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Scaling of Geographic Space from the Perspective of City and Field Blocks and Using Volunteered Geographic Information
Bin Jiang, Xintao Liu
TL;DR
The paper asks how the scaling structure of large geographic spaces can be characterized beyond conventional boundaries and distributions. It decomposes national street networks into city and field blocks, applies a head/tail division rule and border numbers, and finds lognormal block sizes alongside power-law natural-city sizes. The resulting block perspective supports automatic urban-boundary delineation and topological center identification, while organism comparisons remain conjectural.
Problem
The paper investigates how geographic space’s heavy-tailed scaling can be characterized across cities and countryside using large street networks.
Method
The study decomposes national street networks into minimum-cycle blocks, separates them using the mean block size and spatial autocorrelation, clusters smaller blocks into cities, and assigns recursive border numbers.
Results
Block sizes at country and city levels follow lognormal distributions, while natural-city sizes follow power-law distributions; the mean block size divides blocks into a low-percentage head and high-percentage tail.
Takeaways & Limitations
The head/tail division rule provides a simple solution for delineating urban boundaries from large street networks, and border numbers identify topological country centers.
Takeaways & Limitations
The organism-scaling implications remain conjectural, with future work needed to verify and apply the head/tail division rule.
Abstract
from arXiv · showhide
Scaling of geographic space refers to the fact that for a large geographic area its small constituents or units are much more common than the large ones. This paper develops a novel perspective to the scaling of geographic space using large street networks involving both cities and countryside. Given a street network of an entire country, we decompose the street network into individual blocks, each of which forms a minimum ring or cycle such as city blocks and field blocks. The block sizes demonstrate the scaling property, i.e., far more small blocks than large ones. Interestingly, we find that the mean of all the block sizes can easily separate between small and large blocks- a high percentage (e.g., 90%) of smaller ones and a low percentage (e.g., 10%) of larger ones. Based on this regularity, termed as the head/tail division rule, we propose an approach to delineating city boundaries by grouping the smaller blocks. The extracted city sizes for the three largest European countries (France, Germany and UK) exhibit power law distributions. We further define the concept of border number as a topological distance of a block far from the outmost border to map the center(s) of the country and the city. We draw an analogy between a country and a city (or geographic space in general) with a complex organism like the human body or the human brain to further elaborate on the power of this block perspective in reflecting the structure or patterns of geographic space. Keywords: Power law distribution, scaling of geographic space, data-intensive geospatial computing, street networks
1. Introduction
The paper frames geographic space as heavy-tailed, with many more small constituents than large ones, and investigates this pattern through blocks extracted from national street networks. It uses block-size regularities to distinguish city and field blocks and develop objective city-boundary delineation.
- 1. Introduction: Geographic space exhibits many more small constituents than large ones, forming a heavy-tailed rather than normal distribution.Examples include streets, city blocks, cities, and axial lines.
- 1. Introduction: The study analyzes street networks spanning cities and countryside in France, Germany, and the UK by decomposing them into minimum-ring city and field blocks.Block sizes in each country are reported as lognormally distributed.
- 1. Introduction: The mean block size separates smaller city blocks from larger field blocks, supporting a head/tail rule for identifying cities from clustered blocks.The method also uses spatial autocorrelation to exclude smaller blocks adjacent to larger ones.
2. Data and data processing
The study uses volunteered geographic information from OpenStreetMap and extensive topological preprocessing to extract blocks from national street networks. It then assigns border numbers recursively to characterize topological distance from the outermost border and identify deep country centers.
- 2. Data and data processing: OpenStreetMap supplies the street-network data as volunteered geographic information created through collaborative mapping.The data are supported by sources including GPS traces and digital imagery.
- 2. Data and data processing: France, Germany, and UK networks are used, and topological relationships are built before block extraction because the original data lack topology.Processing assigns directions to line segments and creates arcs, nodes, and left and right polygons.
- 2. Data and data processing: Minimum-cycle extraction starts from an outmost arc and follows leftmost or rightmost traversal until a block or maximum outer-border cycle is obtained.Dangling arcs do not constitute blocks and are eventually removed.
- 2. Data and data processing: Border numbers are assigned recursively from border-adjacent blocks outward through neighboring blocks, producing a topological distance from the outmost border.The procedure assigns border number 1 first, then continues until all blocks receive a number.
- 2. Data and data processing: Germany has almost four times as many blocks as France and the UK, while its maximum border number is 125 versus roughly 70–80 for France and the UK.Blocks with maximum border numbers tend to lie in the countries’ deep topological centers.
3. Lognormal distribution of block sizes and head/tail division rule
Block sizes follow a lognormal, heavy-tailed pattern with many small blocks and few large ones. The mean block size separates these groups, supporting the head/tail division rule and its area-based counterpart.
- 90% of UK blocks are small and 10% are large, with comparable proportions across the other two networks.The percentages vary slightly between countries.
- Block sizes in the three countries follow lognormal rather than power-law distributions.
- The mean block size clearly separates small blocks below the mean from large blocks above it.For the UK, the dividing mean is 0.4 square kilometers.
- The head/tail division rule shows that about 10% of blocks form rural areas while 90% form urban areas.
- The area-based division reverses the block-count imbalance: 10% of blocks enclose about 90% of land, while 90% enclose 10%.The rule provides a foundation for delineating cities.
4. Delineating urban boundaries or defining natural cities based on head/tail division rule
The paper delineates natural cities by selecting blocks smaller than the mean, clustering adjacent smaller blocks, and retaining the largest resulting groups. City-size distributions are reported as power laws for France and Germany, but not statistically validated for the UK.
- Head/tail division and clustering: Natural cities are formed by clustering adjacent blocks smaller than the mean and retaining only the largest groups.The procedure applies the head/tail division rule and spatial autocorrelation to distinguish city groups from smaller clustered groups.
- Head/tail division and clustering: Spatial autocorrelation excludes smaller blocks adjacent to larger blocks, although some city-edge blocks may be categorized as countryside.The authors regard this edge effect as minor because those blocks are relatively large.
- City-size distributions: France and Germany pass the power-law goodness-of-fit test for natural-city sizes, whereas the UK does not.The authors suggest the UK result may relate to its island geography and coastline-constrained urban growth, but identify this as an open research issue.
- City-size distributions: The resulting natural-city boundaries are automatically obtained from clustered smaller blocks rather than manually specified.The method summarizes the workflow as all blocks, smaller blocks, clustered groups, and cities.
5. Mapping the image of the country using the border number
Border number maps geographic centrality through topological distance from the outmost border, producing country images that differ from geometric-center representations. The same block-based analysis extends to cities, where high border numbers identify central cores and block sizes remain heavy-tailed.
- Country centers: Border-number maps use warmer colors for blocks farther from the outmost border and identify the highest-numbered regions as country centers.The mapped red spots or patches represent the true centers in the paper’s topological interpretation.
- Country centers: Topological border-number patterns differ from geometric-distance maps, whose centers are symmetric gravity centers equidistant from boundary edges.The geometric representation is described as a distorted image because geometric centers do not match perceived centers.
- City centers: The border-number framework extends to London, where the highest border numbers indicate a true city center.For London, the border is defined relative to the city rather than the UK country border.
- City centers: London city-block sizes follow a lognormal heavy-tailed distribution, and blocks above the mean form city cores that match the city center.The paper relates this result to earlier power-law claims because both distributions are heavy-tailed.
6. Discussions on the study
The study argues that massive volunteered geographic information and computing enable block-based insights into geographic structure, with implications extending from city delineation to organism morphology. It also frames border number and head/tail division as potentially useful beyond the immediate street-network analysis, while marking the organism analogy as conjectural.
- Massive volunteered geographic information and increasing personal-computer power enable large-scale analysis of geographic structure and patterns.The study highlights OSM, GPS traces, and geotagged photos as data sources for scaling analysis.
- The head/tail division rule may support analogous distinctions in other power-law or heavy-tailed phenomena, including wealth and street connectivity.The paper presents these implications as requiring further verification.
- The block perspective is proposed as a topological approach for interpreting organism morphology, with border number potentially identifying centers that geometric medial axes cannot reliably derive.The discussion contrasts topological block structure with geometric skeletonization for locating brain and heart centers.
- The proposed organism analogy compares city and field blocks with cells and suggests that complex organs may contain smaller units, but this remains unsupported speculation.The paper notes that the relevant units in brains and hearts may be unknown and lack scientific references.
- Scaling laws from biology have been reported to hold remarkably precisely for cities, motivating comparisons between cities and biological entities.The paper uses this prior work to relate urban form and function to biological scaling.
7. Conclusion
The conclusion presents block scaling as a way to characterize spatial heterogeneity and delineate urban areas from large street networks. It also extends the perspective to organism morphology as a conjecture requiring further verification.
- Block sizes at country and city levels exhibit lognormal distributions, with the mean dividing a low-percentage head from a high-percentage tail.The paper uses this head/tail division rule to distinguish urban and rural areas.
- The head/tail division rule provides a simple solution for delineating urban boundaries from large street networks.The conclusion presents this as an application of the block perspective.
- Border number captures the structure and patterns of geographic space and is particularly useful for detecting country or city centers.It is defined as part of the paper’s block-based perspective.
- The paper conjectures that similar scaling structures may occur in complex organisms, comparing geographic blocks with biological cells.The authors state that future work will verify these findings and applications of the head/tail division rule.