Characterizing the livingness of geographic space across scales using global nighttime light data

The hierarchical structure of geographic or urban space can be well-characterized by the concept of living structure, a term coined by Christopher Alexander. All spaces, regardless of their size, possess certain degrees of livingness that can be mathematically quantified. While previous studies have successfully quantified the livingness of small spaces such as images or artworks, the livingness of geographic space has not yet been characterized in a recursive manner. Zipf ’ s law has been observed in urban systems and intra-urban structures. However, whether Zipf ’ s law is applicable to the hierarchical substructures of geographic space has rarely been investigated. In this study, we recursively extract the substructures of geographic space using global nighttime light imagery. We quantify the livingness of global cities considering the number of substructures (S) and their inherent hierarchy (H). We further investigate the scaling properties of the extracted substructures across scales and the relationships between livingness and population for global cities. The results demonstrate that all sub-structures of global cities form a living structure that conforms to Zipf ’ s law. The degree of livingness better captures population distribution than nighttime light intensity values for the global cities. This study contributes in three aspects: First, it considers global cities as a whole to quantify spatial livingness. Second, it applies the concept of livingness to cities to better capture the spatial structure of the population using nighttime light data. Third, it introduces a novel method to recursively extract substructures from nighttime images, offering a valuable tool to investigate urban structures across multiple spatial scales.


Introduction
Understanding the complexity of geographic space or cities is a pivotal research goal in the field of geography and urban studies.Sustainable urban planning and design necessitate a profound comprehension of this complexity, essentially due to human activities are shaped the intricate geospatial structure.Spatial complexity is manifested in the hierarchical structure of the urban environment.Central Place Theory (CPT) (Christaller 1933(Christaller , 1966)), for example, initially outlines how urban centers provide goods and services to their surrounding hinterlands, with larger settlements offering higher-order services.A substantial body of literature underscores the importance of hierarchies to better understand the urban environment.For instance, Xu et al. (2021) delineate hierarchical urban boundaries using artificial impervious surfaces, facilitating spatial comparisons across different countries.Wang et al. (2022) quantify urban structural complexity using a hierarchical model.Considering the hierarchy in applications such as urban area delineation and urban structure identification provides deeper insights into understanding the complexity of urban function and structure.However, few studies truly consider taking a series of urban space hierarchies as a whole to capture the essence of urban structure.
With ongoing urbanization and evolving infrastructure, urban size and shape undergo significant changes.This complexity is often characterized by fractal geometry and the scaling properties of cities (Mandelbrot, 1982;Goodchild and Mark, 1987;Batty and Longley, 1994).It has been observed that cities' sizes and their ranks exhibit an inverse relationship, known as Zipf's law (Zipf, 1949).Cities at the global level comply with Zipf's law (Jiang et al., 2015), with some countries and intra-city hotspots also follow suit (Jiang and Jia, 2011;Veneri, 2016;Ma et al., 2021).Many studies concerning Zipf's law about urban systems are based on a single scale or a limited number of scales.For instance, Wu et al. (2018) utilize Zipf's law for mapping urban areas within a country.Wang et al. (2023) reveal the multi-fractal structure within a city.Chen and Wang (2014) discover that Zipf's law also persists through recursive subdivisions inside a city.Researchers have come to recognize that observing underlying scaling patterns through a multiscale lens is more insightful.
The notion of living structure denotes a phenomenon or structure with interconnected elements across scales, later refined into a mathematical model for computing the degree of livingness (Alexander, 2002;Jiang, 2015).Prior research by Jiang andde Rijke (2021, 2023) has demonstrated that livingness can be quantified through either recursive or non-recursive decomposition, relying on the number of substructures, their inherent hierarchies and number of recursions.Both methodologies effectively capture the objective beauty or degree of livingness at relatively small-scale spaces such as photos or paintings.Such livingness profoundly influences human perceptions of the urban environment, shaping what is commonly referred to as the image of cities (Lynch, 1960).The concept of living structure amalgamates notions of hierarchy and the holistic view of urban systems, offering a deeper understanding of urban complexity.This study builds upon previous research by extending its scope from relatively small spaces to the global geographic scale, utilizing nighttime light image data to illustrate the spatial distribution of human presence on the Earth's surface.
Geospatial big data has demonstrated its ability to accurately capture the dynamics of geographic space.Nighttime light data (NTL), a significant source of geospatial big data, has been extensively used to explore the relationships between urban structures and human activities (Jiang et al., 2015;Ouyang et al., 2019;Yang et al., 2021;Ma et al., 2021;Chang et al., 2023;Ren et al., 2024).NTL depicts the brightness and intensity of lights in different areas of a city at night, serving as indicators of activity level, economic prosperity, and development.Generally, areas with more and brighter lights at night are associated with higher levels of economic activity and urban vitality (Jacob, 1961;Liu and Shi, 2022;Wu et al., 2023;Chang et al., 2023).However, several research gaps persist in the current literature: Firstly, there is a lack of hierarchical generation of substructures as a coherent whole from NTL at the global level.Secondly, there has been limited investigation into the statistical examination of Zip's law or scaling law of these substructures collectively and individually.Thirdly, the relationships between the degree of livingness of substructures and the population within those substructures have not been thoroughly explored.
In the present study, we introduce a methodology for recursively decomposing global nighttime light images based on head/tail breaks (Jiang 2013).The resulting substructures at various recursion levels can be associated with conventional spatial scales such as regional, country, city, and intra-urban levels.Subsequently, we investigated the scaling properties of these substructures to uncover spatial patterns across scales.The livingness of substructures was quantified using both nonrecursive and recursive methods.Finally, we observed improved correlations between the degree of livingness and population within substructures compared to pixel values.The contributions of this paper lie in three aspects.Firstly, it marks the first instance of employing global nighttime light images to create substructures through a recursive method, enabling the formation of a cohesive whole for cross-country comparisons.Secondly, it represents the first exploration of Zip's law or power law properties of hierarchical substructures as a collective entity.The findings indicate that the recursive generation of substructures can serve as an effective approach for delineating urban boundaries at various scales.Thirdly, the enhanced correlations between L, LR and population distribution suggest that the living structure offers a superior model for studying urban structures and human activities in geographic space.
The reminder of this paper is structured as follows: Section 2 introduces the concept of living structures and describes why geographic space can be regarded as a living structure.Section 3 introduces the data and data processing methods, including how to generated substructures from NTL and how to quantify its livingness using different approaches.
Section 4 reports the results including the scaling properties at different levels and accessing our approaches using population data.We further visualize the living structure of global NTL using a graph.In Section 5, we delve deeper into the significance of adopting a holistic view of space and discuss its implications.Finally, we draw conclusions and point out future research directions.

The living structure of geographic space
The concept of living structure is closely related to the notion of 'wholeness'.The relationship between wholeness and living structure is analogous to the relationship between temperature and warmth.A structure with a high degree of wholeness is called a living structure (Jiang, 2015).The wholeness is a recursive structure made of centers at different levels of scale.Those centers are inter-connected in a recursive manner.Although Alexander called it 'a quality of without a name', it exists pervasively in an ornament, in a painting, in a room, in a building, and in a city (Jiang, 2015;Salingaros 2000).Since the concept of wholeness or living structure is latent and subtle, Alexander (2002Alexander ( -2005) ) distilled 15 properties to assess whether a structure is living or less living (Table 1).A living structure exhibits some of these properties, but not necessarily all of them.Jiang (2015) made a detailed description on 15 properties using Koch snowflake, we will briefly introduce these properties using an ornament and the NTL of Stockholm.
For example, in Fig. 1(a), an ornament from Alexander's classical work, 'the nature of order', exhibits a high degree of wholeness.Fig. 1(b) shows the abstracted graph in which each center represents the important parts of the ornament.The centers have five levels of scales (first property) and small centers support the large centers (strong centers) forming a coherent whole.Moreover, the centers also exhibit good shape, local symmetries, thick boundaries, deep interlock and alternative repetition.As a result, the structure of this ornament is a living structure.Previous studies have revealed images or paintings are living structures made of far more small centers than large ones (Jiang andde Rijke 2021, 2023).The real geographic world can be also regarded as a living structure based on the 15 properties.Fig. 1 (c) is the nighttime light image of the large Stockholm area.The brighter areas are corresponding to the urban areas with large population and human activities.The level of scales of the Stockholm NTL is represented as the centers in different sizes in Fig. 1 (d).There is one largest center, surrounded by six smaller centers.The smaller centers are supported by the smallest centers.There are three levels in total.The NTL also exhibits some of the 15 properties, including the property of roughness.The roughness indicates the fractal character of the centers.The cities are essentially fractals in terms of shape and distribution (Batty and Longley 1994).There are also properties of strong centers and not separateness, but with less symmetries and repetition.
However, determining whether a structure is a living structure using qualitative judgments is somewhat subjective.Some of the properties are also very latent and abstract because they are summarized from hundreds of examples by Alexander.Therefore, a quantitative approach to measure livingness becomes necessary.Previous studies have demonstrated that the livingness of an image can be computed based on the number and the hierarchy of substructures (Jiang and de Rijke 2023).As depicted in Fig. 1(a), there are 1 + 4 × 4 = 17 centers of five hierarchies, whereas in Fig. 1(d), there are 1 + 6 + 89 = 96 centers of three hierarchies.As a result, the livingness of the ornament is 17 × 5 = The hypothetical case of ornament can be analogized to real cities in geographic space, where centers represent cities, and the hierarchies of centers represent the hierarchies of urban centers and their corresponding hinterlands, as proposed in the CPT (Christaller 1933(Christaller , 1966)).There are also links connecting centers together to represent the interactions between urban elements.Such a model has been utilized in previous work to identify urban centers (Ren et al., 2024).As we can observe in Fig. 1(c), there are still complex structures inside the largest patch.It means that intra-urban structures are still very heterogenous (Ma et al., 2019).We will perform a recursive approach to decompose such centers until there is no more complex structures inside.By doing so, the complexity of urban structures can be clearly revealed across scales, providing us a deeper insight on the urban complexity.In next section, we will illustrate how recursive substructures of a living structure can be derived and how to quantify livingness using NTL.

Methods for characterizing the livingness of space
In this section, we introduce the methodology for characterizing the livingness of geographic space using NTL.The framework and datasets are introduced.Then we delve into how to generate substructures from NTL.Both non-recursive and recursive approaches are illustrated.After deriving the substructures, we first illustrate how can we quantify the degree of livingness using three different measurements (L, LR and V).Finally, we introduce the steps for statistical estimation of power law and Zipf's law.
Fig. 2 shows the framework of the present study.First, the global NTL is recursively decomposed at a series of recursion levels.The boundaries of these substructures are compared with those of FUA and GUC in a spatial context.Substructures at different recursion levels correspond to different spatial scales, such as regions, cities, and intra-city levels.Statistical analysis, including power law statistics, is performed on the substructures collectively and at each recursion level.Subsequently, livingness values are computed using both non-recursive (L) and recursive (LR) approaches for each city.To assess whether both livingness measures accurately represent human presence on the earth's surface, population grids from GHSL are spatially joined or aggregated to the substructure polygons of global cities.Additionally, NTL pixels are spatially joined to substructures to analyze the correlations between NTL and population using substructures as spatial units.The results and detailed analyses are presented in the following sections.

Data
In this study, four datasets were utilized: The first dataset is Global Nighttime Light Imagery (NTL), obtained from the Earth Observation Group of the Colorado School of Mines.This dataset provides highquality global nighttime light coverage using the annual VIIRS nighttime lights.The NTL data underwent initial filtering to remove sunlit, moonlit, and cloudy pixels, resulting in a rough composite.The masked average radiance from the year 2021 was selected for analysis, with background, biomass burning, and aurora filtered out (Elvidge et al., 2021).The resolution is 15 arc seconds (approximately 500 m at the Equator).The dataset comprises 82,170 × 35,531 pixels, re-projected to the world Mollweide equal-area projection for cross-country comparison.
The second dataset is Global Human Settlement Layer (GHSL), released by the European Commission.This dataset illustrates the distribution of residential population, represented as the number of people per cell.The data resolution selected for this study is 3 arc-seconds (approximately 100 m using the World Mollweide projection) for the year 2020 (Schiavina et al., 2023).The third datatset is GHS Functional Urban Areas (FUA), representing the commuting areas of urban centers as of 2015.This dataset aggregates administrative boundaries where people in the centers commute to (Schiavina et al., 2019).The last dataset is Global Urban Center Data from GHSL (GUC): This dataset comprises 13,135 urban areas identified based on a set of integrated multitemporal thematic attributes from GHSL (Florczyk et al 2019).The  datasets ( 1) and ( 2) are the main datasets.The datasets ( 3) and ( 4) serve as references to compare with generated substructures.

Computing the livingness of cities from nighttime light data
Any space, regardless of its size, possesses a certain degree of livingness, which can be quantified in terms of the intensity values of pixels or the areas of substructures or centers.The livingness of space can be measured using head/tail breaks (Jiang 2013), originally developed for classifying heavy-tail distributed data.We illustrate how it works to compute livingness using a set of 100 numbers that strictly follow Zipf's law (1, 1/2, 1/3 to 1/100).The head/tail breaks algorithm is a recursive process that divides the data into head and tail parts around its mean value.Each dataset undergoes its own number of hierarchies and a series of cut-off values without any imposed criteria (Jiang and Yin 2014).For example, in Fig. 3(a), the first head of 100 digits ranges from (1, 1/2, …, 1/19), and the first tail is (1/20, 1/21, …, 1/100), divided by the first mean value of 0.05.The second mean value of 19 numbers is around 0.19, continuously dividing the data into two parts.The second head part is in the range of (1, 1/2, …, 1/5).Finally, the last head (1, 1/2) is derived.Jiang and Yin (2014) suggest a relaxed percentage in the head part for the geographic features to stop iteration.
Previous studies (Yang et al., 2021) chose 30 percent as the threshold, in this study, in order to derive more iterations, we further relax the threshold to 49 precent.In other words, if the number of pixels in the head is larger than 49 % of the total number of pixels at each iteration, these pixel values are considered not to follow heavy-tailed distribution.
The recursive process objectively classifies the dataset into four parts, including three head parts and the first tail part.The ht-index is four that can be used to characterize the heterogeneity or the livingness of data (Jiang and Yin 2014).Statistically, the set of 100 numbers (1, 1/ 2, …, 1/100) is deemed more living than the set ranging from (1, 2, 3, …, 100) based on the ht-index.However, when these 100 numbers are mapped to geographic space, numerous patterns can emerge.For example, the above process generates three substructures (three heads) shown in Fig. 3(b).There can be more substructures if the pixels are clustered into more than one single cluster each time.Determining whether the spatial distribution of these numbers is living or not cannot solely rely on head/tail breaks.To address this issue, the non-recursive method has been introduced to quantify the livingness (L) of images (Jiang and de Rijke 2021).The livingness (L) is formally defined as: where S represents the number of spatial clusters or substructures, and H Fig. 3. Illustration of head/tail breaks as a recursive algorithm using 100 numbers (Note: the intensity values of fictional NTL are 100 numbers, and their substructures are recursively generated using head/tail breaks).shows the decomposable substructures in a graph, the nodes with the out-links are decomposable ones).
signifies the hierarchy of spatial clusters, same as the ht-index.This implies that a structure is considered more living if it contains more heterogeneous sub-clusters.The H reflects the heterogeneity of the substructures, S reveals the spatial pattern of the clusters.NTL serves as a reliable proxy for human activities, with each pixel recording the intensity value of artificial lights emitted from infrastructures (Lou et al., 2019;Wu et al., 2023).Previous research conducted by Jiang and de Rijke ( 2023) revealed that an image can be recursively decomposed based on the pixel values of its substructures.These decomposable substructures effectively capture the skeleton or saliency of images.
Their motivation stems from human perception, where parts of an image with higher degrees of livingness tend to attract more attention.Similarly, in NTL data, areas with higher intensities and heterogeneous intensity values typically exhibit greater urban vitality (Jacobs, 1961;Wu et al., 2023).The recursive method for quantifying livingness focuses on these decomposable substructures.The recursive livingness (LR) is defined as the sum of the livingness of these decomposable substructures induced by a recursive dichotomy process, as shown in Formula (2): where D represents the total number of decomposable substructures, S i and H i denote the number and hierarchy of substructures of the i-th decomposable substructure, respectively.The LR reveals the fact that the more substructures, and the higher hierarchy of the substructures, the more beautiful of an image (Jiang and de Rijke 2023).Alternatively, we can also use the structural beauty (V), to characterize the degree of livingness, V is defined as the number of decomposable substructures (D) multiply their levels of recursion (I): Fig. 4 (a) illustrates the process of recursive generating substructures from NTL.The NTL is dichotomized into bright and dark pixels each time.The brighter pixels are then clustered into three substructures during the first recursion.We assess whether these substructures can be further decomposed.After four rounds of decomposition, we derive the final substructure (shown in red in Fig. 4).Within this substructure, the intensity values cannot be further divided.In other words, the intensity values are similar with ht-index less than two, conforming to Tobler's law (1970).The substructures of four rounds recursive decomposition together constitute a living structure or coherent whole.In total, we derived 3 + 8 + 4 + 1 = 16 substructures, their ht-index is three for example, as a result, the non-recursive livingness L=16 × 3 = 48.Different from non-recursive approach, LR considers hierarchies and number of decomposable substructures at each recursion.Fig. 4 (b) shows the substructures in a hierarchical graph.The arrows point to the child nodes from the decomposable nodes, while dashed lines indicate that the substructures are within the same father node at the same recursion.It's important to note that this is a recursive process, so multiple mean values (M1, M2, …, M8) are created each time shown in Fig. 4 (b).During the first recursion, the original NTL is decomposed, resulting in 3 substructures, all of which can be further processed (represented by blue nodes in the graph in Fig. 4 (b)).In the second recursion, 8 substructures are generated.Four of them can be decomposed.Similarly, one out of four is decomposed in the third recursion until it reaches the red node.The total number of decomposable substructures is calculated as 3 + 4 + 1 = 8, while the total number of substructures is 3 + 8 + 4 + 1 = 16.For the 8 decomposable substructures, we calculate the L for each decomposable structure respectively and the LR is the sum of L of these 8 substructures.The first decomposable patch (largest blue area with M1 in Fig. 4 (b)) contains 6 substructures, assuming their ht-index is 3, so L 1 = 3 × 6 = 18.The rest of seven substructures only contains one patch, as a result, the L for them is 1.As a result, the total recursive livingness, LR=18 + 7 × 1 = 25.In addition, there are 4 recursions, the livingness can be also measured by structural beauty, V: 8 × 4 = 32.This working example shows how L, LR and V are computed from NTL substructures.
We notice that he percentage of decomposable nodes is much higher than in the images reported in Jiang and de Rijke's work (2021,2023).This difference arises because the aim of their work was to capture the degree of beauty of an image, with the algorithm attempting to mimic the human eye's perception process based on the principles of Gestalt Psychology (Koffka et al., 1936).Their decomposing stops if the ht-index is less than three in terms of size.However, in this work, we aim to exhaust all possible substructures at a finer scale so that the algorithm stops when the ht-index of pixels values is less than two.The small substructures in deeper recursion usually have higher intensity values, making them more relevant to the real world, where human activities are more concentrated.After deriving t hose substructures from NTL, next step, we will examine their statistical properties using power law estimation method.

Power law/ Zipf' law detection of substructures
Zipf's law is often employed to characterize the inverse proportional relationship between city size and its rank.This relationship implies that if we rank all cities in a system in descending order, the largest city is approximately twice as large as the second largest city, three times the size of the third largest city, and so forth (Zipf, 1949;Arshad et al., 2018).It is formally denoted by: s(r)∝r − 1 (4) where s presents the size or population of a city, r is the rank.Zipf's law or the Pareto distribution is more commonly observed in a Probability Density Function (PDF), representing the number of cities whose size is exactly x.It should be noted that the power law exponent for Zipf's law in the PDF is two rather than one (Newman, 2005).Generally, a power law is expressed as: Where X is some quantity such as city size or population, c is a constant and α is the power law exponent.Empirical data usually follows power law distribution from a certain lower bound, denoted as x min .The power law exponent α is estimated based on the maximum likelihood estimation (MLE) method (Clauset et al., 2009), formulated as following: where x i , i = 1 … n are the observed values of X such that x i ≥ x min .We can estimate the goodness of fit by comparing the observed data with the fitted power-law distribution.Kolmogorov-Smirnov (KS) distances or visual inspection of the rank-size plot can be employed.According to the KS method, typically, if p-value is greater than 0.05, the power law fitting is considered plausible (Clauset et al., 2009).

Characterizing the livingness of global NTL
In this study, we recursively generated substructures using global NTL as a whole.We examined the scaling properties of the generated substructures at individual recursions and collectively across all recursion levels.The substructures generated at the second recursion align with conventional urban and urban agglomeration boundaries.Subsequent substructures represent smaller areas within cities.The livingness metrics L, LR and V were calculated at the city level, considering all substructures across scales within each city.We also explored the correlations between the livingness metrics and the population of global cities, comparing them with NTL intensity values.

Substructures generated across scales using global NTL as a whole
Christopher Alexander, advocating for a holistic view of space, depicted a city is not tree but a semilattice or a complex network (Alexander 1965), challenging the notion of space as lifeless and emphasizing its potential as a living structure with varying degrees of livingness (Alexander, 2002).Applying this holistic view to the global nighttime light (NTL) data, we can demonstrate why it qualifies as a true living structure.Fig. 5 shows how substructures or centers connected forming a whole using global NTL as input data.Through each iteration of decomposition, high-intensity pixels are clustered, depicted as dots using a spectrum of colors (Fig. 5).The dots are connected if they belong to the same father dot at last recursion.Since all substructures are created from one data source, they are inter-connected with no isolated nodes (Fig. 5 (a)).All nodes in the graph can be located at specific locations shown in Fig. 5(b), (c), (d), (e).Spatially, the dark blue dots represent larger areas while red dots represent the smaller areas such as urban centers.
Despite geographical isolation, cities like Dallas, Los Angeles, the Yangtze River Delta Area (YRD) and the Guangdong-Hongkong-Macau Bay Area (GBA) are interconnected in the graph, as they are derived from the same NTL dataset using consistent criteria, i.e., head/tail breaks.Moreover, the intra-urban structures can be observed from the visualization of substructures.The dark blue dots represent cities or urban agglomerations, while the intra-city structures are depicted by its child dots in subsequent figures (e.g., Fig. 5(f), (g), (h) and (i)).For example, Fig. 5(b) and (f) illustrate the polycentric structure of the YRD,   containing cities like Shanghai and Suzhou.Similarly, Fig. 5 (c) and (h) depict the GBA, including cities like Guangzhou, Shenzhen, and Hong Kong.This visualization highlights how geographic space forms a living structure with interconnected centers at different levels.It's important to note that a city is not a tree but rather a complex network, as highlighted by previous studies (Alexander, 1965;Batty, 2009;Jiang, 2015;Huang et al., 2022).Links between dots exist if they belong to the same father substructure, as illustrated in Fig. 4(b).The visualization not only shows the hierarchical levels of urban system, but also demonstrates the inner relationships between the substructures with the system.

Scaling properties of substructures across scales
We utilized entire global NTL to generate substructures.In total, 102,678 substructures are derived in eight recursion levels (Fig. 6).Generally, the power law exponents exhibit an incremental trend (Table 2).Notably, none of the recursion levels exactly comply with Zipf's law, with α values exceeding 2.0.Surprisingly, we observed that the power law exponent of all substructures closely approximates 2.0 (±0.03), with a p-value of 0.30.Consequently, the global substructures collectively constitute a coherent whole that adheres to Zipf's law.We also employed head/tail breaks at each recursion.In Table 2, the htindex of level six is 7, differing by only one hierarchical level compared to the first recursion.However, the number of substructures accounts for only 2.7 percent of that in the first recursion.It suggests that the substructures exhibit self-similarity across multiple scales.Therefore, it becomes imperative to adopt a multi-scale analysis approach when dealing with heterogeneous geographic data (Chen and Wang, 2014;Chen and Jiang, 2018).Next, we shift our focus to examining parts of the substructures.
In this study, we did not strictly divide the data according to geographic-defined continents.Instead, we partitioned the global data into 8 parts where the substructures are relatively coherent or clustered together.For example, North America, South America, and Africa continents are geographically isolated from other continents, so we regarded these continents as three individual parts.At the continental level, Zipf's law generally holds for most parts, with α values close to 2.0 (±0.05).In Africa, Central and West Asia, the α values are slightly lower than 2.0.In South Asia, the α is very close to 2.0, however, the p-value equals zero (Table 3).The hierarchies of substructures for each part have also been visualized in Fig. 7.The large dots represent the large substructures and the size of dots are classified using head/tail breaks.The ht-indices (H) in East and South Asia, North America, South America are seven.This could be because they contain the largest countries in terms of population, such as China, India, USA and Brazil.The Oceania and South Asia have the least number of substructures, but it has a higher H than Africa, indicating the substructures are more living than those in Africa.The high ht-indices indicating the substructures at continental level is heterogenous and they are self-similar to the whole dataset.After examining the continental level, we zoom into the individual countries to see if the Zipf's law still holds.
We examined the power law for 145 countries with a number of substructures greater than 10.At the country level, the substructures in Fig. 7. Substructures of different regions of world (Note that the dot size represents in the hierarchical levels of substructures at each continent, H is the hierarchical levels of substructures of size).(Note: S denotes number of substructures, H represents their hierarchies induced by head/tail breaks, α is the power law exponent, S-ratio is the ratio between S and area in each city).
most countries violate Zipf's law.In fact, only 23 out of 145 countries exhibit substructures that follow Zipf's law (see Appendix).The average α for the 145 countries is 2.17.We also found that the number of substructures (S) in these 145 countries follows a power law distribution with α equals to 2.28 and a p-value of 0.83.
Table 4 shows that none of the countries strictly comply with Zipf's law.We also found that the ratio of S and area (S-ratio) reveals that Italy and France have the greatest number of substructures per unit size.The violation of Zipf's law at the country level suggests that administrative boundaries for countries might not be legitimate spatial units for Zipf's law (Jiang et al., 2015).Many urban agglomerations are spatially connected, but they are forced to be separated by country boundaries.These occurrences are common in Europe and North America (Fig. 7(a), (b)).In this study, the substructures are objectively defined spatial units at multiple scales, allowing them to better capture urban structure and human activities than conventional city boundaries.In the next section, we will quantify the degree of livingness for substructures at the city level and explore the potential of using the living structure to predict human presence patterns on the earth's surface using NTL.

Correlation analysis on the livingness of substructures
In this section, we quantified the livingness of global cities using both non-recursively metric (L) and recursive metric (LR and V).Previous  studies have shown that NTL has low correlations between intensity values and population count at the grid level (Wu et al., 2023b).To address this, we aggregated the pixels from NTL to the 10,605 decomposable substructures and counted the population.Next, we calculated L, LR and V for these cities using the methods mentioned in Section 3.2.Finally, we performed linear regression between population, NTL pixel values, L, V and LR (Fig. 8).Globally, the R-squared (R 2 ) value between population and NTL is 0.41.This indicates a moderate correlation between population and NTL pixel values.However, when considering the livingness metric L, the R 2 increases to 0.57.We further calculated LR based on the decomposable substructures inside each city.The R 2 of LR and population is 0.71.The structural beauty V, however, has a lower R 2 of 0.44, compared to L and LR.
We further verified the above results using substructures in top ten countries (Table 4).The correlations between population and NTL, L, and LR and V are respectively 0.5,0.62,and 0.80,0.51 (Fig. 8 (b)).The high correlation of L, especially LR compared to NTL, suggests that our approaches to quantify livingness are capable of revealing the underlying spatial structure of geographic space and capturing human activities using NTL data.Overall, these results demonstrate that the living structure offers a more accurate representation of human activities, making it a valuable tool for analyzing urban structures and population patterns.

Discussion and implication
This study verifies that all substructures from global NTL constitute a whole that complies with Zipf's law.Unlike previous studies that commonly examine Zipf's law at an individual scale, such as the city level or intra-city level, this study considers a set of urban agglomerations (recursion 1), cities (recursion 2), and intra-urban hotspots (recursion 3 to recursion 8) as an interconnected or coherent whole.As shown in Fig. 9, the substructures at the first recursion have a similar spatial scale to the FUA that consider human commuting extent outside the city.We further conducted Intersection over Union (IoU) analysis on the substructures, FUA, and GUC.The IoU index indicates the spatial overlapping ratio of two spatial features (Ma et al., 2019) reflecting their geometric relationships.We found that substructures at the first recursion are, on average, 1.7 times larger than FUAs, with an IoU index of 0.39.The substructures at the second recursion are 1.3 times smaller than GHS urban centers, with an IoU index of 0.5 (Fig. 9 (a)).The substructures at the second recursion are close to the GHS city boundaries (Fig. 9 (b)).As a result, we choose the second recursion as the global city boundaries to compute the L and LR for the substructures.
Researchers found that some urban systems deviate from Zipf's law, possibly due to the bias of city definitions (Jiang and Jia, 2011;Veneri, 2016;Ortman et al., 2020;Moreno-Monroy et al., 2021).For example, Chen and Jiang (2018) argue that natural cities better characterize urban structure based on naturally defined units.Cao et al. (2020) define city boundaries using multi-source data based on percolation theory and find that Zipf's law exists in some countries.Moreover, Cristelli et al. (2012) highlight data sample coherence as another important factor for violations.Consequently, the proper definition of cities and considering all scales as a coherent whole when examining scaling properties are vital aspects often overlooked in the literature (Paolo 2016, Lestrade, 2017;Jiang et al., 2015;Moreno-Monroy et al., 2021).The boundaries of FUAs are the aggregation of administrative boundaries that don't follow Zipf's law and power law.On the contrary, like natural cities proposed by Jiang andJia (2011), Jiang et al., (2015), the boundaries of substructures are naturally defined from the intensity values of NTL using head/tail breaks.Moreover, the multi-scale boundaries are essentially self-organized.In this study, the multi-level boundaries are derived based on recursive approach using multiple mean values at each recursion, which is different from the non-recursive approach that based on single cutoff value (Yang et al., 2021).The recursive approach regards those undecomposable structures with high intensity pixels as low degree of livingness.The multi-scale boundaries of substructures provide a more accurate and valuable data source for urban studies.
The correlation results show the potential for comparing the livingness of individual cities based on their population.If a city has relatively fewer populations but a large L or LR, it is structurally more living or heterogeneous.Fig. 10 shows 15 cities with large populations and degrees of livingness (LR).For example, the Yangtze River Delta (YRD) area, including Shanghai, is structurally more living than the Guangdong-Hong Kong-Macao Greater Bay Area (GBA) in China.Bengaluru is more living than Mumbai and Kolkata in India.Tehran and Moscow are more living than Beijing.And New York is more living than Istanbul.The proposed method can effectively identify the different urban structures.Different cities in same country exhibit different urban structures (Ma et al., 2024;Shi et al., 2023).The proposed method can not only differentiate the urban form but also make cities across country comparable.Those polycentric cities such as Shenzhen and Shanghai tend to have higher degree of livingness.On the contrary, those monocentric cities such as Beijing and Moscow tend to have lower degree of livingness.
The night-time light image is a good data source to reflect the human activities at night (Wu et al., 2023b;Hu et al., 2024).Compared to the conventional method such as KDE, this study objectively derives the cutoff values based on the data itself using head/tail breaks.This provides a practical way to delineate urban boundaries and inner-urban hotspots at multiple scales without finding proper thresholds subjectively.The proposed method is born out of and inspired by the CPT (Christaller 1933(Christaller , 1966)).It has been proved that the way using head/tail breaks to delineate urban structures is useful to differentiate urban configurations such as mono-centric cities in Europe and poly-centric cities in China (Loo and Huang, 2023;Hu et al., 2024).To elaborate further, there are two fundamental laws of geography that play a crucial role in shaping human activities: Tobler's law and the scaling law (Tobler, 1970;Anselin, 1989;Goodchild, 2004;Batty, 2008;Jiang and Yin, 2014).Tobler's law, often referred to as the first law of geography, posits that spatial features tend to be similar within the same scale.On the other hand, the scaling law describes the heterogeneity of space across different scales, with a prevalence of smaller entities compared to larger ones (Jiang, 2013;Jiang and Yin, 2014).This study effectively incorporates both Tobler's law and the scaling law.The way to study urban structure is akin to peeling an onion, understanding how this complexity forms through recursive decomposition, urban planners and policymakers can design more resilient urban environments with more substructures.

Conclusion
The concept of livingness, inherent in any space, can be effectively captured in geographic space using NTL.By quantifying the degree of livingness through the number of substructures and their hierarchical organization, we gain valuable insights into the urban environment.These substructures are derived from NTL data through recursive decomposition of pixels based on their intensity values.Unlike conventional definitions of cities or urban boundaries, these substructures emerge naturally and objectively from the data pattern itself, allowing for a bottom-up representation of urban structures.The increased R 2 between population and measures such as L, LR suggest that the livingness of substructures can not only differentiate between structural beauty or degrees of life in different images but also effectively capture human presence patterns with in urban environments.
By examining Zipf's law at different recursion levels collectively and individually, we find that while it may be violated at individual recursion levels, it holds true when considering all substructures together.This suggests the importance of coherence in data samples for observing Zipf's law.Geographic space is not fragmented but interconnected, with a prevalence of smaller, low-intensity centers compared to larger, highintensity ones.Collectively, all substructures form a coherent whole, enabling comparisons of livingness across different geographic scales, from continents to cities and inner-city areas.The graph visualization of substructures vividly depicts the geographic space as a whole, portraying cities as complex networks.This study opens up potential research avenues for integrating social-economic factors such as population, GDP, CO 2 emissions, and energy consumption with the living structure of geographic space from a complex network perspective.By doing so, it can provide valuable insights for sustainable and resilient urban planning practices.

Data and code availability statement
The data used and generated in this study are packed and put publicly accessible at the Figshare site: https://figshare.com/s/b046f8f8844f759576ce.The package includes (1) Python scripts for recursively decomposing NTL, (2) Excel files with results of global NTL, and (3) Shapefiles for the results.The scripts use python 3.10 (https://www.python.org/download/releases/3.10/).The visualization and analysis software used is ArcGIS 10.8 (https://www.esri.com/en-us/arcgis/products/arcgis-desktop/overview), and Gephi 0.10 is used for network analysis (https://gephi.org/).

Table A1
Power law statistics for 145 countries in the world (Note: Alpha is the power law exponent, X min is the lower bound, S is the number of substructures and p is the goodness of fit).
(continued on next page) Z. Ren et al. International Journal of Applied Earth Observation and Geoinformation 133 (2024) 104136 Table A1 (continued ) Z. Ren et al. International Journal of Applied Earth Observation and Geoinformation 133 (2024) 104136

Fig. 1 .
Fig. 1.Living structure of an ornament from Alexander's book (2002-2005) and living structure generated from real geographic space.(Note: (a) is the original ornament, (b) is the graph of the ornament, in which hierarchies are represented by dot size, (c) is the NTL of large Stockholm area, (d) is the graph of Stockholm and its surrounding cities and towns).

Fig. 2 .
Fig. 2.Framework for characterizing the livingness of geographic space using NTL.(Note: the three main analyses are listed including power law estimation, correlation and IoU analysis).

Fig. 4 .
Fig. 4. Recursive generation of decomposable substructures from nighttime light data.(Note: Fig. 4 (a) shows the recursive way to decompose the NTL, Fig. 4 (b)shows the decomposable substructures in a graph, the nodes with the out-links are decomposable ones).

Fig. 5 .
Fig. 5.The living structure of global cities from NTL forming a whole (Note: the nodes in different colors represent the substructures generated at different recursion level, the connected links show the spatial relationships between those substructures).

Fig. 6 .
Fig. 6.Recursive substructures from global nighttime light image and their power law plots (Note: (a) shows global substructures at eight recursions, (b), (c), (d), (e) are zoomed-in areas for USA, Brazil, Europe and China).

Fig. 8 .
Fig. 8.The correlations between population and livingness metrics (Note: (a) shows the correlations of all cities; (b) shows the correlations of substructures in top ten countries).

Fig. 9 .
Fig. 9. IoU overlay of substructures, FUA and GHS urban centers (Note: (a) denotes the spatial differences between substructures at first recursion, R1 and FUA boundaries; (b) shows the spatial patterns of substructures from second recursion, R2, and GUC boundaries).

Fig. 10 .
Fig. 10.The scatter plot between population and LR for top ten countries.(Note: the cities in different countries are shown in different colors, the lower population with higher LR indicates the city is more living in terms of structure of NTL).

Table 1
15 properties of living structure.

Table 2
Power law statistics for global substructures.

Table 3
Power law statistics for substructures at continent level.

Table 4
Power law/living structure statistics for substructures in 10 countries.