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ABSTRACT Urban scaling models assume homogeneous stochastic processes across a city system exhibiting power-law behavior. Strong persistent systematic factors may, however, cause deviations

from scaling curves and reflect local heterogeneities in processes that govern individual cities’ conformity to system mean trends. We conduct a scaling analysis across several quantitative

properties of municipal pollution performance in Chinese cities and verify the findings using data from Japan, South Africa, India, and European cities. A generalized truncated power-law

model is proposed to capture the region where performance metrics for small cities fall below the linear-fitted curve. We hypothesise that the truncation arises from limited capability of

smaller cities in performing pollution abatement activities requiring high process complexity. Pollution control strategies that facilitate the transition from a fragmented response of each

small city to a collaborative mode can provide a solution to even-out lags and inequalities across the national urban system. SIMILAR CONTENT BEING VIEWED BY OTHERS WORLDWIDE SCALING OF

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Article Open access 24 January 2025 URBAN AND AIR POLLUTION: A MULTI-CITY STUDY OF LONG-TERM EFFECTS OF URBAN LANDSCAPE PATTERNS ON AIR QUALITY TRENDS Article Open access 29 October 2020

INTRODUCTION The unprecedented increase in number and size of cities since the 1950s1,2,3 has reshaped the Earth’s surface and changed the complex relationship between humans and nature4,5.

Urban population growth is both humanly productive and environmentally damaging. More people on the planet means more consumption of resources6 and higher waste generation, especially

municipal pollution (e.g., domestic sewage and solid waste)7. Meanwhile, as more of the human species cluster in cities, their interactions drive nonlinear increments in production,

productivity, and output of all social products, good and bad8. The urban sprawl, rising population density, and greater energy input and industrial output associated with urbanization all

exacerbate man-environment conflicts. Optimum city size has long fascinated urban scholars9 as a normative issue. Scaling studies provide an approach for examining the question empirically

in respect of a human population’s revealed preference for clustering, and size-adjusted environmental performance10. It remains unclear, with the present state of knowledge, whether large

or small cities perform better in managing their pollution. The processes are clearly complex. While the question can be examined by econometric modeling to disambiguate contributing

factors, in this paper, we limit ourselves to capturing whole-system complexity effects in the urban scaling tradition, taking population as an indicator of complexity. We depict urban

environmental performance by a general power-law function \(Y={Y}_{0}{N}^{\beta }\), where \(Y\) represents the quantity of an environmental indicator, \({Y}_{0}\) is a constant, \(N\) is

urban population, and \(\beta\) is scaling exponent11,12,13. Scaling studies of urban environmental attributes are few14,15, and primarily focused on emissions16. Mixed findings have been

reported, with scaling exponents of carbon emissions ranging from 0.31 to 1.5117,18,19,20, municipal wastes from 0.85 to 1.1521,22 and air pollutants from 0.18 to 1.227,23,24,25, varying

across study context. Emissions are only part of the story, however. A city’s accumulated infrastructural capability for pollution abatement, meaning the technological and operational

capacities dedicated to mitigating environmental externalities, also matters when determining city size-performance relationships26. The widely-validated sublinear scaling of generic urban

infrastructure (e.g., road, urban impervious surfaces), with exponents of around 5/627,28,29, is driven by economies of scale. However, pollution control infrastructure operates under

fundamentally different driving factors and constraints. Whereas road and impervious land expand to accommodate existing and projected growth of human interactions, pollution control

infrastructure may respond to accumulated, sometimes unsolved environmental pressures or regulatory shocks. Its development is demand-driven and contingent upon waste generation patterns,

treatment targets, and socioeconomic adaptive capacity. A lack of understanding of this scaling leads to questions about how city size tends to intensify or alleviate the balance between

pollution generation and pollution control. It may be, for example, that some kinds of waste grow linearly with population size while the corresponding waste treatment infrastructure grows

less than per capita. There are two opposing effects as cities of increasing size attempt to manage their waste. Larger cities have been found to generate higher absolute volumes of

pollution at an aggregate level30,31. Meanwhile, larger cities also exhibit stronger economic and technical capacities to tackle pollution as a whole, since cities’ wealth and innovation

increase at a faster rate than population growth15,32,33 due to agglomeration economies. They foster knowledge spillovers, labor-market pooling, and closer linkages among transaction

participants compared to smaller cities. This is especially so for activities with higher complexity (i.e., those requiring a deeper division of knowledge, labor, capital, as well as

government and market institutions34,35). This implies spatial inequality in treatment capacity and outcome across cities. Urban scaling laws of these interconnected environmental attributes

are essential for revealing the opposing relationships at a whole city system level, which remain underexamined. Doing so, as we show, reveals an issue that is not yet fully addressed in

urban performance modeling: the treatment of parts of an urban system curve that deviate systematically from the expected log–log linearity. Our paper investigates this for environmental

performance quantities that deviate negatively from the system trend at the lower end of city size. We explore the use of truncated scaling models and caution against over-simplified

interpretation of traditional power-law models. By definition, scaling methods embed a strong homogeneity assumption across scale32,36, deriving the exponent \(\beta\) by a linear fitting of

log-transformed urban attributes versus population size. This will result in biased estimates of the scaling exponent when the performance-size scaling relationship is heterogeneous across

the city size range. If urban environmental performance varies with size-related factors such as affluence and technology, captured by systematic deviation from a power-law trend at certain

size ranges, then there is an argument for endogenising the effect in a modified scaling model37,38,39. Strong persistent systematic factors can lead to truncations in the heads or tails of

the distribution for many attributes in natural or anthropogenic systems40,41,42,43,44, normally regarded as a fall-off region and reflecting start-off or fall-off dynamics. Under this

circumstance, a simple log–log OLS (ordinary least squares) model is misleading since the truncation is ignored by, and is likely to distort, the fitting process42, thus limiting our

understanding of heterogeneities in stochastic processes across a city system. Researchers have proposed different methods to address issues in the tail of the power-law distribution, such

as adopting an unbiased test method45 or a switching model46 between the lognormal and the power-law; fitting with stepwise functions47; and proposing new functions with truncated parts to

capture nonlinearities40,41. This paper aims to propose a functional form for scaling in urban pollution control that exhibits satisfactory performance in terms of goodness-of-fit, broad

applicability to pollution-related metrics, and high interpretability which can inform urban environmental management strategies. We compared the proposed model with the traditional scaling

model and several other truncated ones, and discussed their performance with multi-year datasets. Does population size enhance or reduce a city’s ability to realize pollution control? Using

population size to represent the aggregated motivations and resourcefulness that generate social products, is a fundamental assumption in city scaling. It links to city centrality, function,

and tax revenues that can affect resource allocation for environmental enhancement. Environmental waste is a by-product of the collective endeavors of an urbanized population and

environmental waste management systems are a collective response. We conduct a scaling analysis across a set of quantitative properties relating to municipal pollution in China and verify

the findings using data from Japan, South Africa, India, and European cities. We find that the traditional homogeneity hypothesis in city scaling fails to characterize the statistical

properties of urban pollution control features as the process complexity increases, and we propose a truncated power-law model to estimate scaling exponents which better depict the pattern

with a fall-off region corresponding to small cities. The identified threshold size of a national urban system implies cities that need special policy attention. In sum, modeling the

truncation rather than ignoring it, provides a more nuanced understanding of the size-related insufficiency of cities to cope with the complexity of pollution control activities. RESULTS

SCALING OF MUNICIPAL POLLUTION GENERATION The scaling of urban pollution generation is an appropriate first step in assessing whether pollution control capacity matches pollution treatment

demand as population increases. We found sublinear scaling properties (Fig. 1a), which imply that inhabitants living in larger Chinese cities generate less solid waste per capita on average.

We also found similar sublinear scaling patterns in German cities (\(\beta =0.92\)) and Japanese cities (\(\beta =0.97\); Figs. S1 and S2), indicating that solid waste production appears to

display economies of scale. This is surprising, since city-level per capita solid waste generation has widely been found to correlate positively with GDP48,49, as a wealthier economy tends

to have more extravagant and resource-consuming lifestyles. But urban scaling models capture multiple effects of human interaction simultaneously. The result corroborates previous research

that lower levels of demand management, environmental regulation, and public education in smaller and poorer cities lead to higher per capita waste generation compared to larger cities50,51.

This indicates that the weaker capacity in small poor cities to collectively implement advanced waste management overrides the income effect (demand for more waste in larger and richer

cities), exhibiting overall increasing returns in costs of urban economy. From 2000 to 2010, there was a significant decrease in solid waste generation in small cities in China (as indicated

by the transition from the purple to green fitting line in Fig. 1a), suggesting a more environmentally friendly consumption behavior. During 2010–2020, waste generation increased more

substantially in large cities (evidenced by the shift from the green to pink fitting line), which suggests a stronger income effect. These two trends contribute to a shift toward a more

linear scaling exponent over the observation period. In contrast to the pattern of solid waste, larger cities generate municipal wastewater in disproportionately higher quantities with

respect to population size (Fig. 1b). This is another surprising finding since water consumption is expected to scale linearly with population size, being an urban attribute closely related

to individual needs11,52. Our empirical observations for 144 European cities (\(\beta =1.04\)) and 414 Class I cities in India (\(\beta =1.06\)) also support a mildly superlinear

relationship (Figs. S3 and S4). Water resources are essential for maintaining high-quality urban amenities, and social interconnectivity in large cities can intensify the need for extra

water addition to per capita private consumption demand53. Cities attract inhabitants by the public shared goods supplied. The larger the cities, the greater the ratio of public to private

goods54, and most public goods, like any other product, generate wastewater. We test this assertion in Fig. S5. In addition, the underpricing of water may amplify its consumption in urban

production as cities grow. Similar to the trend of solid waste generation, a more linear scaling exponent can be observed for wastewater over time (from 1.18 to 1.08). From 2000 (purple

line) to 2020 (pink line), a reduction in water demand can be observed across all cities, with a more pronounced water-saving effect in larger cities, likely due to their advanced

facilities, water prices adjustments, and water-consciousness campaigns. In addition, we note that for both solid waste and wastewater, the fit becomes tighter over the 20-year period, as

evidenced by higher values of _r_2, indicating a maturing of China’s urban system in terms of waste production. Both have become more systematically related to population size as urban

inhabitants and governments learn how to live more efficiently in cities of all sizes. We note that this was a very dynamic period in China’s urban history in terms of personal enrichment,

economic deepening, and the rapid evolution of municipal government in a mixed-market economy55. SCALING OF MUNICIPAL POLLUTION CONTROL Pollution treatment can be expected to exhibit a

similar scaling pattern to pollution generation if pollution control capacity does not vary significantly across cities. As Fig. 2 depicts, compared with municipal solid waste generation in

2020 (\(\beta =0.97\)), the quantity, capacity, and infrastructural support for municipal solid waste treatment all indicate a fit within the bounds of expectation. All three wastewater

treatment exponents also correspond closely to the generation exponent (~1.08). These consistencies, especially the same exponents for waste production and quantity treated, indicate

efficient and adequate treatment across city sizes in China, evidencing the governments’ claim of 99.7% and 97.5% treatment rates in 2020, respectively56. We note that solid waste treatment

capacity and trucks and equipment quantity are more varied across cities than actual treatment, with a greater spread of under- and over-capacity compared to the population-size-related

means. The latter may be attributed to variations in the capacity of equipment which affect their deployment quantity. For wastewater treatment the fits are generally tighter, and we suggest

that the difference reflects the difficulties that municipalities have in siting and upgrading municipal waste facilities, be they landfills or processing plants. TRUNCATION IN THE SCALING

OF POLLUTION CONTROL WITH HIGHER PROCESS COMPLEXITY In this section, we show that other urban waste treatments with varying complexity display deviations from scaling laws, which can be

better captured by a truncated power-law model. Sewage sludge is a main secondary pollutant in wastewater treatment, which needs to be properly treated57 in every city. Historically, the

sludge treatment rate has been much lower than wastewater in China due to factors such as incomplete separate drainage systems, strict selection criteria of techniques, and limited options

for final disposal58. Figure 3a depicts the scaling of sewage sludge treated in Chinese cities (green triangles) in 2020. A traditionally fitted power-law function yields a superlinear

scaling relationship with a \(\beta\) of 1.29 (black straight line in Fig. 3a). However, there is a fall-off region, where small cities tend to fall below the national trend. To capture the

deviation, we investigated the power-law with exponential cutoff59 and fitted a truncated power-law model as follows, adding an exponential term \({e}^{-\frac{k}{N}}\) to the traditional

urban scaling model to characterize the fall-off in small cities. $$Y=a{N}^{\beta }{e}^{-\frac{k}{N}}$$ (1) where \(\beta\) represents the underlying power-law exponent when the truncated

region has been corrected for via \(k\); \(k\) is a parameter linking to environmental performance shortfalls in small cities compared to what they should behave; and \(a\) is a

normalization constant. \(k\) does not indicate a specific cut-off point, but rather characterizes the extent of underperformance experienced by small cities, estimated under the condition

of best fit using the Levenberg–Marquardt algorithm. For a given value of \(k={k}_{0}\), cities with population \({k}_{0}\) would, on average, suffer an evident decay rate of 63.2%

(\(1-{e}^{-1}\)), while cities with population \(10{k}_{0}\) correspond to a considerably lower decay rate of 9.5% (\(1-{e}^{-0.1}\)). Urban systems with higher values of \(k\) have a more

truncated pattern, that is, smaller cities perform worse, because of the K/N term in the second exponent expression. When \(k\)=0, Eq. (1) is reduced to the traditional power-law function,

indicating no truncation. Equation (1) thus has a universal form that applies to distributions that are truncated or not. We also proposed two alternative truncated models (Eqs. (2) and (3))

based on a review of the literature, and compared the fitting performance of all truncated and non-truncated traditional scaling models from the perspectives of goodness-of-fit (using _r_²

and Adjusted-_r_²), prediction accuracy (using Mean Squared Error and Root Mean Squared Error), model parsimony (using Akaike Information Criterion), and residual normality (using

Kullback–Leibler divergence) (see methodological details in the “Methods” section). $$Y=a({N}^{\beta }-{c}^{\beta }),$$ (2) $$\begin{array}{c}Y=\left\{\begin{array}{cc}a{N}^{\beta 1} &

(if\,\mathrm{In}\,N\,\ge \,c)\\ a{N}^{\beta 1}{e}^{\beta 2(\mathrm{ln}N-{\rm{c}})} & (if\,\mathrm{In}\,N\, < \,c)\end{array}\right.\end{array},$$ (3) where \(\beta\) in Eq. (2) and

\(\beta 1\) in Eq. (3) represent the underlying power-law exponent; \(a\) is a normalization constant; \(c\) in Eq. (2) is the parameter modeling the performance decay, and \(c\) in Eq. (3)

is the cut-off point of two functions. Table 1 reports the fitting performance of each model using empirical data in Fig. 3a, with parameter estimation results and fitted images shown in

Fig. S6. All truncated models exhibit better goodness-of-fit and higher prediction accuracy compared to the traditional scaling model. The improvement in fit outweighs the penalty for added

model complexity (lower AIC value), and the residual distribution is closer to the standard normal distribution (lower Kullback–Leibler divergence). Among the three truncated models,

although Eq. (3) performs slightly better than Eq. (1), we choose the latter for two reasons. First, Eq. (3) shows poor performance in other empirical cases discussed later, whereas Eq. (1)

can be widely applied. Second, Eq. (3) yields a specific cut-off city size (\(c\)), excluding cities below this size from the scaling analysis entirely. This approach does not align with

understanding urban dynamics from a whole-system perspective, while Eq. (1) applies over the entire range of empirical data, capturing continuous changes. As Fig. 3a shows, the sludge

scaling distribution for all cities is better approximated by Eq. (1) (red curve) compared to a non-truncated scaling curve (black line). We further performed a stepwise exclusion of small

cities and re-estimated scaling exponents using the traditional method, generating a series of descending values approaching the estimated underlying power-law exponent as more cities were

excluded (Fig. 3b). This comparison reveals that city groups of different size ranges have different scaling performance using traditional urban scaling analysis. This contradicts the

scale-free homogeneity assumption central to simple allometric theory, necessitating incorporation of scale-dependent mechanisms. A traditional scaling model, in this case, overestimates the

exponent due to systematically lower performance of the smallest cities. The bias risks misinterpretation and misunderstanding, such as false determination of superlinear or sublinear

scaling classification, overestimation of agglomeration economies of scale and returns to scale, and inflated expectations for large cities when estimating city performance according to

population size60,61. The more general issue is the stability of \(\beta\) exponents and appropriate functional form61,62,63. Responding to nonlinearities in log-transformed scaling models

and to instability in scaling exponents with urban boundary definition, researchers have modeled fluctuations in \(\beta\) with modifiable urban units61; identified scaling region only in

part of the distribution64; and fitted multiple functions to different portions of the distribution47. Rather than excluding deviant cases, our approach rectifies \(\beta\) by modeling

deviations directly in the fitted system curve. This has the advantage of parameterizing the truncation as part of the system, at the same time as preserving a best fit for the non-truncated

part. Figure 3b illustrates that under a size threshold of approximately 500,000 persons, cities tend to underperform in pollution control as expected. Figure 3c–e illustrate the

application of Eq. (1) and non-truncated models to multi-year datasets (2006, 2010, and 2015), with comparisons of fitting performance across alternative models reported in Table S1–S3.

Although Eq. (1) does not always perform best across all indicators in different cases, it exhibits good performance regarding goodness-of-fit and prediction accuracy. More importantly, the

change of parameter \(k\) in multiple years shows a system-wide improvement of sludge treatment performance and has policy relevance. Over the 2006–2020 period, \(k\) declines from 19.91 to

9.12 (Fig. 3a, c–e), indicating a shrinkage of the truncation effect and revealing that small cities in China continued to catch up with the average level of sludge treatment. This is seen

in the trajectory of quantity of sludge treated for individual cities (Fig. S7). We interpret this to mean that the complexity of sludge treatment no longer exerted the same level of

constraint on small cities that it once did. The truncation threshold population of around 0.5~1 million with possible fluctuations, corresponds to small- to medium-sized cities in China’s

official classification of urban hierarchy. Adding a policy focus on smaller cities, such as by extra subsidy for their sludge treatment, is likely to improve the environmental performance

of the whole Chinese urban system. This exemplifies how urban scaling studies can guide national and regional urban and environmental policy, by regulating the systematic influences on the

residual curve65. Comparison of truncated features (existence and degree) in scaling patterns reveals discrepancies in the sufficiency level of pollution control. Figure 4a indicates a

truncation point in municipal wastewater secondary and tertiary treatment capacity in the 2000 distribution, with truncated exponent \(\beta =1.10\) and \(k=12.03\). The non-truncated

\(\beta\) of 1.28 is misleadingly high due to the low values of the smaller cities bending the curve upward61. In 2000, establishing adequate treatment capacity was still a challenge for

many small cities. By 2010, the \(\beta\) estimate remains almost the same at 1.09, but truncation is barely evident, indicating that small cities in China had eliminated the capacity gap,

developing sufficient capacity to deal with complex secondary and tertiary wastewater treatment and better conforming to the overall superlinear scaling. The trend is captured in the plotted

trajectory for individual cities (Fig. S8), the 2020 \(\beta\) of 1.10, and an even tighter fit with no truncation. We take the consistency of \(\beta\) over time and the disappearance of

small-city technology disadvantage as evidence of a real and stable underlying scaling dynamic. This pattern may arise from cross-system knowledge spillovers that accelerate the diffusion of

advanced technologies and innovation, economic diversification enhancing resource allocation efficiency, and the maturing of practices relating to urban public goods65 and services66. We

note that a technological capacity argument only holds for complex technologies likely to be affected by threshold effects. The ability of small and less-developed communities to deal with

waste and other ‘commons’ problems through traditional institutions and technologies, is well understood67. Figure 4b compares the distributions of two properties related to wastewater

treatment: quantity of wastewater treated and quantity of recycled water. The former does not reveal truncation, while the latter does. This suggests that wastewater reuse, which is more

costly, requires more advanced technologies than basic wastewater treatment, and has higher social production process complexity, still poses challenges for small cities in China. There is a

city size threshold of approximately 300,000–500,000 persons under which cities require targeted improvements to their wastewater reuse capabilities. ADDITIONAL COST TO SMALL CITIES IN

PERFORMING COMPLEX POLLUTION CONTROL ACTIVITIES Apart from population, financial and technical capacity greatly influence a city’s capability for handling pollution, as captured in the

classic IPAT model68. This provides us with a perspective for interpreting truncation in national system scaling laws for complex pollution control. Scaling patterns of municipal investments

in pollution control and the number of green patents across Chinese cities exhibit truncations (Fig. 5a, b). This challenges the urban scaling idea of municipal infrastructure trending as

log–log sublinear and investments and ‘inventing’ activity as log–log superlinear. An estimated \(k\) of 8.08 in Fig. 5a indicates that Chinese cities below a population threshold of 500,000

persons, can be expected to suffer an under-capacity penalty of 14.9% (1-\({e}^{(-\frac{8.08}{50})}\)) in terms of fixed assets investment in urban wastewater treatment and environmental

sanitation facilities, on average, compared with 1.6% (1-\({e}^{(-\frac{8.08}{500})}\)) for a city of 5 million persons. Graphing per capita GDP in Fig. 5 reveals that the difference in

environmental economic and technical capability between large and small cities is not fully an income effect. Smaller cities in China are not necessarily poorer cities, measured by per

capita GDP, but they nevertheless show a systematic deficiency in investment, low technical innovation, and pollution control under-capacity. This is also true of the very smallest cities,

which are not the poorest and located mostly in Tibet, Yunnan, and Gansu provinces, where GDP has been enhanced relative to size, by natural resources and tourism and by national transfer

payments. The observation strengthens the interpretation of truncation being due both to urban complexity-related capacity and complexity of advanced pollution technology. In public goods

production, smaller cities are more vulnerable not only to cost factors (economic, technological, and others) but organizational, political, networking, and labor-market factors too, all of

which affect municipal management, innovation, and investment capacity. Besides institutional capacity factors, there are scale-economic factors65. As cities adopt more sophisticated

pollution control measures — requiring specialized expertise, advanced technologies, increased funding, and coordinated management — marginal cost and benefit change and without

institutional friction, pollution control would, in theory, stabilize to equate marginal cost and benefit69. Smaller cities have basic institutional limitations as noted, but also tend to

have higher marginal cost due to absence of economies of scale in infrastructural investment and lower capacity for technological output (Fig. 5). Higher treatment efficiency and lower

per-unit treatment costs of larger facilities have been verified across countries and waste types70,71. The capacity disadvantage of obtaining economies of scale contributes a threshold

effect, related to resource mobilization across multiple domains, including economic strength, technological support, specialized labor skills, public environmental awareness, and so on. As

a result, smaller cities only perform well in pollution control at a lower level of process complexity with lower marginal cost; and this is likely to manifest in the truncated part of the

curve. We thus capture an exponentially increasing vulnerability to capability lag in environmental management in the Chinese urban system. As the whole urban system becomes more mature, the

vanishing truncation indicates improved pollution control performance of small cities in addressing higher levels of complexity, either through gains in intrinsic capital and labor

capacities66, or spillover from larger cities of elements needed to overcome the threshold effects72. The threshold analysis of economic development and underlying infrastructure is well

studied73,74,75,76,77, but not yet so in scaling studies, partly because of lack of attention to heterogeneity in residuals. DISCUSSION Our study demonstrates that the tacit assumption of

homogeneity, or universality, in traditional urban scaling theory may not hold true across the entire system and is likely to lead to biased estimation of scaling exponents61,62,78 and we

have discussed the systematic bias in \(\beta\) exponents for urban properties relating to pollution control. There is a small literature on systematic deviations from the urban performance

power-law functions. Residual analysis of urban scaling models is now an accepted approach to identifying cities that are under- and over performing79,80,81,82, with the Scale-Adjusted

Metropolitan Indicator (SAMI) being a notable example, but not all look at the patterns of heterogeneous conformity to system mean trends systematically. Bettencourt et al.80 plotted a

cross-correlation matrix of time-series profiles of ranked standardized residuals from performance scaling models of US cities. This amounts to using patterns in residuals trends to identify

families of similarly performing non-contiguous cities. Arcaute et al. 61 used systematic truncation of scale-free performance curves to demonstrate how \(\beta\) exponents can move from

superlinear to sublinear as outliers are removed. More attention has been given to deviations from Zipf’s law and Gibrat’s law in national78,83,84 and global85,86 city size distributions,

particularly concerning the heads and tails of the distribution and their implications. Since the debate between Pareto distribution87 and lognormal distribution88, research has continually

refined testing methods45 and enhanced city boundary definitions46,78 that can substantially influence the scaling exponent89, as measuring urban population under different spatial contexts

is associated with heterogeneous growth dynamics90. Outliers in skewed distributions can also disrupt scaling laws, which can be detected using an adjusted boxplot91 and analyzed

separately92. Spatial dependence is another widely-concerned error-inducing factor93, as spatial proximity can greatly alter the number of effective interactions among individuals94. Studies

have adopted eigenvector spatial filtering models79 and generative models94 to address the influence of spatial autocorrelation. Our truncated power-law model provides estimations of

scaling exponents across all city sizes, statistically capturing small-city underperformance on pollution control goals. Speculatively, we interpret this as capturing intrinsic system

threshold effects in the capacity to supply local public goods65. A similar model could truncate at the upper size end to capture upper outliers, upper-end threshold effects, and log–log

nonlinearities, including diminishing economies and returns to scale as saturation thresholds are approached. As mentioned, another approach to tackling inverted hockey-stick log–log

distributions, is fitting a power function only for identified scaling region64. Another is to fit multiple functions to different portions of the distribution42,47. These approaches

encounter challenges due to the subjectivity in selecting demarcation points and justifying the choice of upper and lower bounds of the city size for scaling analysis. Comparatively, our

model is a robust and general approach to reveal evolution over the entire range of empirical data with a fixed form. We chose to model truncation via an exponential term

(\({e}^{-\frac{k}{N}}\)) that ‘bends’ the overall national urban system power-law, forming smoothened distribution over the entire curve, rather than as an alternative function for the

lowest city size range. Justification for this approach lies in the gradual transformation to a more linear curve over time – indicating better conformity to scaling laws (Figs. 3 and 4a,

Figs. S7 and S8). We conjecture that these are system-wide adjustment processes, not processes that are independent for fixed city size ranges. The generality of our model is evidenced by

its application in the case of urban wastewater treatment in South Africa and solid waste treatment in Japan (Fig. 6). Worse performance of smaller municipalities in South Africa regarding

complex technology capacity and total current asset value for wastewater treatment can be demonstrated by the truncation. In Japan’s case, waste recycling scaling demonstrates truncation,

with recycling being a relatively more complex activity that presents significantly higher challenge for the smallest prefectures compared to waste incineration (no truncation in scaling).

The much tighter fit regarding waste incineration also indicates a more equitable development across city size range, which is similar to the case of China’s wastewater reuse (Fig. 4b). The

decrease of \(k\) in the scaling of waste recycling capacity from 29.47 in 2010 to 16.49 in 2018 also indicates technological progress over time in small prefectures of Japan. The challenges

of small cities in pollution control are observed in other developing countries around the world. Table S4 compares the distributions of urban wastewater treatment capacity among cities of

different sizes in India (2005), Brazil (2017), and South Africa (2021), covering a total of 2547 municipalities. The insufficiency of treatment capacity is always most severe in the

smallest group of cities, while the largest 25% of cities all show significantly larger treatment capacity. One way of looking at this, in terms of truncated scaling laws, is that the

\(k\)-coefficients kick in at a large population size, and the under-capacity effect caused by small population size has affected most cities. Only the largest cities — often the core urban

centers - have sufficiently powerful political, economic, and technological strength to mobilize resources across sectors for better pollution reduction. Agglomeration economies of scale

produce inequalities as well as growth. Our results show that the gap in pollution control services between China’s larger and smaller cities over two decades is more significant in

less-developed urban systems (Figs. 3 and 4a). We conclude that truncations in the scaling patterns are typical for environmental technology and that this finding has significant

implications for urban environmental management. Strategies that enhance the transition from separated and fragmented responses by each small city, to a collaborative and interconnected

mode, may point to a fundamental solution for speeding the evening-out (levelling-up) of the overall systems curve over time. Our analysis supports the case for re-allocation of capital and

technical resources to small cities to overcome threshold effects. In addition, in-depth cooperation among cities can hasten technical diffusion and enhance multiplicative processes of

innovation adoption, highlighting the influence of spatial interactions on system scaling. This can be further verified through spatial econometric models93. City innovation networks

focusing on pollution control demand and capacity, can help generate synergistic, learning, and even federation effects. Joint construction and sharing of treatment facilities among

neighboring cities, especially for pollutants that require significant economic and technical support to manage, could be an effective way of breaking the constraints of small population in

respect to construction cost, capacity, and economic sustainability of pollution treatment95,96. This can be particularly beneficial for cities in western and northeastern China clustered

around mineral and other resources, which have low population density and capacity gap in developing sufficient pollution control infrastructure. In summary, our work proposes a general

model of truncations in scaling of environmental performance indicators that captures spatial and size inequalities and urban developmental dynamics over time. We highlight two specific

advantages of the approach. First, it better depicts the statistical properties of urban pollution control features and allows modeling of \(\beta\) convergence. It adds to the current

literature on urban allometry, particularly discussion about inconsistency of \(\beta\) and how to recover systematic information from deviations to system means. Second, it identifies

regions that deserve special policy attention, and reveals the necessity of system-wide management strategies for pollution treatment. We demonstrate that truncations in power-law rules,

governing urban environmental performance reliant on highly complex organization of human-technology-environment interactions, are better understood rather than ignored. Our truncation

parameter _k_ yields important information otherwise lost. It reveals the rate (over size and over time) at which social products are generated within the black-box of a city’s complex

self-organizing networks of knowledge and resource exchange. This study has limitations that should be considered. Firstly, the model proposed primarily addressed the unconformity of small

cities to system-wide mean trends. If outliers in the upper tail (e.g., largest cities) were removed, the fitting model and scaling laws might differ significantly. However, as urban

pollution control is driven by demand, although larger cities have additional resources, they are unlikely to overinvest in treatment facilities only to meet their specific needs, which

prevents them from being outliers in the upper tail. Second, city definition could influence the scaling coefficients, which can be tested for robustness by adopting alternative measurement

of urban population in future research. Finally, spatial interactions between cities may also impact the findings, which can be further examined using spatial econometric models93. METHODS

DATA For Chinese cities, we collected the data relating to urban population, pollution generation, pollution treatment, and fixed assets investment for sewage and environmental sanitation

from _China Urban Construction Yearbook_ in 2000–2020, published by the Ministry of Housing and Urban-Rural Development of the People’s Republic of China. The administrative division of

prefecture-level cities was set based on the National Bureau of Statistics in China. Specifically, city size is characterized by the urban permanent population, which is calculated as the

sum of the urban registered population and the urban temporary population. The green patent data was obtained from the Chinese Research Data Services (CNRDS) platform. The selection

criterion for green patents refers to the IPC (International Patent Classification) Green Inventory, which provides classification for environmentally sound technologies. For our modeling of

Indian cities, data on urban population, urban sewage generation, and existing sewage treatment capacity came from an official report titled _Status of Sewage Treatment in India_ published

by the Central Pollution Control Board of India. All 414 Class I Indian cities (having more than 100,000 persons) and 489 Class II towns (having 50,000–100,000 persons) are included. In the

case of Brazil, we acquired data on urban population and urban sewage treatment capacity for 1501 municipalities in 2017 from the _Diagnosis of Water and Sewage Services_ released by the

Ministry of Regional Development, based on data from the National Sanitation Information System. For South Africa, urban population data for 143 municipalities came from the _Community

Survey 2016_ provided by Statistics South Africa, and data on sewage treatment came from _The Green Drop 2022 report_ published by the Water and Sanitation Department, South Africa. Data for

European cities and Japan came from Eurostat and the Ministry of Environment Japan, respectively. More details about data collection are provided in Supplementary Note 1. MODEL COMPARISON

CRITERIA To evaluate the performance of different fitting models (i.e., the non-truncated traditional urban scaling model and the truncated models shown in Eqs. (1–3)), we employed multiple

statistical metrics and information-theoretic measures, including _r_² and Adjusted-_r_² for comparing goodness-of-fit, Mean Squared Error and Root Mean Squared Error for comparing

prediction accuracy, Akaike Information Criterion for balancing model fit and complexity, and Kullback–Leibler divergence for evaluating residual normality. GOODNESS-OF-FIT MEASURES _r_²

reflects the proportion of variance explained by the model, as Eq. (4) shows. $${{\rm{r}}}^{2}=1-\frac{\displaystyle {\sum }_{i=1}^{n}{({y}_{i}-{\widehat{y}}_{i})}^{2}}{\displaystyle {\sum

}_{i=1}^{n}{({y}_{i}-\overline{y})}^{2}}$$ (4) where \(y\) denotes the observation vector, \(\widehat{y}\) represents the model predictions, \(n\) indicates the number of observations, and

\(\overline{y}\) is the mean of observed values. Adjusted-_r_² is particularly useful for comparing multiple models as it accounts for the number of parameters (\(m\)), penalizing

unnecessary complexity (Eq. (5)). $${\rm{Adjusted}}-{{\rm{r}}}^{2}=1-(1-{r}^{2})\frac{n-1}{n-m-1}$$ (5) PREDICTION ACCURACY The Mean Squared Error (MSE) and Root MSE (RMSE) are calculated as

Eqs. (6) and (7) to provide a measure of predictive performance. $${\rm{M}}{\rm{S}}{\rm{E}}=\frac{1}{n}\displaystyle \sum _{i=1}^{n}{({y}_{i}-{\widehat{y}}_{i})}^{2}$$ (6)

$${\rm{RMSE}}=\sqrt{{\rm{MSE}}}$$ (7) MODEL PARSIMONY The traditional urban scaling model is nested within the truncated models, leading to concerns about whether the increased model

complexity can justify the improvement in fit. We address this with Akaike Information Criterion (AIC), shown in Eq. (8). $${\rm{A}}{\rm{I}}{\rm{C}}=-2{\rm{I}}{\rm{n}}\,{\mathcal{L}}+2m$$

(8) where \(\mathrm{ln}{\mathcal{L}}\) is the log-likelihood for Gaussian errors, calculated as Eq. (9). $${\rm{I}}{\rm{n}}\,{\mathcal{L}}=-\frac{n}{2}\left[\mathrm{ln}\,\left(2\pi

\frac{1}{n}\sum _{i=1}^{n}{({y}_{i}-{\widehat{y}}_{i})}^{2}\right)+1\right]$$ (9) Model with lower value of AIC is preferred since it performs better in the trade-off between explaining

variation and model simplicity. RESIDUAL NORMALITY In classical linear regression, the assumption of residual normality is fundamental to the reliability of statistical inference. If the

residuals of the truncated model are closer to a normal distribution, it may indicate that the model better satisfies the regression assumptions, thereby enhancing statistical robustness.

Consequently, we consider residual normality as a supplementary criterion for model comparison, evaluated by quantifying the Kullback–Leibler (KL) Divergence between residual distribution

(\(P\)) and standard normal distribution (\(Q\)), as Eq. (10) shows. $${D}_{KL(P\parallel Q)}=\sum _{x\in \chi }P(x){\rm{l}}{\rm{o}}{\rm{g}}\frac{P(x)}{Q(x)}$$ (10) \(P\) and \(Q\) are

estimated using kernel density estimation with Silverman’s rule for bandwidth selection. Probability densities were normalized for numerical stability. DATA AVAILABILITY All data used in

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Science Foundation of China (72304072) and the PDF funding from Faculty of Architecture, The University of Hong Kong. The work described in this paper was conducted in part by Dr. Chen CHEN,

Jockey Club Global STEM Post-doctoral Fellow supported by The Hong Kong Jockey Club Charities Trust. AUTHOR INFORMATION AUTHORS AND AFFILIATIONS * Department of Urban Planning and Design,

The University of Hong Kong, Hong Kong, China Chen Chen, Xiaohu Zhang & Chris Webster * School of Geography and Remote Sensing, Guangzhou University, Guangzhou, Guangdong, China Chen

Chen * Urban Systems Institute, The University of Hong Kong, Hong Kong, China Xiaohu Zhang & Chris Webster Authors * Chen Chen View author publications You can also search for this

author inPubMed Google Scholar * Xiaohu Zhang View author publications You can also search for this author inPubMed Google Scholar * Chris Webster View author publications You can also

search for this author inPubMed Google Scholar CONTRIBUTIONS Z.X. and C.C. co-designed the research. C.C. collected the data and performed the quantitative analysis. W.C. and Z.X. supervised

the work and provided guidance. C.C. wrote the original manuscript. W.C., Z.X., and C.C. revised the manuscript. All authors engaged in the discussion and contributed significantly to this

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