Entropy, complexity, and spatial information

Abstract

We pose the central problem of defining a measure of complexity, specifically for spatial systems in general, city systems in particular. The measures we adopt are based on Shannon's (in Bell Syst Tech J 27:379-423, 623-656, 1948) definition of information. We introduce this measure and argue that increasing information is equivalent to increasing complexity, and we show that for spatial distributions, this involves a trade-off between the density of the distribution and the number of events that characterize it; as cities get bigger and are characterized by more events-more places or locations, information increases, all other things being equal. But sometimes the distribution changes at a faster rate than the number of events and thus information can decrease even if a city grows. We develop these ideas using various information measures. We first demonstrate their applicability to various distributions of population in London over the last 100 years, then to a wider region of London which is divided into bands of zones at increasing distances from the core, and finally to the evolution of the street system that characterizes the built-up area of London from 1786 to the present day. We conclude by arguing that we need to relate these measures to other measures of complexity, to choose a wider array of examples, and to extend the analysis to two-dimensional spatial systems.

Keywords: Density; Entropy; Information; London population; London street system; Spatial complexity.

Fig. 1

The two-event and three-event systems

Fig. 2

Hierarchical aggregation–decomposition in the three-event system

Fig. 3

Entropy distributions from 1901 to 2001

Fig. 4

Entropy differences from 1901 to 2001

Fig. 5

Spatial aggregation of zones according to a spiral from the two most central to the outer boroughs

Fig. 6

Entropy statistics associated with the spiral aggregation

Fig. 7

Entropy and information differences associated with the spiral aggregation

Fig. 8

The complete metropolitan region, organized into distance bands

Fig. 9

Left Entropy H and right spatial entropy S

Fig. 10

Left Information I and right Tribus–McIrvine H(p/q)

Fig. 11

The information difference H _max—H and the information difference minus spatial entropy

Fig. 12

The density of street intersections in Greater London 1786–2010

Fig. 13

Changes in information and complexity in the population of streets