  Long and stable timescales are often observed in complex biochemical
networks, such as in emergent oscillations. How these robust dynamics persist
remains unclear, given the many stochastic reactions and shorter time scales
demonstrated by underlying components. We propose a topological model that
produces long oscillations around the network boundary, reducing the system
dynamics to a lower-dimensional current in a robust manner. Using this to model
KaiC, which regulates the circadian rhythm in cyanobacteria, we compare the
coherence of oscillations to that in other KaiC models. Our topological model
localizes currents on the system edge, with an efficient regime of
simultaneously increased precision and decreased cost. Further, we introduce a
new predictor of coherence from the analysis of spectral gaps, and show that
our model saturates a global thermodynamic bound. Our work presents a new
mechanism and parsimonious description for robust emergent oscillations in
complex biological networks.
