# Redefining Urban Centrality: Integrating Economic Complexity Indices into
Central Place Theory

Jonghyun Kim  Donghyeon Yu  Hyoji Choi  Dongwoo Seo  Bogang Jun  Research
Center for Small Businesses Ecosystem, Inha University, Incheon 22212, South
Korea  Department of Statistics, Inha University, Incheon 22212, South Korea
Department of Data Science, Inha University, Incheon 22212, South Korea
Department of Economics, Inha University, Incheon 22212, South Korea

###### Abstract

This study introduces a metric designed to measure urban structures through
the economic complexity lens, building on the foundational theories of urban
spatial structure, the Central Place Theory (CPT) (Christaller, 1933). Despite
the significant contribution in the field of urban studies and geography, CPT
has limited in suggesting an index that captures its key ideas. By analyzing
various urban big data of Seoul, we demonstrate that PCI and ECI effectively
identify the key ideas of CPT, capturing the spatial structure of a city that
associated with the distribution of economic activities, infrastructure, and
market orientation in line with the CPT. These metrics for urban centrality
offer a modern approach to understanding the Central Place Theory and tool for
urban planning and regional economic strategies without privacy issues.

###### keywords:

Complexity , Central Place Theory , Market Boundary

††journal: arXiv

##  1 Introduction

Understanding the dynamic nature of urban structures is crucial for
elucidating the economies of a city and for implementing effective urban
planning policies. The urban areas have transformed their structure
ceaselessly. The structure, for example, co-evolved with the introduction of
new transportation and logistics systems, such as railways in the 19th century
and automobiles in the early 20th century, along with the emergence of new
industries [Anas et al., 1998]. These transformations underscore the critical
need to comprehend the underlying spatial structures, as these structures are
associated with the distribution of economic activities, infrastructure, and
market orientations—key components in urban planning and regional economic
strategies. To keep pace with these evolutionary changes, it is imperative to
develop metrics that can accurately measure and reflect the current status of
a region’s urban structure.

This study introduces a metric designed to measure urban structures through
the economic complexity lens, building on the foundational theories of urban
spatial structure, particularly Central Place Theory (CPT) [Christaller,
1933]. Proposed by Walter Christaller in the 1930s, CPT has significantly
influenced the field of urban studies and geography by offering a novel
perspective on how central functions are distributed across an urban
landscape, driven by consumer behavior and market forces. According to
Christaller, settlements are organized in a hierarchical manner where central
places fulfill specific roles by providing goods and services, which in turn
shape the geographical and economic context, and, recursively, are shaped by
the context. While Christaller’s CPT is well-known for its hexagonal market
boundaries, his primary focus was, in fact, on the interaction between product
centrality and location assuming a homogeneous space and uniformly distributed
consumers [Pacione, 2009]. Over the decades, CPT has provided an analytical
framework for understanding the clustering of economic activities,
particularly in retail trade and service industries, and has profoundly
influenced urban and regional planning [Mulligan et al., 2012, Bunge, 1962,
Boussauw et al., 2014, Dale and Sjøholt, 2007, Neal, 2011, Shearmur and
Doloreux, 2015].

The incorporation of big data and advanced analytical techniques in recent
years has reinvigorated CPT, allowing researchers to explore the theory’s
micro-foundations through detailed data analysis, thus reaffirming its
relevance in contemporary urban studies. Van Meeteren and Poorthuis [2018],
for example, show the relationship between threshold and range of goods and
the various central functions for the case of Louisville, Kentucky, by using
Twitter data, confirming the micro-foundation of CPT. Kii et al. [2023]
estimate the urban spatial structure based on remote sensing data of Tokyo,
Japan, by estimating trip attraction. These researches have opened pathways to
reconsider the main idea of CPT in the new light of the emergence of big data.
Although the literature shows the applicability of CPT with big data, it is
limited in developing a universal metric for CPT that highlights on a region
and a product in terms of centrality. Moreover, literature tends to focus on
the hexagonal configuration itself, rather than the interaction between the
products and the locations that was a key emphasis of Christaller.

To suggest a metric for the universal measure of the centrality of products
and locations, this study revisits the core principles of CPT, with more
paying attention to the interaction between the the characteristics of goods
and services, and the location that provides the goods and services. When we
focus on this fundamental aspect of CPT, which is the recursive interaction
between the centrality of products and places, we can find the resemblance of
CPT to the concept of economic complexity as depicted in Figure 1.

Figure 1: Similarity of definitions between the Central Place theory
[Christaller, 1933] and the Economic Complexity [Hidalgo and Hausmann, 2009]

The concept of economic complexity was introduced by Hidalgo and Hausmann
[2009]. In their seminal work, they derived two metrics, which are the Product
Complexity Index(PCI) and the Economic Complexity Index(ECI), by looking at
world trade data. The two complexity metrics show the recursive structure in
their definition: a country with a high complexity measured by ECI is the
country that produces complex products, while a product with a high complexity
measured by PCI is the product that is produced in a country with high
complexity. Hidalgo and Hausmann [2009] derived the PCI and ECI by solving the
eigenvalue problem of the recursive equations. Thanks to the flexibility and
applicability of two metrics, ECI and PCI, it has become popular tools in
economic geography, international development, and innovation studies
[Hidalgo, 2021].

In this work, we suggest a more universal way of measuring the centrality of
products and locations in a city by combining the theoretical framework by
Christaller and recent advancements in the field of Economic Complexity
[Hidalgo and Hausmann, 2009, Hidalgo, 2021]. We aim to show how the ECI and
PCI reveal the urban structure in Seoul, which is a megacity, combining
various big data, such as data on all the geo-locations of small businesses in
Seoul; residential, labor, and floating population data; and credit card data
with high spatial resolutions. Specifically, our new measures for the
centrality of products and places, PCI of a product and ECI of a region, are
calculated by using data on the geo-location of small businesses, which can be
free from privacy issues so that it can be applied for the other cities
easily.

##  2 Results

### Detecting the spatial unit of analysis

To explore the ECI and PCI of an urban area, we look at Seoul, which is a
megacity, exhibiting a gross regional domestic product (GRDP) of $\$3,739$
million (20% of all GDP) and involving 2,860,562 households, 823,624 firms,
5,226,997 workers and 9,662,041 residents in 2019. Then, based on the location
data of the small businesses shops, we define the spatial unit of analysis.
While administrative districts are often used as a spatial unit of analysis,
this administrative boundary within the city level does not determine
consumption behavior. Therefore, we define the spatial unit of analysis based
on data showing how the small businesses are located [Hidalgo et al., 2020,
Jun et al., 2022], resulting in 523 amenity clusters, whose average radius is
241 meters. The following analysis is based on this detected spatial unit of
analysis that is depicted with polygons in Figure 3.

### The optimal distance among markets over products

We investigate the relationship between PCI and the spatial range of the
market, using PCI in two ways. One focuses on products and services and
examines their market boundaries–which reflect Christaller’s concept of the
demand threshold, the minimum number of customers necessary for a central
function to exist–by using the location data of all the small businesses in
Seoul. The demand threshold is captured by the minimum distance among markets
of a product. The other approach focuses on consumer behavior by using credit
card data, by looking at how the distance between consumers’ residential area
and their places of their consumption differs over products. This reflects the
maximum distance that a consumer is willing to travel to obtain a central
function (the range in Christaller’s concept). We construct two empirical
models, which are Equation (1) and (2), to investigate economic-geographical
characteristics of central place goods and central places in terms of the PCI
and ECI.

First, the demand threshold in Christaller’s idea is captured by the minimum
distance among markets of a product. To calculate the minimum distance among
markets, we start with calculating Revealed Comparative Advantage of product
$p$ in cluster $i$
$(RCA=\frac{shops_{cp}/\sum_{p}shops_{cp}}{\sum_{c}shops_{cp}/\sum_{cp}shops_{cp}})$
by looking at the number of shops in the cluster, $shops_{cp}$, and figure out
whether a product in a cluster has the comparative advantage [Balassa, 1965].
$RCA_{cp}$ is greater or equal to 1 when the share of production of cluster on
a given product $p$ is larger than the share of that product on the entire
city. When $RCA_{cp}$ is greater or equal to 1, we regard the market of
product $p$ exists in cluster $c$. Then, we look at the distance between every
pairs of the market and pick the minimum distance among them. That is the
minimum distance between clusters for a product $p$, $Dist_{p,c,c^{\prime}}$,
which is our dependent variable. We construct the empirical specification in
Equation (1) to investigate the spatial range of market by each product:

$\displaystyle\begin{split}Dist_{p,c,c^{\prime}}&=\beta_{0}+\beta_{1}PCI_{p}+\beta_{2}\Delta{ECI_{c,c^{\prime}}}+\beta_{3}\Delta{Diversity_{c,c^{\prime}}}+B\cdot
X\\\ &+\mu_{location}+\mu_{industry}+\varepsilon_{p,c,c^{\prime}}\end{split}$
(1)

where the dependent variable, $Dist_{p,c.c^{\prime}}$, is the minimum distance
between two spatial units that shows comparative advantages in product $p$.
Our variable of interests is the product complexity, $PCI_{p}$. We also
control for the difference in economic complexity of two spatial units,
$\Delta{ECI_{c,c^{\prime}}}$, and product diversity,
$\Delta{Diversity_{c,c^{\prime}}}$. $B\cdot X$ includes other control
variables, such as the demographic characteristics, difference in labor,
floating, and residential population in thousands of people, and the
difference in land price between the spatial units. Additionally, we add
vectors of dummy variables to control location-specific and industry-specific
fixed effects and $\varepsilon_{p,c,c^{\prime}}$ is the error term 111The
administrative district dummy variable was measured at the ward level gu,
which is an intermediate step in the city’s administrative division. The dummy
variable for industry classification is measured at the highest level, with a
total of nine classifications.. The result is depicted in Columns (1) and (2)
of Table 1.

Figure 2: Market boundaries over the level of PCI. (A) Hexagonal configuration
the product centrality and (B) schematic market boundaries over different
products suggested by Christaller (1933). (C) Calculated market boundaries of
convenience store (PCI = 0.26, average market distance = 0.8 $km$), Second
handed luxury goods (PCI = 0.46, average market distance = 1.6 $km$), and
Diamond trader shop (PCI = 0.84, average market distance = 2.9 $km$). Black
dots represent the centers of small business clusters that exhibit the
comparative advantage of the product.

Before moving to the second empirical specification, we visualize our result
on the relationship between the market boundaries of products and their PCI as
shown in Figure 2. Figure 2 (A) is the conceptual framework suggested by
Christaller [1933]. He argues that the spatial configuration of a city depends
on the centrality of places. In turn, central places are those providing
products and services with high centrality according to the iterative
structure of his argument. Based on Christaller’s idea, we can draw the
schematic market boundary of different products as depicted in Figure 2 (B).
This shows that the more central products are likely to have a longer distance
between the markets. As expected by the conceptual framework of Christaller
[1933], Figure 2 (C) shows that the more complex product, which corresponds
with the higher centrality, tends to exhibit the larger market boundary. For
example, in the case of convenience stores, which have lower centrality, the
average distance among the market is 0.8 $km$, and its product complexity is
0.26. In contrast, diamond trader shops, which have higher centrality, have a
long market distance, which is 2.9 $km$, with a PCI of 0.84. Although the
spatial configuration is not the exact hexagonal shape as the conceptual
framework expected in Figure 2 (A) and (B), which is based on the assumption
with homogeneous space and uniformly distributed consumer, our results show
the materialized world of the idea.

Next, we construct the second empirical specification, Equation (2), to
explore relationships between consumer’s purchasing mobility and the product
centrality measured by PCI. Because of privacy issue, tracking individual
consumer’s purchasing behavior is impossible using our data. Instead, given
information aggregated by purchasing and residential location, and the
demographic information on consumers, we constructed consumer group $i$ by
looking at their age (aggregated by 10s), gender, purchasing and residential
location based on 50m $\times$ 50m cell units. Our empirical specification to
see the purchasing behavior of consumer group $i$ for product $p$ is
following:

$\displaystyle\begin{split}Dist_{i,d}&=\beta_{0}+\beta_{1}PCI_{p}+\beta_{2}Count_{i,d}+\beta_{3}Female_{i}+B\cdot
X\\\ &+\mu_{location}+\mu_{industry}+\varepsilon_{i,p}\end{split}$ (2)

where the dependent variable, $Dist_{i,p}$, is the travel distance of
consumers group $i$, who purchase goods of product type $p$ in a region from
their residential area. Our main variable of interest is again product
complexity $PCI_{p}$, while $Count_{i,p}$ is the aggregated purchase count for
product $p$ by consumer group $i$. $Female_{i}$ is a binary indicator of
consumers’ gender that takes 1 if consumers are female and 0 otherwise.
$B\cdot X$ includes demographic controls, such as age groups of consumers. We
also add the location and industry fixed effects, which are $\mu_{location}$
and $\mu_{industry}$, to control for the time-invariant characteristics of
consumers’ residential location and the industry field of the goods222The
administrative district dummy variable was measured at the ward level gu,
which is an intermediate step in the city’s administrative division. The dummy
variable for industry classification is measured at the highest level, with a
total of nine classifications.. $\varepsilon_{p,o}$ is the error term. The
result is depicted in Column (3) and (4) of Table 1.

As reported in Table 1, a high PCI product is likely to have a broader market
boundary, pulling consumers from a further distance. In Columns (1) and (2),
the PCI is significantly and positively correlated with the minimum distance
between two markets of product $p$. Since the ECI and PCI are normalized, the
coefficients of $PCI_{p}$ in Columns (1) and (2) indicate that the minimum
distance between markets is, on average, 0.297- 0.324 km further apart when
the PCI value increases 1 unit of standard deviation. The difference between
the case of minimum and maximum PCI is 2.1-2.3 kilometers further apart.

Table 1: Relationship between the market boundary and the PCI of products

| Market boundary focusing on product | Market boundary focusing on consumer’s behavior
---|---|---
| (1) | (2) | (3) | (4)
$PCI_{p}$ | 2.272∗∗∗ | 2.082∗∗ | 8.942∗∗∗ | 9.228∗∗∗
| (0.092) | (0.089) | (0.011) | (0.011)
$\Delta{ECI_{c,c^{\prime}}}$ |  | 0.583∗∗∗ |  |
|  | (0.069) |  |
$\Delta{D_{c,c^{\prime}}}$ |  | 0.001∗∗∗ |  |
|  | (0.0005) |  |
$Count_{i,d}$ |  |  |  |  $-$0.388∗∗∗
|  |  |  | (0.001)
$Female_{i}$ |  |  | 0.195∗∗∗ | 0.098∗∗∗
|  |  | (0.004) | (0.004)
Demographics controls | ✓ | ✓ | ✓ | ✓
Land price controls | ✓ | ✓ |  |
$FE_{industry}$ | ✓ | ✓ | ✓ | ✓
$FE_{location}$ | ✓ | ✓ | ✓ | ✓
$FE_{consumerage}$ |  |  | ✓ | ✓
Observations | 22,215 | 22,215 | 4,201,653 | 4,201,653
R2 | 0.220 | 0.274 | 0.277 | 0.294
Adjusted R2 | 0.201 | 0.256 | 0.277 | 0.294
Residual Std. Error | 0.9156 | 0.883 | 4.200 | 4.155
F Statistic | 11.169∗∗∗ | 14.768∗∗∗ | 39,316.160∗∗∗ | 41,569.570∗∗∗
Note: |  ∗p$<$0.1; ∗∗p$<$0.05; ∗∗∗p$<$0.01 |  |

Column (3) and (4) of Table 1 shows the second results that are associated
with the empirical specification of Equation 2. Again, the dependent variable
is the distance between the consumers’ residential area and their consumption
places. As shown in Columns (3) and (4), a product with a higher PCI is likely
to be purchased by consumers who are visiting the market from longer apart.
Interestingly, as shown by the positive and significant coefficient of
$Female_{i}$, the mobility of female consumers is higher than that of male
consumers. The results in Columns (3) and (4) of Table 1 show that on average,
consumers travel 8.9 – 9.2 $km$ further to purchase the more complex product
(or service), when the PCI of the product increases 1 unit of standard
deviation, and as suggested Christaller [1933], the higher centrality goods
measured by the PCI are associated with the longer boundary of the market of
the product.

### Economic complexity of a region reveals the central place of a city

Figure 3:  Economic complexity of amenity clusters in Seoul. Amenity clusters
with high, medium, and low level of ECI are in red, gray, and blue,
respectively.

We visualize ECI for each spatial unit in Seoul in Figure 3. Although we
calculate the ECI by just looking at the locations of small businesses’ shops
in a city, interestingly, the result shows that high ECI regions, which are in
red, are mostly well-known place for city’s economically and politically
central functions. For example, spatial units labelled $H1$ in Figure 3,
primarily include the headquarters of major conglomerates or big companies. In
these area, which has high ECI, complex products and services are more likely
to provided/traded, as suggested by Christaller [1933]. Likewise, other
clusters with high ECI, which are in red, are also responsible for central
function of city Seoul: regions labelled H2 include Seoul city hall and
leading media companies, those with H3 are university town areas, those with
H4 are financial hub in nation, those with H5 are the small businesses’
cluster of small electronic shops in Yongsan, and those with H6 are the area
where are the hub of foreign cultures or expensive amenity goods in Iteawon.

Figure 4: The level of ECI (A) and the density of floating (B), labor (C), and
residential (D) population in Seoul. The high, medium, and low level of the
values in a cluster are depicted in red, gray, and blue color, respectively.

Other regions with high ECI with yellow boxes are located in relatively
peripheral area in Seoul but they are also serve for central city function
with including central facilities, such as domestic airport, science park,
cluster of private academies, enterprise zone for ICT companies, central
market of agricultural and marine product in a country, administrative office
of sub-district of Seoul, a central express bus terminal, amusement park,
market for office supplies, and logistics $\&$ distribution hub. As
demonstrated previously, the spatial concentration of economically and
functionally important facilities on high ECI regions.

Further, focusing of production side, we explore another spatial concentration
on the high ECI regions, since according to Christaller [1933], not only do
central places play a crucial role in consumption, but they are also pivotal
in production. To examine this assertion, we analyzed the spatial
concentration patterns of the labor force and potential consumers in Seoul by
visualizing the number of labor, floating, and residential populations within
the city’s clusters. These patterns are depicted in Figure 4. The results show
that regions with high ECI correspond to areas with bigger number of both
floating and labor populations. However, these high ECI regions do not
necessarily coincide with a bigger number of residential population. This
finding supports Christaller’s theory by illustrating that central places
primarily support production activities, whereas surrounding areas with lower
centrality predominantly serve residential functions.

Table 2: Correlations between the various characteristics of an area and the
ECI of a region. $ECI_{c}$ is the economic complexity of the cluster,
$Diversity_{c}$ is the diversity of small business shops, $Shops_{c}$ is the
number of shops in the cluster, and $Price_{c}$ is the average land price of
the area. $Labor_{c}$, $Float_{c}$, and $Resi_{c}$ are labor, floating, and
residential population numbers, respectively.

| $ECI_{c}$ | $Diversity_{c}$ | $Shops_{c}$ | $Labor_{c}$ | $Float_{c}$ | $Resi_{c}$ | $Price_{c}$
---|---|---|---|---|---|---|---
$ECI_{c}$ | $1$ |  |  |  |  |  |
$Diversity_{c}$ | 0.056 | $1$ |  |  |  |  |
$Shops_{c}$ | 0.449 | 0.706 | $1$ |  |  |  |
$Labor_{c}$ | 0.564 | 0.250 | 0.384 | $1$ |  |  |
$Float_{c}$ | 0.267 | 0.638 | 0.667 | 0.320 | $1$ |  |
$Resi_{c}$ |  $-$0.574 | 0.413 | 0.184 |  $-$0.281 | 0.271 | $1$ |
$Price_{c}$ | 0.661 |  $-$0.034 | 0.222 | 0.442 | 0.064 |  $-$0.481 | $1$

To delve into the production side, we check the Pearson correlation
coefficients between the population type in regions and the ECI of them (Table
2), and the point-biserial correlation between the labor share of industrial
sector in a region and the ECI in the region333The industrial sectors are
classified by the Korean Standard Industrial Classification. (Table 3). Table
2 shows that the high correlation between the ECI and the labor population of
the area, while negative correlation between the ECI and the residential
population. Furthermore, the place with high ECI is corresponds with
relatively higher land price as well. Interestingly, the diversity of amenity
shops is not necessarily correspond with the high ECI of the area.

Table 3: Correlation coefficients between labor share of industrial sector and
the ECI of a region

| The level of ECI in a region
---|---
Industrial sector | High | Intermediate | Low
Financial and insurance | 0.386 | 0.036 | -0.355
Professional, scientific and technical services | 0.383 | 0.138 | -0.440
Information and communication | 0.280 | 0.208 | -0.415
Wholesale and retail trade | 0.222 | 0.012 | -0.196
Business facilities management and business support services; rental and leasing activities | 0.075 | 0.180 | -0.219
Electricity, gas, steam and air conditioning supply | 0.042 | -0.056 | 0.013
Accommodation and food service activities | 0.035 | -0.083 | 0.042
Agriculture, forestry and fishing | 0.014 | -0.024 | 0.009
Public administration and defence; compulsory social security | -0.014 | -0.026 | 0.034
Mining and quarrying | -0.028 | 0.084 | -0.049
Transportation and storage | -0.099 | -0.135 | 0.199
Construction | -0.109 | 0.077 | 0.025
Real estate activities | -0.131 | -0.050 | 0.153
Manufacturing | -0.162 | -0.060 | 0.188
Water supply; sewage, waste management, materials recovery | -0.193 | -0.065 | 0.218
Arts, sports and recreation related services | -0.224 | -0.000 | 0.187
Education | -0.311 | -0.059 | 0.3152
Human health and social work activities | -0.378 | -0.193 | 0.483
Membership organizations, repair and other personal services | -0.386 | -0.167 | 0.468

Table 3 shows that population working in “financial and insurance sector” and
“professional, scientific and technical services” are more likely to be
concentrated in high ECI regions, by showing relatively high value of
correlation 0.386 and 0.383, respectively. By contrast, labor shares in
construction, manufacturing, and sectors related with infra structure, which
are a bit far from the central role of a city, are higher in regions with
relatively low ECI, with scoring negative correlation.

According to Table 2 and 3, the high ECI regions exhibit the high ECI not only
in terms of city function as shown in Figure 3 but also in production side.
Our results confirms the hypothesis suggested by Christaller that the cenral
places plays a crucial role in consumption and at the same time, in production
as well.

##  3 Discussion and Conclusion

In the preface of his doctoral thesis, Christaller stated that the purpose of
Central Place Theory (CPT) is to analyze the economic and geographical
principles that determine the spatial arrangement of urban functions. Rather
than merely describing these arrangements, as typified by the hexagonal
configuration, he emphasized the interactions between central
functions—represented by sets of goods and services—and their centrality
within the city. Despite of the theoretical contribution of CPT, it has been
limited to apply for the real world, since CPT was developed with assumption
the uniformly distributed consumer in a homogeneous space. When focusing on
the key principle of CPT, we could find the similarity between the concept of
economic complexity, which is already widely used thanks to its flexibility
and applicability, and suggest the two economic complexity indices, PCI and
ECI, as a metric for the centrality in CPT. We calculate the PCI of products
and the ECI of places by using the geo-location data of small businesses in
Seoul and test whether these two metrics well capture the centrality of
products and places.

First, to capture the effect of PCI of a product on the market boundary of the
product, we calculate the minimum distance among markets of the product by
using the shop location data, and see the effect of the PCI on the minimum
market distance. Our result show that a high PCI product is likely to have a
broader market boundary. When 1 unit of the standard deviation of PCI
increases, the minimum distance among the product’s markets increases, on
average, 0.297-0.324 km. The difference in market distance between a product
with the minimum PCI and that with the maximum PCI is 2.1-2.3 km. In addition,
to capture the demand threshold of a product, which is the minimum number of
customers necessary for a central function to exist, we look at the effect of
PCI on the distance between the residential area and the shop the consumption
behavior happen by using the card transaction data. Our results show that a
consumer who purchase a high PCI product is more likely to be from further
away from the shop. When the one standard deviation of the PCI of products
increases, consumers travel 1.782-1.838 km further to purchase the product.
The difference in market range between a product with the minimum PCI and that
with the maximum PCI is 8.2–9.2 km.

Second, following the logical process in the Christaller’s theory, we look at
the economic complexity of places. According to him, the centrality of
location is based on the accumulated functions provided by products and
services on the place, given that there exist various products in the place.
We expected that the recursive structure of economic complexity index well
capture this aspect and see whether the ECI of places successfully identify
the central places in terms of city function and the production. When we
visualized the ECI of places, we found that high ECI places are the well-known
for Seoul’s economically and politically central places including, for
example, the central bank of Korea, a terminal for high speed rail and express
bus terminal, financial hub, headquarters of big firm, like Samsung. When
checking the production side by looking at the population composition of a
place and the industrial sector that the labor share is working at, we find
that the high ECI places play a central role for more complex economic
activities. Furthermore, the high ECI places are strongly correlated with the
labor population, especially those associated with knowledge intensive
industries. Our results confirm that ECI is well representing the level of
centrality of places.

Figure 5: Robustness of economic complexity indices as a measure for the
centrality using the location data of all shops in Seoul. (A) ECI and (B)
Diversity over the spatial units in Seoul. (C) Contingency matrix between ECI
rank and PCI rank and (D) between Diversity rank and Uniqueness rank.

One might argue that an alternative metric, such as the diversity of economic
activity in a place, could effectively represent the centrality of a place.
This notion could be supported by Christaller’s argument that the most unique
goods are produced in the most central locations, whereas ubiquitous goods are
produced universally, independent of a place’s centrality. To explore this, we
examined the diversity of goods and their spatial distribution in Seoul,
comparing to these findings with the results of PCI and ECI. As illustrated in
Figures 5 (A) and (B), the regional diversity rankings in Seoul present a
different picture to that depicted by the complexity index, with regions of
high diversity not necessarily correlating with the central functions of
Seoul.

Further analysis involved examining a contingency matrix for the diversity
versus the uniqueness of goods, as shown in Figure 5 (D), to investigate the
recursive relationship between uniqueness and diversity. The dark area on the
left side of Figure 5 (D), regardless of the level of diversity, indicates
that the uniqueness of products does not correlate with diversity. A recursive
structure, if present, would likely show products in the upper right corner of
the matrix, indicating regions where high uniqueness products are produced
alongside high diversity, as the relationship between PCI and ECI in Figure 5
(C). For the principle of low PCI/uniqueness products being produced in low
ECI/diversity regions to hold, the top left corner of the contingency matrix
would need to be dark. However, this is observed only in the case of the
economic complexity index (Figure 5 (C)), but not for diversity. These
findings confirm that the economic complexity indices, PCI and ECI,
effectively embody the recursive structure of centrality defined in CPT.

Taken together, we present a metric to measure the centrality of a product and
a region with borrowing the theoretical foundation of Christaller’s CPT. Given
the core principle of CPT is the interaction between the centrality of
products and places, our metrics, stemming from the field of economic
complexity, are conceptually consistent with the centrality of products and
places. Our findings supports our arguments that PCI of products are
positively correlated with the market size of the products, and high ECI
places is the place that provides the central city functions and complex
products. Considering that we use the location data of shops that are free
from the data privacy issue, our findings can contribute to the body of
literature on urban economy and policy makers by capturing the dynamic nature
of urban structure more precisely and easily.

##  4 Methods

### Detecting the spatial unit of analysis

By using the location data of all small business shops in Seoul, first, we
calculate the effective number of small businesses, $A_{i}$ and detect the
amenity dense neighborhood Hidalgo et al. [2020], Jun et al. [2022].

$A_{i}=\sum^{N}_{j=1}e^{-\gamma d_{ij}}$ (3)

where $N$ denotes the total number of stores in the city and $d_{ij}$ is the
geodesic distance between stores $i$ and $j$. With the stand point of small
business shop $i$, the effect of the existence of other store is diminishing
over distance. $\gamma$, decay parameter capture this diminishing influence of
distant stores. In our analysis, we set $\gamma$ into 7.58.

Second, by looking the spatial distribution of the effective number of shops,
we identify the local peaks of $A_{i}$ on the map. Then, as capturing the
center of the cluster with the local peak, we allocate neigh by shops to the
cluster until the boundaries of the cluster are overlapped, resulting in 523
amenity clusters, whose average radius is 241 meters. Following analysis is
based on this detected spatial unit of analysis that are depicted in Figure 3.

### Data

On the detected spatial unit of analysis, we accumulated various big data. To
detect the amenity clusters and examine the composition of various amenity
types they provides, we use data on small businesses sourced from the Korean
Small Enterprise and Market Service (KSEMS). This dataset encompasses detailed
information on the location and business categories of 401,071 small
businesses observed to be in operation during the period from March 2019 to
June 2019. To investigate the urban structure without COVID-19 effects, we
choose the data of 2019.

To examine the structure of economic activities in urban areas, we use spatial
statistics data on residential, labor, and floating populations in 2019 and
credit card transactions in June 2019. First, the residential and labor
population data is from Korea National Statistical Office (KOSTAT). The
residential and labor populations represent the number of people living in
households and working within a given area based on 100m $\times$ 100m cell
unit, respectively. Second, human mobility big data, a floating population, is
collected from the cell phone activity of personal subscribers to SK Telecom,
the biggest network provider in South Korea. Here, floating population refers
to the daily average of people staying or moving within a given area based on
50m $\times$ 50m cell unit.

Finally, we also use credit card data provided by BC Credit Card Corporation
for the robustness check. The credit card data includes information on the
total number and amount of purchases (in KRW), number of stores, location, and
industry classification provided by BC. Due to privacy concerns, the
information is provided with aggregated based on 50m $\times$ 50m cell unit.

We reconfigure location information of the residential, labor, and floating
population data and credit card data into our detected amenity clusters in
Section 2. Again, there are 523 small business clusters in our analysis.

### Calculating complexity index of region and product

Motivated by the Christaller’s conceptualization of place centrality
Christaller [1933], whose bears resemblance to the principles underlying the
Economic Complexity Index, we utilize economic complexity indices as proxies
for the assessing the centrality of locations and goods. Figure 1 describe the
concepts of centrality by Christaller and that of economic complexity, showing
the similarity between the two in terms of recursive structure. Hidalgo and
Hausmann [2009] introduced new metric on economic complexity, with describing
that complex economies are those involved in complex activities, while complex
activities are those performed by complex economies [Hidalgo, 2021]. By using
the method of reflection, Hidalgo and Hausmann [2009] derived the complexity
indexes that the economic complexity index (ECI: the complexity of embedded
production capability of a place), which is normalized average of the product
complexity, and that the product complexity index (PCI: the complexity of
required production capability for a product), which is normalized average of
the economic complexity. Following Hidalgo and Hausmann [2009], we calculate
ECI of a region and PCI of a product as shown in Equation 4 and 5.

$\displaystyle\begin{split}ECI_{c}&=K_{c}=\frac{1}{M_{c}}\sum_{p}M_{cp}K_{p}\end{split}$
(4)
$\displaystyle\begin{split}PCI_{p}&=K_{p}=\frac{1}{M_{p}}\sum_{c}M_{cp}K_{c}\end{split}$
(5)

where $M_{cp}$ is a binary matrix consisted of 0 or 1, representing subscript
$c$ is cluster of small businesses, while $p$ is product. To determine an
element of the matrix, we calculate Revealed Comparative Advantage of a
product $p$ in the shops’ cluster $c$
$(RCA=\frac{shops_{cp}/\sum_{p}shops_{cp}}{\sum_{c}shops_{cp}/\sum_{cp}shops_{cp}})$
by looking at the number of shops in a cluster, $shops_{cp}$ [Balassa, 1965].
When $M_{cp}$ equals to 1, we regard a cluster c has the comparative advantage
in product $p$.444 We normalize the complexity indexes, between 0 and 1 for
the convenience of interpreting results:
$ECI_{c}=\frac{K_{c}-min(K_{c})}{max(K_{c})}$,
$PCI_{p}=\frac{K_{p}-min(K_{p})}{max(K_{p})}$

## Acknowledgement

This work was supported by the National Research Foundation of Korea (NRF)
Grant through the Korea Government [Ministry of Science and ICT (MSIT)] under
Grant NRF-2022R1A5A7033499. We also thank the support from Inha University.

## Data Availability Statement

The small business firm data is provided by Korean Small Enterprise and Market
Service (KSEMS). The labor and residential population data is provided by
Korea National Statistical Office (KOSTAT). The original small business firms
and labor and residential population data can be found in the OPENapi portal:
https://www.data.go.kr/ and https://sgis.kostat.go.kr/.

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