Structural selection of phase regimes in locally timed networks

We study phase allocation in networks with an assumed cyclic timing frame. Each node chooses where in a cycle of length P to update; directed edges reward useful phase offsets, and a crowding term penalizes excessive use of the same phase. The model therefore does not explain the origin of the cycle itself. It asks how graph structure shapes the stable allocation of nodes across phases once such a frame is available. We show that the resulting finite timing game converges under asynchronous local improvement moves. Its equilibria are not a single kind of clock: they include diffuse timing, phase clustering, hub- or core-anchored timing, and distributed or defected scaffolds. We analyze these outcomes through circular phase modes, normalized phase entropy, and degree-weighted timing fields. In simulations across synthetic graph families, high-degree momentum terms provide compact diagnostics of the converged phase-entropy organization and improve within-ensemble prediction relative to mean degree alone. Within the controlled graph ensembles studied here, these quantities diagnose phase-regime organization; they should not be read as universal graph-family-independent predictors. Degree-weighted order parameters, interpreted against frozen-random baselines, distinguish learned phase organization from structural concentration artifacts in highly heterogeneous graphs. Additional local-bandit experiments show that sampled decentralized learners can recover similar regime-level outcomes without evaluating all phases at every update.