Biological functions rely on ordered structures and intricately controlled collective dynamics. Such order in living systems is typically established and sustained by continuous dissipation of energy. The emergence of ordered patterns of motion is unique to non-equilibrium systems and is a manifestation of dynamic steady states. Many cellular processes require transitions between different steady states. Can general principles of statistical physics guide our understanding of such cellular self-organization? In this talk, I will show model actomyosin cortices, in the presence of rapid turnover, self-organize into three non-equilibrium steady states as a function of network connectivity. The different states arise from a subtle interaction between mechanical percolation of the actin network and myosin-generated stresses. I will discuss our experimental and theoretical approach to identify governing principles of collective dynamics, spatiotemporal stress pattern formation and energy dissipation in such far from equilibrium biological systems.