scaledependent

Scale-dependent, or scale dependence, describes phenomena whose properties change when the scale of observation or measurement changes. A quantity is scale-dependent if its value or behavior varies with spatial, temporal, or organizational scale. This contrasts with scale-invariant properties that look the same across scales.

In physics, scale dependence is a central concept of the renormalization group, where effective theories change with zoom level and certain laws emerge only at long distances. In climate, ecology, and geography, many patterns are scale-dependent: species-area relationships, landscape structure metrics, and population densities can change when measured at different grid sizes or time intervals. Similarly, turbulence, river networks, and urban form often exhibit different characteristics at different scales.

Data analysis uses scale-aware methods to identify and model scale dependence. Wavelet transforms, multi-resolution analysis, and detrended fluctuation analysis help separate processes operating at different scales. Multiscale or multi-resolution modeling aims to link fine-scale processes to coarse-scale outcomes through homogenization or renormalization-like techniques.

In social science and economics, scale dependence can affect market dynamics, network structure, and policy impacts, making it important to specify the scale of inference. The term is sometimes used interchangeably with scale variance or scale sensitivity.

See also scale invariance, multiscale modeling, and fractals.