Emergence of double scaling law in complex systems

2014

Nexus Netw J

There have been many studies on the spatial configuration of cities, but few attempts to quantify the difference in building patterns between the old and new parts of cities. This may be partly attributable to lack of suitable study methods. This paper presents a new application of statistical methods for quantifying the geometric difference between different parts of a city using, as a case study, the old (historical) and new parts of the city of Yazd in Iran. We measured 341 edge lengths of 4 bazaars, 302 edge lengths of 5 mosques and tombs, and 239 edge lengths of 3 schools. We also measured 6,804 edge lengths and the areas of 1,243 well-preserved courtyard houses in the old part and 4,948 edge lengths and the areas of 1,237 houses in the new part of the city. In the old part, all edge-length and house-area frequency distributions, to a first approximation, follow power laws, indicating that there are many small and very few large buildings. By contrast, in the new part the edge-length and house-area frequency distributions follow bimodal (two-peak) distributions. The calculated entropies (measures of dispersion) of the house edge lengths and areas in the old part are much higher than of those in the new part and provide a clear, quantitative measure of the geometric differences between the builtup structures of old and the new parts of the cities. The power-law distributions in the old part indicate a gradual and continuous variation in sizes of edge lengths and house areas, whereas the bimodal distributions in the new part indicate abrupt (discontinuous) changes in the edge lengths and house areas. The entropy results imply that the size distributions of houses in the old part are more dispersed than those in the new part, indicating more interconnected structures in the old part of the

Section: Size Distributionsmentioning

confidence: 99%

Quantifying the Differences in Geometry and Size Distributions of Buildings Within Cities

2014

Nexus Netw J

“…There are commonly one or more breaks in the log-log plot, yielding multiple straight lines with different slopes, suggesting that the data can be divided into sub-populations rather than treated as a single population [49]. The straight-line slopes on the log-log plots represent different scaling regimes which may not necessarily follow power laws.…”

Section: Street Length Distributionmentioning

confidence: 99%

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

French

2013

Entropy

Geographical location and landforms of various types have strong effects on the developments of many cities and associated street networks. This study presents new results of landform effects, in particular the geometry of shorelines, on the grid street networks (a total of 10,442 streets) of three Brazilian coastal cities. The results are compared with the street networks of three coastal cities in Britain (a total of 22,002 streets) that have evolved through a more natural "bottom-up" process. Gibbs/Shannon entropy (a measure of dispersion) generally has a positive linear correlation with length ranges and the average lengths of the street, and for the power-law tails in particular. The geometry of the adjacent shorelines has great effect on the spatial orientation of streets in the Brazilian cities but less so for the networks of the British cities. More specifically, the more curved the shoreline, the greater is the dispersion in the street orientation and the greater the associated entropy. The results also show that the length-entropies of the outer parts of the Brazilian cities are generally lower than those of the inner parts, whereas the entropies of outer parts of the British cities are higher than those of the inner parts, indicating dispersion during streetnetwork growth in the British cities.

“…On log-log plots, however, the deviations from the straight line are large and two straight lines (two power laws, Figures 7c, 9c1,c2), or, occasionally, three straight lines (Figures 8c, 10c1,c2), fit the data much better, suggesting that both segments and named streets follow power laws that have different slopes (scaling exponents) for different street length ranges. Different slopes of these kinds, referred to as double scaling laws when involving two straight lines, are observed in many power-law data sets [35], including those on fractures [36,37]. …”

Section: Power Law Street-length Distributionsmentioning

confidence: 99%

Entropies and Scaling Exponents of Street and Fracture Networks

2012

Entropy

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).