Topology Protects Robust Global Cycles in Stochastic Systems

Living systems can exhibit time-scales much longer than those of the underlying components, as well as emergent dynamical and collective behavior. How such robust global behavior is subserved by stochastic constituents remains unclear. Here, we present biologically plausible motifs from which two-dimensional stochastic networks can be constructed. The motifs represent out-of-equilibrium cycles on the microscopic scale, which support macroscopic edge currents in configuration space, a consequence of the topological Zak phase. Uniquely non-Hermitian properties of the system are seen in the emergence of exceptional points or the non-zero vorticity and doubled periodicity of edge states. Our framework enables a wealth of dynamical phenomena such as a global clock, dynamical growth and de-growth, as well as synchronization, similar to observations that are quite prevalent in biology. Our models suggest new insights into the theoretical framework of non-Hermitian physics, and pave the way for the prediction of new states in both classical and quantum systems.