This document explores the theoretical foundations and practical implementations of distributed consensus mechanisms in blockchain systems.
The consensus problem in a distributed system of n nodes can be formalized as follows:
Where x_i(t) represents the state of node i at time t. The Byzantine Fault Tolerance (BFT) property ensures that consensus can be reached even if f nodes are faulty, provided that:
The efficiency of consensus mechanisms is heavily influenced by the underlying network topology. In a fully connected network with n nodes, the communication complexity is O(n²), which can become a bottleneck for large-scale systems.
Alternative topologies, such as tree-based structures or gossip protocols, can reduce this complexity to O(n log n) or even O(n), at the cost of increased convergence time.