A Lyapunov function is a real-valued function over system configurations that (1) decreases by a fixed amount whenever the system is not at equilibrium, and (2) has a minimum value. If you can find such a function, you PROVE convergence to equilibrium. The rate of decrease bounds convergence speed -- like a ball rolling downhill, potential energy is the Lyapunov function.
Even without central coordination, systems can self-organize to equilibrium. The existence of a Lyapunov function proves this mathematically. Watch how V(t) monotonically decreases -- it can never increase -- guaranteeing the system stops changing.
Coordination games have multiple equilibria. Which equilibrium society reaches depends on history, critical mass, and sometimes luck. Once established, equilibria are self-reinforcing -- everyone drives on the right because everyone else does. Switching between equilibria requires coordinated collective action (like Sweden's "Dagen H" switch in 1967).
Initial conditions matter enormously. A small early advantage for one standard or convention can snowball through positive feedback into total dominance. The "basin of attraction" diagram shows which starting conditions lead to which equilibrium -- the boundary is the tipping point.
Each agent has multiple cultural features (language, food preference, music taste, etc.). Agents interact with neighbors with probability proportional to their cultural similarity. When interacting, they adopt one differing trait from each other. This creates homophily + social influence: similar people interact more and become even more similar, but dissimilar neighbors rarely interact and remain different -- creating stable cultural boundaries.
More features = more possible barriers = more stable diversity. With few features, global convergence is likely. With many features, the world fragments into stable cultural regions. This explains why cultural diversity persists despite globalization -- the more dimensions of culture there are, the harder it is for two groups to be similar enough to interact.