Each round: draw a ball at random, return it plus one more of the same color. Early random draws get amplified by positive feedback (rich-get-richer). The final proportion is entirely determined by early luck. Unlike a fair coin (which converges to 50%), the Polya process LOCKS IN to an unpredictable ratio. This is the essence of path dependence: history matters, and identical processes produce very different outcomes.
The final distribution across many urns is uniform on [0, 1]. Any final proportion is equally likely! This means we cannot predict where any single urn will end up. The histogram should flatten as you run more urns -- every outcome is equally probable.
Each new user evaluates: Value = Quality + NetworkEffect * (fraction of adopters). With increasing returns, early adopters influence later ones. A slightly inferior technology can dominate if it gets an early lead, because network effects make it more valuable. Once locked in, switching costs prevent change even when a better alternative exists.
Try setting A's quality lower than B's but running many simulations. Even the inferior technology wins sometimes! Increase network effects to see how they amplify early randomness. The "tipping point" is when network effects overwhelm quality differences and lock-in becomes inevitable.
Random (Erdos-Renyi): Each pair connected with probability p. Short paths but low clustering. Small-World (Watts-Strogatz): Start as ring lattice, randomly rewire edges. Both short paths AND high clustering -- "your friends know each other." Scale-Free (Barabasi-Albert): Nodes added one at a time, connecting preferentially to high-degree nodes. Creates power-law degree distribution with hubs.
Scale-free networks are robust to random failure (removing random nodes barely affects connectivity) but vulnerable to targeted attack (removing hubs shatters the network). Six degrees of separation emerges in small-world networks: most pairs can be connected in surprisingly few hops.