2. Defining Complexity in Cities

This chapter defines a series of measures of complexity that pertain to the spatial structure of cities. First the development of complexity theory is sketched with specific reference to the ways in which it has and is being exploited in applied areas such as urban science and city planning. The key signatures of a complex system are outlined, with a focus on the various subsystems of the city manifesting self-similarity over a range of scales, thus invoking ideas about fractal geometry. A series of scaling relations are then defined that are adopted as these signatures. These scaling relations are power laws and they emerge in many different contexts with respect to cities. Here our focus is on spatial scale which serves to define these relationships. As well as basic fractal scaling, these include allometry, spatial interaction, mass, distance and area relations, and rank-size rules that pertain to city size and related distributions. Once these functions have been introduced, we examine ways in which the classic entropy or information formula first stated by Shannon (1948) can be used to measure the degree of complexity in a city. These measures focus on the shape of these scaling distributions such as population that define a city and the number of objects or components that count the size of such distributions. We conclude with some challenges for defining complexity further.