Ants, birds and traders each follow local rules and respond to neighbors or feedback. Group patterns emerge with no leader, a process called self-organization.
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Feedback loops either stabilize or amplify change. When amplifying loops overpower stabilizing ones past a threshold, a system can shift abruptly into a new, often hard-to-reverse state.
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Failures spread because parts of a network depend on one another: load shifts to neighbours, dependent nodes shut down, and links between networks carry damage across.
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Normal distributions cluster around an average with very rare extremes. Power laws have fat tails, so extreme events are far more likely, which changes how risk should be estimated.
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The butterfly effect is sensitive dependence on initial conditions: tiny differences in a deterministic system can grow into large ones. Lorenz found it in 1961 when rounding a number changed his simulated weather.
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Conway's Game of Life is a zero-player cellular automaton devised by John Horton Conway in 1970, in which four simple rules applied to a grid of cells produce complex patterns.
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Above a critical vehicle density, small fluctuations in speed grow instead of fading, producing a jam that travels backward along the road with no obstacle needed.
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Termites build mounds through self-organization: each worker responds to local cues, and the structure that results channels air so the colony's gases are flushed out.
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A phase transition is an abrupt change in a system's macroscopic properties as conditions like temperature or pressure vary. It happens when competing energy and entropy terms switch which one dominates.
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Schelling's model is an agent-based simulation showing that mild preferences for same-group neighbors can produce highly segregated neighborhoods, even when no one wants strict separation.
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Self-organized criticality is the tendency of some systems to drive themselves to a critical state without tuning. The sandpile model of 1987 was the first system shown to do this.
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The small-world effect is the finding that large networks, including social ones, have very short paths between nodes despite being highly clustered. A few long-range links produce it.
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A trophic cascade is a predator's effect rippling down the food web. Wolves did change Yellowstone, but how far that reached plants is still debated.
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The cobra effect names a reward that worsens the problem it targets. The Delhi story behind it is probably apocryphal, but the incentive pattern it describes is real and well studied.
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