Power law

A relationship in which one quantity varies as a power of another, often producing scale-free patterns. Its defining feature is that rescaling the input changes the output by a constant multiplicative factor.

Scaling: A transformation that changes the size of a system while preserving or systematically changing its properties. Power laws keep a simple form under rescaling, linking exponent values to scale behavior.
Exponent: A number specifying how many times a base is multiplied by itself, or the power to which it is raised. The exponent sets the rate at which a power-law relationship grows or decays.
Scale-free network: A network whose degree distribution has a heavy tail, often approximated by a power law. Power-law degree patterns distinguish hubs-rich networks from networks with typical node degrees.
Lognormal distribution: A distribution whose logarithm is normally distributed. It can resemble a power law over a limited range while having a different tail form.
Self-organized criticality: A proposed mechanism by which slowly driven systems evolve toward critical states without external parameter tuning. It offers one explanation for power-law event sizes in systems such as sandpiles.
Homogeneous function: A function whose value changes by a fixed power when all inputs are multiplied by the same factor. This mathematical property gives multivariable power laws their characteristic scaling rule.
Proportionality: A relation in which one quantity varies in a fixed ratio to another. Direct proportionality is the special power law whose exponent is one.
Gutenberg–Richter law: An empirical relation stating that earthquake frequency decreases exponentially with magnitude. Because magnitude is logarithmic, this law corresponds to a power-law relation in seismic moment.
Weibull distribution: A probability distribution used to model lifetimes and failure rates with flexible tail behavior. Its stretched-exponential tails can look power-law-like in finite datasets.
Preferential attachment: A growth process in which new connections favor nodes that already have many connections. This mechanism can generate heavy-tailed degree distributions in growing networks.

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