历性及其向平稳分布的收敛性，并给出了尖端池规模的量化界。最后，我们通过基于智能体的仿真实验，验证了尖端池规模的收敛性及其对通信延迟与去中心化程度的依赖关系。