This chapter provides an introduction to how power-laws are used to characterize cities and urban systems. On the one hand, four power-law representations are used to describe city-size distributions.

Here we detail how the respective exponents are related. For numerical simulations it might be necessary to study power-law distributed random numbers.

The chapter includes some basics how to generate them. As another important aspect, the relation between power-law distributions and power-law correlations is presented.

On the other hand, essentials on fractals and dimensions are provided. E.g. one section outlines why box-covering (aka box-counting) leads to an estimate of the fractal dimension.

D. Rybski, An Introduction to Mathematical Concepts of Power-Laws in Cities and Urban Systems, in: D. Rybski (ed.), Compendium of Urban Complexity (2025), Springer, Cham.