Mathematical Modelling of Dual–Layer Byzantine Fault–Tolerant Consensus Process for Optimal Sharding and Mitigation of Blockchain Trilemma

The dual–layer consensus process enhances the transaction processing performance of Byzantine fault–tolerant consensus blockchains through a method called sharding. In this sharding method, the network of validator nodes is partitioned into multiple subsystems, each consisting of a group of nodes at a lower level, known as shards. Consensus formation within these shards is aggregated at a higher level among the leaders of each shard, thereby reducing the total communication load throughout the process. This paper theoretically analyzes the impact of node group partitioning in the dual–layer consensus process on average throughput using a stochastic model of Byzantine fault–tolerant consensus blockchains. Consequently, we derive a formula that qualitatively explains the simulation results of existing research. Additionally, we determine the optimal number of shards that maximizes average throughput in dual–layer Byzantine fault–tolerant consensus processes. Furthermore, we demonstrate that at this optimal value, the blockchain trilemma—balancing decentralization, scalability, and security—can be mitigated. Specifically, we demonstrate that both security and scalability are enhanced when the degree of decentralization is increased.