Forks in the Bitcoin network result from the natural competition in the blockchain's Proof-of-Work consensus protocol. Their frequency is a critical indicator for the efficiency of a distributed ledger as they can contribute to resource waste and network insecurity. We introduce a model for the estimation of natural fork rates in a network of heterogeneous miners as a function of their number, the distribution of hash rates and the block propagation time over the peer-to-peer infrastructure. Despite relatively simplistic assumptions, such as zero propagation delay within mining pools, the model predicts fork rates which are comparable with the empirical stale blocks rate. In the past decade, we observe a reduction in the number of mining pools approximately by a factor 3, and quantify its consequences for the fork rate, whilst showing the emergence of a truncated power-law distribution in hash rates, justified by a rich-get-richer effect constrained by global energy supply limits. We demonstrate, both empirically and with the aid of our quantitative model, that the ratio between the block propagation time and the mining time is a sufficiently accurate estimator of the fork rate, but also quantify its dependence on the heterogeneity of miner activities. We provide empirical and theoretical evidence that both hash rate concentration and lower block propagation time reduce fork rates in distributed ledgers. Our work introduces a robust mathematical setting for investigating power concentration and competition on a distributed network, for interpreting discrepancies in fork rates -- for example caused by selfish mining practices and asymmetric propagation times -- thus providing an effective tool for designing future and alternative scenarios for existing and new blockchain distributed mining systems.

翻译：比特币网络中的分叉源于区块链工作量证明共识协议中的自然竞争。其频率是衡量分布式账本效率的关键指标，因为分叉可能导致资源浪费和网络不安全。我们提出了一个模型，用于估计异构矿工网络中自然分叉率与其数量、算力分布以及点对点基础设施上区块传播时间的函数关系。尽管模型采用了相对简化的假设（例如矿池内零传播延迟），但其预测的分叉率与经验性陈旧区块率具有可比性。过去十年间，我们观察到矿池数量减少了约三分之二，并量化了其对分叉率的影响，同时揭示了算力分布中截断幂律分布的出现，这可由受全球能源供应限制约束的"富者愈富"效应解释。我们通过实证和定量模型证明，区块传播时间与挖矿时间的比值是分叉率的充分准确估计量，同时也量化了其对矿工活动异质性的依赖关系。我们提供了实证和理论证据，表明算力集中度和更低的区块传播时间均能降低分布式账本的分叉率。本研究建立了一个稳健的数学框架，用于研究分布式网络中的算力集中与竞争，解释分叉率差异（例如由自私挖矿行为和不对称传播时间引起），从而为设计现有及新型区块链分布式挖矿系统的未来方案与替代场景提供了有效工具。