title: 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.
authors.affiliation: Centre for Advanced Spatial Analysis (CASA), University College London (UCL), 90 Tottenham Court Road, London, W1N 6TR UK.
authors.forename: Michael
authors.initials: M
authors.lastname: Batty
authors.affiliation: Centre for Advanced Spatial Analysis (CASA), University College London (UCL), 90 Tottenham Court Road, London, W1N 6TR UK.
authors.forename: Robin
authors.initials: R
authors.lastname: Morphet
authors.affiliation: Centre for Advanced Spatial Analysis (CASA), University College London (UCL), 90 Tottenham Court Road, London, W1N 6TR UK.
authors.forename: Paolo
authors.initials: P
authors.lastname: Masucci
authors.affiliation: Department of Architecture, The Martin Centre for Architectural and Urban Studies, 1-5 Scroope Terrace, Trumpington Street, Cambridge, CB2 1PX UK.
authors.forename: Kiril
authors.initials: K
authors.lastname: Stanilov
keywords: Density; Entropy; Information; London population; London street system; Spatial complexity