Generalize the reduction scheme to structured, networked populations beyond all-to-all mixing

Generalize and validate the effective one-dimensional reduction based on the Doi-Peliti Hamiltonian to structured, networked populations with non-all-to-all interactions; develop a framework that incorporates network topology into the constraint selection and reduced Hamiltonian H⋆ while preserving the ability to characterize fixed points, phase transitions, and fluctuations.

Background

The current development and demonstrations largely address well-mixed (all-to-all) systems. Many real-world systems are structured by network topology, where interactions are not homogeneous and mean-field assumptions break down.

The authors explicitly identify extending their reduction method to networked populations as an open question with potentially large impact due to broad applicability in complex systems with structured interactions.

References

An open question that remains that we foresee to have a great impact due to the wealth of potential applications, is the generalization of this dimension reduction scheme to structured, networked populations that interact beyond the all-to-all limit.

— Effective one-dimension reduction of multi-compartment complex systems dynamics  (2404.11366 - Visco et al., 2024) in Conclusions

Several directions remain open. First, the present work focuses primarily on systems with a stable balanced-growth composition. Systems with multiple balanced-growth states would require a global quasipotential constructed from least-action transitions between competing growth modes . Second, although genetically engineered cells can in some cases be maintained for prolonged periods in a balanced-growth regime, natural cells generally undergo repeated cycles of growth and division. Incorporating division would connect the present framework to lineage and population statistics and could provide more systematic null models for single-cell variability . Third, our moment calculation uses a symplectic reduction of an underlying contact eigenvalue problem. To our knowledge, the variational and geometric structure of such contact eigenvalue problems has not been systematically developed in the large-deviation setting. A more complete treatment would clarify the role of the contact eigenvectors, the accuracy of the reduced approximation used here, and the construction of driven processes that realize atypical fluctuations in supercritical growing systems. Finally, extending the theory to dilution, environmental fluctuations, and interacting populations would make it more directly applicable to microbial growth, metabolic ecology, and experimental studies of cellular stochasticity.

— Large-deviations theory for growing chemical reaction networks  (2609.30970 - Gagrani et al., 25 Sep 2026) in Section Discussion