How applicable are scaling laws in predicting slum populations in urban systems? Evidence from India

Abstract

Recently, we have seen new developments in our understanding of the emergence and organization of cities and urban systems, including application of scaling laws to urban areas. A recent wave of studies has observed consistent behavior of multiple urban measures that scale with city size across geographic and sectoral contexts. However, the extant evidence is lacking in two important ways: first, a wide variety of urban measures still remain unexplored, and second, there is limited evidence from developing countries. This paper offers new evidence on both these fronts: i) applying scaling laws to predict slum population in cities, an urban measure that remains largely unexplored, and ii) applying them in the context of a developing country, India. Results suggest that population alone is not sufficient to predict slum population in India. Conversely, I use empirical results from scaling laws to test established slum growth theories that have influenced policymaking globally for decades, despite having limited empirical evidence to support them. I also show that scaling exponents are sensitive to the way we define urban systems, of which cities are a part, an issue that has been raised in the ongoing methodological debate on urban scaling laws. I believe that findings presented in this paper have implications for advancement of slum theories as well as urban scaling laws by offering new empirical evidence and mechanisms through which such scaling might happen in the context of slums.

Keywords

slums developing world city size urban scaling laws India

Introduction

Theories and models originally developed to understand physical and biological systems have been extended to study behaviors and properties of social systems, primarily with a notion that social systems follow a similar evolutionary pattern (Paulus et al., 2006). This idea got traction after advancements in complexity science, which cut across both natural and social systems (Batty, 2013). One such theory about scaling laws states that the sizes of physical and biological systems scale in a predictable manner (West, 2017). In an impressive transference from natural sciences into social sciences, many recent studies have shown that scaling laws hold for many social systems and for a variety of parameters of interest within those systems (e.g., Cesaretti et al., 2016; Rybski et al., 2009).

Validation of these laws in the context of urban systems is considered important for its potential implications for our understanding of cities. Studies have shown that multiple urban metrics, such as innovation and income, scale positively with city size, while such urban metrics as length of networked infrastructure, pollution, and crime rate scale negatively with city size. This indicates that cities are a great platform for achieving sustainable development for humanity. While it has been proposed that the urban scaling laws provide a unified theory of cities, a predictive framework applicable to the cities around the world (Bettencourt et al., 2010), most empirical studies have only corroborated this theory for urban systems in the developed world. There are only a handful of studies testing these laws in the developing world (e.g., Alves et al., 2013, 2015; Gomez-Lievano et al., 2012; Meirelles et al., 2018). None of the extant studies, including those studying urban systems of developing countries, have applied urban scaling laws to the growth of slums with a single exception of Sahasranaman and Bettencourt (2021).

Slums present one of the major global concerns, especially for cities in the global South. Consequently, there is great interest in understanding how slums emerge and evolve within cities (Patel et al., 2012), and scholars have approached this from a variety of substantive analytical frames. For example, studies on the growth of slums range from understanding their evolution with agent-based modeling (for a review see Roy et al., 2014), detecting slums from space using remote sensing techniques (for a review, see Kuffer et al., 2016), identifying and predicting slums with machine learning (Ibrahim et al., 2019), deep learning and artificial intelligence (Ibrahim et al., 2021), and exploring the causes of slum formation and growth in cities from an urban economics perspective (for a review, see Stokes, 1962; UN-Habitat, 2003). However, there is a general lack of a unifying theory of slum growth that could explain one of the most policy-resilient issues of the global South. In this paper, I conduct an empirical test of one such candidate theory, namely, that of urban scaling laws, in one of the world’s largest and fastest growing urban systems that is also a home to a sixth of the world’s slum population. The main purpose of this paper is to apply scaling laws to understand the relationship between city size and slum population in India.

It is worthwhile to note that both the Millennium Development Goals (MDG) and Sustainable Development Goals (SDG) declared in 2000 and 2015, respectively, by the United Nations (UN) included specific targets for improving the lives of slum-dwellers, a strong indication of the salience of this issue globally. Many national-level programs with similar goals (e.g., “Housing for All by 2022” declared by Government of India in 2015) are dedicated to improving living conditions of slum-dwellers, indicating similar interest among policymakers in rapidly urbanizing countries. A greater understanding of slums could be thus instrumental to improve the lives of over one billion people who currently reside in slums globally (United Nations, 2019).

Urban scaling laws: A brief introduction

One of the prominent urban scaling theories draws from scaling laws observed in physical and biological systems and applies them to understanding the behavior of cities that dates back to the law of proportional effect (Gibrat, 1931) and Zipf’s law (1941) as cited in Pumain (2012). Scaling laws are shown to predict multiple urban measures of importance as a function of a single property of the cities, namely, their size measured in terms of population, to the extent that they are proposed to be “a unified theory of urban living” (Bettencourt and West, 2010). This urban scaling law is parsimonious in the sense that it uses a sole parameter, city’s population as its input, and powerful in the sense that it can predict diverse properties of cities, ranging from economic measures such as innovation reflected by number of patents to measures of infrastructure services such as electrical cable length in a city.

This urban scaling law is presented as a relationship between outcomes of interest and the size of the city as measured by its population and takes the general form of a power law or allometry

Y_{t} {= Y}_{0} P_{t}^{ß}

(1)

where Y_t denotes socioeconomic characteristics such as number of patents in cites at time t; Y₀ represents a normalization constant; P_t represents city size as measured by population at time t; and ß is the scaling exponent that is of analytical interest and reveals the nature of the urban dynamics under study. The proposed qualitative interpretation of this quantitative measure categorizes it into three universal regimes (Bettencourt et al., 2010): i) if ß < 1, it represents sub-linearity of allometric relationships, indicating economies of scale as observed in several networked infrastructures, such as per capita length of electricity cable in a city, that tend to decline with city size; ii) if ß > 1, it represents super-linearity of allometric relationships, indicating agglomeration economies as observed in number of patents, which tend to grow exponentially with city size; and iii) if ß = 1, it represents linearity of allometric relationships or isotropy, indicating one-to-one relationships with population and outcomes such as housing and employment that tend to proportionately grow with city size.

This law has been tested in multiple urban systems in the USA, Germany, China, and Australia to name a few (e.g., Bettencourt et al., 2010; Fragkias et al., 2013; Galeser and Kahn, 2010; Sarkar, 2019), with a limited number of applications in the context of developing countries, mainly Latin American countries such as Brazil, Mexico, and Colombia and African countries such as South Africa (e.g., Alves et al., 2013, 2015; Gomez-Lievano et al., 2012; Meirelles et al., 2018). To the best of our knowledge, urban scaling laws have not yet been tested in the South Asian context with a notable exception of Sahasranaman and Bettencourt (2021) that has studied infrastructure in slums of India’s 210 urban agglomerations in 2011. Our study joins this effort by expanding the application of the urban scaling laws to the entire urban system of India that includes 2613 cities.

Furthermore, the law has been corroborated by empirically testing it in multiple sectoral domains spanning social, economic, and environmental aspects of urban life. These aspects have been studied using observable urban measures such as CO₂ emissions, GDP, number of patents, personal income, unemployment, electrical cable length, length of street networks, number of hospital beds, gas stations, housing, electricity consumption, water consumption, homicides, violent crimes, and child labor (e.g., Alves et al., 2015; Bettencourt et al., 2007, 2010; Gomez-Lievano et al., 2012; Meirelles et al., 2018). While Meirelles et al. (2018) have emphasized the need to include a broad range of indicators to advance the universality proposition of the scaling laws, this theory has not been tested to predict the number of people living in slums with one notable exception mentioned above. The present study aims to fill this gap in terms of testing the validity of the urban scaling laws in the domain of slums, an important measure of housing shortage and deprivation in the developing world. Evidence from such an application could be useful to corroborate the universality of the theory.

“Deconstructing” scaling laws: Major critiques

The scaling laws are cautiously termed “loose laws” (Batty, 2013: 40), and urban scaling laws have been particularly challenged as a unifying theory of cities on at least three grounds in the extant literature. First, there is contention about the lack of mechanistic insights in the observed scaling in empirical studies, which often leads to fallacies and misinterpretations of the findings. For example, (Gudipudi et al., 2019) have emphasized in their commentary that the results from scaling studies on CO₂ emissions have come up with opposite and contradictory results solely because they lack the mechanisms governing the phenomena under study. While Louf and Barthelemy (2014) recognize the value of empirical studies for testing and validating theorical models, they advise against drawing policy implications, especially if these studies lack mechanistic explanations of the observed phenomena. To this end, I attempt to interpret and corroborate my findings from scaling analysis with the help of established slum formation and growth theories.

Second, it has been shown that scaling exponents are sensitive to the way cities are defined. For example, as Arcaute et al. (2015) have shown with an extensive sensitivity analysis of the urban systems of England and Wales, that urban measures fluctuate considerably by the way cities are defined. The study also shows that when cities are defined using commuting-to-work and population-density thresholds as defining criteria, most urban measures scale linearly, thus challenging the notion that a theory of cities cannot rely on scaling laws alone. Cottineau et al. (2017) and Cottineau et al. (2019) reached the same conclusion in their studies of urban systems in France. While I could not test the sensitivity of scaling exponents by varying parameters of urban definition due to a lack of disaggregated data on slums at the enumeration block level in India, I could replicate a strategy similar to Louf and Barthelemy (2014), who used two different US Census city definitions (urban areas vs. metropolitan statistical areas). I show the implications of two different urban definitions on scaling exponents by the Census of India: cities with and without outgrowth included in their geographic extent and resultant aggregated urban measures.

Third, the scaling studies make an assumption that the urban systems are ergodic, in the sense that a cross-sectional understanding of the cities in urban system could provide insights about the longitudinal and evolutionary behavior of a given city. While the assumption of ergodicity makes the scaling laws appealing, it is problematic since it is well-known in urban geography that the urban systems evolve over time and that the “instantaneous” explanation via mathematical model of relationship between two variables does not provide sufficient understanding of the processes that lead to such relationships (see Pumain, 2012 for a detailed treatment of ergodicity in the context of urban scaling laws).

The implications of defining urban systems

While critiques of urban scaling laws have raised concerns about their sensitivity to urban definitions and how the urban indicators themselves are measured, the issue of the sensitivity of urban system definitions is also beginning to get attention. A large number of urban scaling studies have taken national urban systems as default systems (e.g., Alves et al., 2015; Bettencourt et al., 2007; Sarkar, 2019). However, there are a few notable exceptions in scaling studies that considered multiple countries as part of a single urban system: Arcaute et al. (2015) studied England and Wales, and Pumain and Rozenblat (2019) included cities from 28 member states of the European Union and Sweden and Norway in their study. Pumain and Rozenblat (2019) used scaling exponents to inductively identify the existence of two different regions within this system (Eastern and Western Europe).

With one exception of data consistency within a country, there are no explicitly discussed rationales for selecting national urban systems as appropriate networks of cities for scaling analysis. Furthermore, the lack of international standards for urban measures often makes it difficult for studies to consider cities in multiple countries as part of a single transnational urban system, even if they might functionally be a single system, such as cities in the European Union or a transnational system of global cities, as Sassen (2018) has conceptualized. Pumain et al. (2015) has suggested a way to create harmonized databases by defining cities that follows from common principles while allowing for local adaptation by demonstrating it in seven zones across the globe that includes both the developing and the developed countries. Conversely, some urban systems might function at subnational scale, such as cities on the East Coast versus cities in the West Coast in the United States, but this aspect has not been tested in the extant literature on scaling. I should note that intra-city scaling behavior has been tested though, for example in Shanghai (Xu et al., 2020).

This paper complements the current works on the sensitivity of exponents to the urban system definitions (Pumain and Rozenblat, 2019). I demonstrate that scaling exponents fluctuate considerably depending on how urban systems are defined within a national system. I use subnational regions and individual states of India to show the implications of urban system boundaries on scaling exponents. I argue that defining urban systems for scaling analysis is not trivial, and these definitions have significant implications for the validation of urban scaling laws.

Slum population as an urban measure

Slum population as urban measure reflects two overlapping constructs: i) as a measure of housing, it represents lack of formal and affordable housing supply for urban poor; and ii) as a measure of access to basic services, it represents lack of networked infrastructure such as water and sanitation. Bettencourt et al. (2010)’s theory suggests that housing follows a linear regime, whereas networked infrastructure follows a sublinear regime. Characterizing slums into a single category poses a challenge in terms of placing them in suggested regimes. If we also take legal aspect of slums into consideration (many slums illegally occupy land or violate building codes), this urban measure could be conceptualized like crime rates, which would place it in a superlinear regime. However, I take the former view and consider slum population as a measure of housing and basic infrastructure which results from interaction in housing market of a city (Patel et al., 2012) rather than living in a slum as an act of criminal behavior.

Data

I used the two most recent decadal censuses, 2001 and 2011, which were the first to systematically enumerate slum population in India. While earlier censuses reported slum population estimates for cities, they were not the result of proper canvassing using standardized questionnaires and data collection procedures. They also suffered from inconsistent definition of slums across the country (see Patel et al., 2014, for discussion on the effect of definitions on slum enumeration). However, Census of India (2001) enumerated slum population using a consistent definition across the country and included all the cities that reported 50,000 or higher population in the previous Census. There were 640 cities that met these population criteria in 2001. In the subsequent Census (2011), the coverage was expanded to all urban areas, defined as areas with populations of 5000 or higher in the previous Census (2001) that met following two criteria: i) 75% or more workers are engaged in non-agricultural activities and ii) population density is 400 people per sqkm or higher. In addition, Census included statutory towns even if they had population of lower than 5000 persons. Consequently, Census of India (2011) had 2613 cities that were included in slum enumeration.

I then created a panel dataset for 640 cities that had slum populations available for both Census years, 2001 and 2011. In addition, Census provided urban agglomeration level data for cities that had outgrowth in their topological periphery. Consequently, to test the sensitivity of scaling exponents, I partitioned cities into: A) urban agglomerations with outgrowths included, and B) cities with no outgrowth present in peripheries. While the universe of cities was mutually exclusive with such a partition (i.e., data for a city was included in either set A or set B, and not both), it gave an opportunity to test the sensitivity of urban definition on scaling exponents to some extent. See Appendix A in supplemental material for a detailed note on data.

To test whether scalar coefficients were subject to different ways of defining urban systems, I took two distinct approaches. I first partitioned the data by major regions of India. For this purpose, I used the commonly used regions that divide the country into six zones reflective of climatic and cultural similarities along with geographic contiguity. The number of cities in these regions varied from 76 cities in the Northeastern India and 864 cities in Southern India. Second, I used states as individual urban systems. India’s state boundaries were historically drawn along linguistic lines and largely represent linguistic and cultural homogeneity within a state, except in Northern India, where Hindi is the commonly spoken language across multiple states. In this sense, India’s states are like the European Union’s nation states, thus providing a unique opportunity to test the sensitivity of scaling exponents to regions.

Methods

Scaling analysis is performed using equation (1) discussed earlier: Y = Y₀ P^ß where Y is slum population, P is city size in terms of total population, and ß is the scaling exponent. While housing is a household-level phenomenon, I have used persons as opposed to households as a measure for convenience of interpretation since substantive slum literature prefers to use number of persons as a metric rather than number of households. While mathematically trivial exercise for aggregate measures, I also performed the scaling analysis using number of households and verified that the results were not sensitive to this choice.

The popular method to estimate scaling exponents is Ordinary Least Square (OLS) regression after log transformation of Y and P. However, recent research has suggested that such an approach has limitations (Leitao et al., 2016); an alternative is to compare the scaling model with the fixed linear model (ß = 1) using the Bayesian Information Criterion (BIC), as is routinely done for model selection in econometrics. In particular, the proposed conservative approach suggests that if $Δ ΒΙ C$ between scaling and linear model is 0, the linear model is preferred; if the $Δ ΒΙ C$ is between 0 and 6, the evidence is inconclusive; and only if $Δ ΒΙ C$ > 6, scaling model is preferred (for details on this test, see Appendix F in Leitao et al., 2016). While this test has not yet become a standard procedure, scholars have begun to use it (Sarkar, 2019), and it is my hope that such tests will become standard in future scaling studies.

Scaling analyses

India: 2011

The results of the scaling analysis for slum population in 2011 show that the pattern demonstrates a sublinear relationship with city size (ß = 0.90 < 1), with the BIC lower than that of fixed linear model $Δ ΒΙ C$ = 28.15. Considering slum population as an indicator of lack of affordable and formal housing, this finding is in disagreement with Bettencourt et al. (2010)’s proposed classification, in which housing is considered an individual need and is expected to scale linearly with city size. It is also different from Sahasranaman and Bettencourt (2021)’s study who have found slum population varying super linearly (ß > 1) for the 210 urban agglomerations, defined as cities with population 100,000 and above. This super-linear relationship may be a construct of excluding a long tail of smaller cities (akin to Zipf’s law) from scaling analysis. By contrast, this study that includes all the Census defined cities, suggest that formal housing supplies in larger cities better meet affordable housing demand and, thus, fewer people are forced to live in slums. It seems that scaling exponent is sensitive to inclusion and exclusion criteria used to select cities in study sample.

Furthermore, Figure 1 clearly indicates a problem in making conclusions based on scaling pattern. The system’s overall sub-linearity is primarily driven by many mid-sized cities, such as Botad (Population: 130,000), that has exhibited much better performance on the housing front compared to very small and very large cities. It is worth noting that smaller cities such as Nayabazar (Population: 1235) and Banjar (Population: 1414) exhibit similitude with large cities such as Mumbai (Population: 12.44 million) and Kolkata (Population: 4.5 million). These nuances are missed when we rely on a system-level exponent and its significance, while ignoring the model fit that clearly indicate that the goodness-of-fit is modest (R² = 0.55). However, literature on urban scaling laws often draw conclusions despite modest model fit (e.g., R² of 0.67 for patents in Bettencourt et al., 2010, and R² of 0.53 for slum population in Sahasranaman and Bettencourt, 2021).

Figure 1.

Scaling Analysis for Slum Population 2011. (Note: Gray dotted line indicates ß = 1; red solid line indicates model fit. NOTE: $Δ BIC$ > 6 indicates that scaling model is preferred over linear model).

The issue of heteroskedasticity also needs to be taken seriously, as emphasized by Leitao et al. (2016). I posit that such heteroskedasticity in scaling studies is introduced partly because of our lack of attention to how we define the extent of our urban systems. I tackle this issue later with sensitivity analyses, demonstrating how model fit fluctuates with varying definitions of urban systems.

Next, I tackle the question of how scaling laws can be utilized to make effective policies. Bettencourt et al. (2010) posit that an obstacle to effective policies is the lack of meaningful urban metrics. The main issue is that traditional per capita measures assume implicit linearity when comparing and ranking cities and completely ignore that many urban measures scale with city size. They suggest that to overcome this limitation, relative measures are more useful than absolute measures, proposing to use the difference between an observed value and the expected value as predicted by urban scaling laws for a given system. Building on Batty and March (1976)’s method of residues in urban modeling, Bettencourt et al. (2010) propose that scaling law parameters could be taken as a null model depicting the average behavior of urban measures, with deviations considered as characteristics of individual cities. The difference between actual and predicted values are termed “Scale-Adjusted Metropolitan Indicators (SAMIs)” that are dimensionless and are independent of city size and represent “the true local flavor” that captures successes and failures of cities relative to other cities in the urban system (Bettencourt et al., 2010: 2). This approach has been demonstrated by Bettencourt et al. (2010) in the context of United States Metropolitan Statistical Areas for urban measures of economic productivity and used in Sahasranaman and Bettencourt (2021) in the context of slums in India. It has also been utilized by studies in other contexts and domains (e.g., Alves et al., 2015; Lobo et al., 2013). I replicate this method and use SAMI to classify cities as relative successes (SAMI<0) and failures (SAMI>0) and to rank cities based on how they perform on slum population measures by using a scaling model as a reference (Figure 2(a)).

Figure 2.

Alternatives for Ranking Performance of Cities. (a) SAMI (red denotes above and blue denotes below average) and (b) LQ.

There are 1480 cities that overperformed (slum population lower than expected) and 1133 that underperformed (slum population higher than expected). While the approach is useful and the ranking of cities is an accessible measure for policymakers, we believe that dimensionless values of residues themselves are harder to interpret. Another possibility is to use Location Quotient (LQ), which is more intuitive and easier to implement for policymaking. The LQ is not a new measure—there is a long tradition of using LQ in regional science, economic geography, and demography (see Isserman, 1977; Miller et al., 1991; Morrill, 1991) and recently in residential segregation studies (e.g., Brown and Chung, 2006; Wong, 2002). These disciplines use LQ to intuitively capture the local flavor of a constituent unit in direct comparison with the system of which it is a part. In essence, the LQ is a measure of how concentrated or dispersed an urban metric in a city is compared to the overall system of cities. In the context of this paper, proportion of slum population in a given city is compared with the proportion of slum population nationally. Formally,

L Q_{i} = \frac{\frac{Y_{i}}{P_{i}}}{\frac{Y_{N}}{P_{N}}}

(2)

where

L Q_{i}

is LQ of city i;

Y_{i}

is slum population of city i (this could be any urban measure of interest);

P_{i}

is city size as measured by total population of city i;

Y_{N}

is total slum population of all the cities (i.e., slum population of India); and

P_{N}

is the total population of all the cities (i.e., total urban population of India). There are 1088 cities that overperformed (percent of slum population lower than the national average) and 1525 that underperformed (percent of slum population higher than the national average) reaching to roughly an opposite distribution than that obtained by using SAMI. Figure 2(b) shows a ranking of cities using LQ as the relative measure. We propose that LQ is a viable alternative to produce classification and ranking of cities that has two clear advantages for policymaking over scaling law residues: i) ease of implementation; and ii) intuitive interpretation for policymaking due to a simple comparison with national average. In addition, it maintains two positive qualities of SAMI: i) it can be used to dichotomize cities into successes and failures (LQ<1 or LQ>1); and ii) it can be used to rank cities in relation to one another. It should be noted that SAMI relies on the assumption that urban systems indeed follow scaling laws. No such assumption is necessary in calculating LQs. In the case presented in this paper, the scaling exponent is significant at the p < .01 level, but the goodness-of-fit for the model is modest (R² = 0.55), as discussed earlier. Calculating SAMI using a model that does not fit data well is problematic because it is impossible to know whether a city is truly exceptional, or a deviation is a construct of a poor model fit.

In addition, I tested whether classifications and rankings produced by LQ are significantly different from those produced using SAMI. I found that 83 cities that were considered underperforming cities by LQ criteria were considered overperforming by SAMI, while 38 cities that were considered overperforming cities by LQ criteria were considered underperforming using SAMI. I also found that over 99% of cities ranked with LQ remained within 10 positions range (higher or lower) of SAMI rankings, but some of them exhibited a vast difference, ranging from a drop of 616 positions to a rise of 398 positions, a rather large jumps in a system of 2613 cities. It will be helpful to develop a better understanding of how these rankings behave and how reliable they are for decision-making purposes. For instance, if national slum improvement programs use these rankings to prioritize the 100 most deprived cities (relative “failures”), many cities that might be eligible in one list may not be eligible in another. If the ranking is a construct of the methods employed, it is worth outlining the conditions under which they are useful for policymaking. I suggest that while SAMI is a useful approach for models with very high R², such as for Gross Metropolitan Product in the Bettencourt et al. (2010) study, where 90–93% variation is explained with scaling laws, LQ might be a better measure when model fit is poor, such as number of patents reported in the same study where only 60–70% variation is explained by scaling law.

Shalizi (2011) has argued that urban scaling laws are the construct of statistical aggregation at the city level. The author shows theoretically and empirically that if the scaling law depicts the relationship for aggregate or extensive measures, this relationship should also hold for per capita or intensive measures, and the exponent for that relationship should be ß – 1. I conducted the test suggested by Shalizi (2011) and constructed a model with LQ as a dependent variable, finding that the scaling exponent for this new model is indeed – 0.10 (p < .01), or roughly (ß – 1), as one would expect theoretically. While the relationship was statistically significant, the model was weak, with an R² as low as 0.011. This is similar to Sarkar (2019)’s study, which was the first to use LQ as an intensive measure and found that such relationships are weak. While Sarkar (2019) found a stronger relationship after removing outliers, I believe that if one finds a strong relationship with extensive measures, a weak relationship with intensive measures for the same phenomena should be expected. To demonstrate this, I compared LQ and SAMI and found that they are both highly correlated with each other (r = 0.85; p < .01). This positive and strong correlation is expected because, in essence, they both capture deviation from the system-level average: LQ directly measures such deviations as a ratio, whereas SAMI measures residues after constructing a statistical model of the system.

India: 2001–2011

I used panel data to test a popular notion in slum literature that posits that rapid urbanization leads to higher slum population growth (e.g., Ooi and Phua, 2007). A Hausman test confirmed that no significant heterogeneity bias existed; hence, I selected an efficient Random Effects model. The scaling exponent (ß = 0.91) suggested that slum population has a sublinear relationship with city size, which is very similar to the finding from cross-sectional data in 2011. However, with the help of the panel data, I could further investigate several properties pertaining to the change over time. Using SAMI, I created a classification of cities into underperforming (higher slum population than expected by scaling law) and overperforming (lower slum population than expected by scaling law) cities for both time periods. I observed that 96 cities that overperformed in 2001 were underperforming in 2011 (negative switch), 103 cities that underperformed in 2001 were overperforming in 2001 (positive switch), and 441 remaining cities did not change their status. There were no clear regional patterns and most states observed similar numbers of positive and negative switches, with exception of Delhi and Kerala, which had all negative switches, and Haryana, which had all positive switches in their cities. There was no statistically significant difference in initial city sizes between groups that positively or negatively switched, thus suggesting size invariance of the change in performance status. However, cities with smaller initial slum populations were more likely to have a negative switch and cities with larger initial slum populations were more likely to have a positive switch. The difference was statistically significant in terms of both absolute population and percentage slum population. This finding is in line with what is known as “regressing towards the mean” (Lovallo and Kahneman, 2003), a concept in psychology that suggests that both underperformers and overperformers regress towards the mean over time. This pattern of positive and negative switching of cities over time might suggest that as India’s urban system matures over time, all cities might regress towards the mean in the future, eventually following the pattern that scaling law has predicted in highly urbanized societies.

Next, I tested the rapid urbanization hypothesis for slum growth. In particular, I investigated the relationship between decadal growth in city size and corresponding growth in slum population. The relationship is statistically significant (r = 0.19; p < .01), and regression results suggest that faster-growing cities added more people to their base slum population than slower-growing cities. Specifically, for each additional 1% population growth, average slum population grew by 6472 people in a decade, thus landing support to rapid urbanization hypothesis.

Sensitivity analyses

Sensitivity to the definitions of cities

I tested the sensitivity of the scaling exponent using two different definitions of cities used by Census of India. One set constitutes 233 urban agglomerations that reported population for a city and its outgrowth as a single unit, whereas the remaining 2380 cities reported measures that did not include their outgrowth in their population data. Both sets had underperforming and overperforming cities distributed uniformly, as confirmed by a chi-square test (p > .05). The urban agglomerations (i.e., with topological periphery included) had larger slum populations and city size on average, but that is expected given that outgrowth adds to overall aggregate measures. Scaling analysis suggests that slum population scales sub-linearly with city size for both sets, but the scaling exponent is comparatively smaller for urban agglomerations (ß = 0.73) than for cities without outgrowths in their peripheries (ß = 0.92). These fluctuations complement Arcuate et al. (2013) and Cottineau et al. (2017)’s observations about how the definitions of cities impact scaling exponents. However, this finding is in contrast with classic slum peripherization theories that build on Alonso (1964)’s logic and predict that slum populations will be pushed out from inner cities toward topological peripheries (e.g., Barros, 2012; Harvey, 2008; Jain et al., 2015). If these theories hold, urban agglomerations should have higher than expected slum population, but application of scaling theory did not support it. More research is needed especially when theories from specific urban fields such as slum studies are in contrast with general theory of cities such as scaling laws.

Sensitivity to the definitions of urban systems

I used six regions of India that have cultural and climatic similarities and geographic contiguity, treating them as separate urban systems to test the sensitivity of scaling exponents to the definitions of urban systems. It is clear from scaling analyses that while India as a whole shows sublinear scaling of slum populations, exponents vary considerably when subregions are taken as systems within a system (Table 1). All exponents are statistically significant (p < .05) and allometry is determined using a conservative approach, where I considered CI for ß as well as

Δ B I C

as guides for model selection. Four regional systems are linear, whereas two regional systems are sublinear. I posit that both Northern India and Northeastern India that shows sublinear relationship include mountainous terrains that may have less interactions within regional labor markets and, consequently, lower labor migration and fewer resulting slums.

Table 1.

Scaling exponents for six regions of India.

Region (Urban system)	Cities (N)	Scaling exponent ß [95% CI]	Allometry	R²
Northern India	557	0.58 [0.73–0.85]	Sublinear	0.53
Western India	402	1.02 [0.93–1.11]	linear*	0.55
Eastern India	317	0.88 [0.77–0.98]	linear*	0.45
Northeastern India	76	0.64 [0.44–0.84]	Sublinear	0.36
Central India	397	0.99 [0.92–1.06]	linear*	0.66
Southern India	864	1.08 [1.04–1.14]	linear*	0.68
INDIA	2613	0.90 [0.87-0.93]	sublinear	0.55

All scaling coefficients are statistically significant (p < 0.01). Models marked with * have $Δ B I C$ < 6, suggesting fixed linear model (ß = 1) should be preferred even if ß < 1, whereas linear in Bold suggests models in which 95% CI for ß includes 1.

In addition, both regions have lower urbanization levels, which may explain lower slum populations than we would expect from scaling laws. It should be noted that India as a whole also has sublinear scaling of its slum population because the nation is only 31% urbanized. The regions with higher levels of urbanization, such as Western and Southern India, have higher scaling exponents, but those relationships are only linear as indicated by ∆BIC. None of the six regions showed super-linear scaling, including those that have large cities such as Mumbai (Western Region), thus negating the generally accepted theory that larger cities will have disproportionately more slums than smaller cities. In fact, they might have fewer slums, as sublinear scaling of the country level model suggest.

To further corroborate these findings, I performed sensitivity analysis taking individual states as individual urban systems. Census data suggests that the majority of rural-to-urban migration is within a state, and thus, it is plausible to conceive of states as closed urban systems de facto even if borders between them are open de jure. It is well-established that rural-to-urban migration is a contributing factor to slum growth in cities. The results showed that ß fluctuated even more and ranged from 0.48 in Himachal Pradesh to 1.10 in Chhattisgarh (For state level scaling analysis see Appendix B in supplementary material). Three states showed sublinear scaling for their slum populations, one of which is in a region (Southern India) that showed a linear relationship at regional analysis, as presented above. The other two showed linear scaling even if they are part of a national system that shows a sublinear relationship overall. These findings indicate that scaling law studies should not take national urban systems as a given. In fact, regions should be defined using a theoretical underlying process, such as migration patterns in the case of urbanization and slum population. In other example, scaling studies using measures of economic activities, flow of goods and trade volumes might determine boundaries for the system under study. While providing a systematic approach to define such regions for scaling studies is not within scope of this paper, I have proposed a conceptual underpinning towards such a goal.

Discussion

This paper showed that the population size of a city alone does not explain the variation in slum populations in the urban system of India. I also demonstrated that scaling exponents are sensitive to definitions of cities, as well as to definitions of urban systems, as previous studies have found. In comparing SAMI and LQ as alternatives to classify and rank cities for purposes such as prioritizing development investments through national urban development policies, I found that both have relative strengths and limitations that should be considered before using them for policymaking. I also tested two established slum theories that posit that rapid urbanization causes slum growth and that slums are pushed to the topological peripheries of cities. There is some support for the rapid urbanization theory, but I did not find evidence to support peripherization theory.

While I have not developed any new analytical methods, this case study presents a combination of several new approaches proposed in a recent wave of scaling studies. In particular, I replicated processes suggested by Leitao et al. (2016) for model selection, used intensive measures to verify scaling suggested by Shalizi (2011), tested sensitivity to urban definitions, as suggested by Arcuate et al. (2013), and tested sensitivity to the definition of urban systems as suggested by Pumain and Rozenblat (2019).

Overall, findings indicate that scaling theory in its elegant form (Equation 1) does not capture variation in slum population, and the model fit remains poor. In terms of policy implications, these findings suggest that slum population scale sub-linearly with city size and hence larger cities are better at creating affordable and formal housing for various income groups and thus have fewer slums than expected. This is not only in contrast with scaling law theory but also a popular notion among policymakers that favor urban investments in larger cities often referred as metropolitan bias (Palat Narayanan, 2020). For example, in line with Ferré et al. (2010)’s finding that poverty is concentrated in smaller cities, I show that small- and medium-sized cities have disproportionately higher numbers of people living in slums. Thus, policy responses to those cities are equally important. However, slums in smaller cities are rarely the focus on national-level slum policies such as housing for all in India. For instance, two most recent urban programs targeted a small number of cities as priority cities for development and investments in India. Government of India’s Jawaharlal Nehru Urban Renewal Mission (JNNURM) selected 63 cities and Atal Mission for Rejuvenation and Urban Transformation (AMRUT) selected 500 cities for development and urban renewal. These programs mainly select large cities with few exceptions. My findings indicate that it will be better to select worse performing cities that could be identified with SAMI or LQ for priority regardless of their size. Second, cities could use SAMI and LQ for their performance measurement on slum improvements over time. Third, planners often project population for a long horizon. Scaling laws could help them predict future slum population in accordance with their city size projections and help guide resource allocations in long-term.

Finally, it is unclear if scaling theories are truly universal, given that I find contrasting results for the same phenomena in the same geographic and sectoral domain by simply changing the definition of a region. I believe that it is important to test theories of urban living such as scaling laws in the context of developing countries and for the urban phenomena of significance in those contexts, such as slums. I hope that this case study precedes many such applications in other parts of the developing world to study not only slums but other urban phenomena of interest for the global South.

Supplemental Material

Supplemental Material - How applicable are scaling laws in predicting slum populations in urban systems? Evidence from India

Supplemental Material for How applicable are scaling laws in predicting slum populations in urban systems? Evidence from India by Amit Patel in Environment and Planning B: Urban Analytics and City Science

Footnotes

Acknowledgments

I am grateful to Brian Beauregard and Sokha Eng from University of Massachusetts Boston for their assistance. I am thankful to three anonymous reviewers who provided valuable feedback on earlier version of this manuscript.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Amit Patel

Supplemental Material

Supplemental material for this article is available online

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