Superlineal

Superlineal, or superlinear, describes a growth rate that outpaces linear growth as the input size increases. Formally, a function f defined on natural numbers (or positive reals) is superlinear if f(n) grows faster than c n for every fixed constant c>0 as n→∞; equivalently, the limit of f(n)/n as n approaches infinity is infinity. This contrasts with linear growth, where f(n) grows proportionally to n, and sublinear growth, where f(n)/n tends to zero.

Common examples of superlinear growth include f(n) = n log n, f(n) = n^p with p>1, and f(n) = a^n with a>1. Functions like f(n) = n log log n also exhibit superlinear behavior, since the logarithmic factors eventually exceed any constant multiple of n. In practice, superlinear growth encompasses a range of rates from gentle to very rapid, all of which dominate linear growth for large n.

Applications and usage vary by field. In algorithm analysis, superlinear time means the running time grows faster than input size, as in many sorting or graph algorithms. In network science and urban studies, quantities such as GDP, innovation counts, or infrastructure usage often scale superlinearly with city population, reflecting increasing returns to scale. In mathematics and asymptotic analysis, identifying superlinear growth helps compare functions and derive bounds.

Notes: In Spanish or Portuguese contexts, the term is often translated as “superlineal.” The precise meaning depends on context, but it generally denotes growth that eventually surpasses any linear benchmark.