Entropies and Scaling Exponents of Street and Fracture Networks

Abstract

Many natural and man-made lineaments form networks that can be analysed through entropy and energy considerations. Here we report the results of a detailed study of the variations in trends and lengths of 1554 named streets and 6004 street segments, forming a part of the evolving street network of the city of Dundee in East Scotland. Based on changes in the scaling exponents (ranging from 0.24 to 3.89), the streets can be divided into 21 populations. For comparison, we analysed 221 active crustal fractures in Iceland that (a) are of similar lengths as the streets of Dundee; (b) are composed of segments; and (c) form evolving networks. The streets and fractures follow power-law size distributions (validated through various statistical tests) that can be partly explained in terms of the energies needed for their formation. The entropies of the 21 street populations and 9 fracture populations show strong linear correlations with (1) the scaling exponents (R 2 = 0.845-0.947 for streets, R 2 = 0.859 for fractures) and with (2) the length ranges, that is, the differences between the longest and shortest streets/fractures, (R 2 = 0.845-0.906 for streets, R 2 = 0.927 for fractures).

Strengths and strain energies of volcanic edifices: implications for eruptions, collapse calderas, and landslides

“…For longer fractures, more energy is needed than for short fractures (Mohajeri and Gudmundsson, 2012). As indicated, the available energy sources, however, depend on the boundary conditions at the time of fracture initiation and propagation.…”

Entropy Measures of Street-Network Dispersion: Analysis of Coastal Cities in Brazil and Britain

“…If, however, the short streets maintained their lengths as the city expanded while the maximum lengths of the long streets (and thus their average lengths) increased, the entropy would have increased during the city evolution [9]. In some coastal cities [9,61], however, only the most "favourably oriented" (i.e. coast-parallel) streets, have a chance of becoming long as the city expands [61].…”

“…For some of the British cities, the street network gradually expanded inland from a nucleus, the initial village, which may be a natural harbour [9,61]. If the network remained unchanged as regards range in street length, then the scaling exponent of the long streets and the entropy would have remained the same [9,61].…”

“…In some coastal cities [9,61], however, only the most "favourably oriented" (i.e. coast-parallel) streets, have a chance of becoming long as the city expands [61]. This can be related to many parameters including the street functionality within the city, the city shape, and the geographical location.…”

“…For some of the British cities, the street network gradually expanded inland from a nucleus, the initial village, which may be a natural harbour [9,61]. If the network remained unchanged as regards range in street length, then the scaling exponent of the long streets and the entropy would have remained the same [9,61]. If, however, the short streets maintained their lengths as the city expanded while the maximum lengths of the long streets (and thus their average lengths) increased, the entropy would have increased during the city evolution [9].…”