A Game of Nodes

This preprint introduces "A Game of Nodes", a gamified recursive network architecture designed to model fast-growing mathematical sequences—specifically targeting tetration-tier growth—using graph-traversal mechanics. The architecture functions as a dynamic system wherein each node traversal triggers the embedding of a new network of equivalent order. This recursive spawning mechanism couples the iteration count with the input value, aggressively driving growth beyond standard factorial scaling. Motivated by the challenge of making the visualisation of fast-growing functions, such as TREE(3), more accessible, the system demonstrates how applying recursive operations to conceptual benchmarks (such as 10^80, the estimated number of atoms in the observable universe) can be surpassed almost immediately. The paper presents the formal ruleset, the recursive constraints, and the algebraic sequence derived from a base network of order 3, providing a conceptual framework for translating abstract hyperoperations into a clear, iterative, and reproducible process.Future planned expansions within this framework include "A Feast for Graham" and "A Dream of Tree".