Approximate lumping yields exact density-dependent limits for edge-based mean-field models on arbitrary networks

The framework begins with a continuous-time Markov chain whose states track every vertex configuration. Partitioning by the macroscopic observables (numbers of vertices and edges in each state) produces a lumped process whose transition rates depend only on those observables. In the large-N limit these rates converge to deterministic flows, recovering classic edge-based equations for SIR, SIS and voter models as special cases without assuming degree homogeneity or tree-like structure. On Erdős-Rényi graphs the lumped rates match the exact moment closure; on heterogeneous graphs the same construction supplies a systematic correction term whose magnitude scales with degree variance.

Existing derivations either invoke ad-hoc moment closures or restrict attention to configuration-model ensembles. The lumping route makes the averaging step explicit and shows that accuracy is controlled by the variation of transition rates inside each partition. Numerical checks on single graphs confirm that the ODE trajectories stay within 3 % of stochastic simulations once N exceeds 5000, with larger deviations only when state-dependent rewiring is present.

Because the method applies to any finite-state vertex dynamics whose rates depend on neighbour counts, it immediately extends to threshold models, coupled oscillators and adaptive networks. Future work can quantify partition variance to obtain rigorous error bounds rather than empirical validation alone.