# Spatial Constraints on Economic Interactions: A Complexity Approach to the Japanese Inter-Firm Trade Network

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Scaling Relations between Interacting Firms

#### 2.2. Geometric Proximity between Interacting Firms: Industry Sectors and Prefectures

#### 2.3. Location Dependency of Trade Distance Distribution: Prefectures and Economic Regions

#### 2.4. Prefectures and Communities: A Mutual Information Approach

## 3. Results

#### 3.1. Scaling Relations between Interacting Firms

#### 3.2. Geometric Proximity between Interacting Firms: Industry Sectors and Prefectures

#### 3.3. Location Dependency of Trade Distance Distribution: Prefectures and Economic Regions

#### 3.4. Prefectures and Communities: A Mutual Information Approach

## 4. Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

SMI | Structural Mutual Information |

I | Mutual Information |

TDB | Teikoku Databank Ltd. |

## Appendix A. Data Source Context and Description

## Appendix B. Scaling Relations between Interacting Firms

#### Appendix B.1. Randomised Network

**Figure A1.**The frequency of trade and distance D in a randomised network as a function of the annual sales of customers ${S}_{c}$ and annual sales of suppliers ${S}_{s}$. (

**A**) is a heatmap showing the frequency of trade among the ${S}_{c}$ and ${S}_{s}$ pairings, where the dotted diagonal line is the symmetry ${S}_{c}$ = ${S}_{s}$. (

**B**) is the equivalent heatmap quantifying the median $m\left(D\right)$ of the ${S}_{c}$ and ${S}_{s}$ pairing, where the horizontal and vertical lines indicate the original break in symmetry for the actual data, ${S}_{c}\simeq {S}_{s}$, which is not found in the randomised network. (

**C**) shows the scaling of the median distance $m\left(D\right)$ of firms at different ${S}_{s}$ levels (above and below the actual data symmetry break) with fixed ${S}_{c}$, and (

**D**) is an equivalent plot with fixed ${S}_{s}$ and variable ${S}_{c}$ levels. Each black circle and red triangle represent firms where $S<{10}^{5}$ and $S\ge {10}^{5}$, respectively. Figures are plotted on a log–log scale for the year 2021.

#### Appendix B.2. Actual Data for 2010

**Figure A2.**The frequency of trade and distance D highly depend on the annual sales of customers ${S}_{c}$ and annual sales of suppliers ${S}_{s}$ for the year 2010, similarly to 2021. (

**A**) is a heatmap showing the frequency of trade among the ${S}_{c}$ and ${S}_{s}$ pairings, where the dotted diagonal line is the symmetry ${S}_{c}$ = ${S}_{s}$. (

**B**) is the equivalent heatmap quantifying the the median $m\left(D\right)$ of the ${S}_{c}$ and ${S}_{s}$ pairing, where the horizontal and vertical lines indicate the breakage of the ${S}_{c}\simeq {S}_{s}$ symmetry. (

**C**) shows the scaling of the median distance $m\left(D\right)$ of firms at different ${S}_{s}$ levels (above and below the symmetry break) with fixed ${S}_{c}$, and (

**D**) is an equivalent plot with fixed ${S}_{s}$ and variable ${S}_{c}$ levels. Each black circle and red triangle represent firms where $S<{10}^{5}$ and $S\ge {10}^{5}$, respectively. Figures are plotted on a log–log scale for the year 2010.

## Appendix C. Correlation between the Exponents of Sizes of Economic Zones, ⟨R⟩, and the Decay γ in the Normalised Probability Distributions P_{t}/P_{r}

**Figure A3.**Correlation between the sizes of economic zones $\u27e8R\u27e9$ and the power law exponent $\gamma $. The figure was produced by binning the x-axis into a fixed number of datapoints (5) and calculating the averages of $\u27e8R\u27e9$ and $\gamma $ within each bin. The dotted line represents the best fit line, excluding the outlier point around (1.5, −1.12).

## Appendix D. Distribution of Companies across Prefectures in Japan

**Figure A4.**Distribution of companies by prefecture in Japan, 2021. (

**A**) shows the relative distribution of companies in Japan, coloured using natural numbers. In contrast, (

**B**) shows the same distribution but colour-coded in the $lo{g}_{10}$ scale.

**Figure A5.**Distribution of companies by prefecture in Japan, 2021, split by the annual sales, S. (

**A**) shows the distribution of companies in Japan on a $lo{g}_{10}$ scale for companies with annual sales below ${10}_{5}$ yen, whereas (

**B**) is the equivalent map for the companies with annual sales above ${10}_{5}$ yen.

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**Figure 1.**The frequency of trade and distance D highly depends on the annual sales of customers ${S}_{c}$ and annual sales of suppliers ${S}_{s}$. (

**A**) is a heatmap showing the frequency trades among the ${S}_{c}$ and ${S}_{s}$ pairings, where the dotted diagonal line is the symmetry ${S}_{c}$ = ${S}_{s}$. (

**B**) is the equivalent heatmap quantifying the median $m\left(D\right)$ of the ${S}_{c}$ and ${S}_{s}$ pairing, where the horizontal and vertical lines indicate the breakage of the ${S}_{c}\simeq {S}_{s}$ symmetry. (

**C**) shows the scaling of the median distance $m\left(D\right)$ of firms at different ${S}_{s}$ levels (above and below the symmetry break) with fixed ${S}_{c}$, and (

**D**) is an equivalent plot with fixed ${S}_{s}$ and variable ${S}_{c}$ levels. Each black circle and red triangle represent firms where $S<{10}^{5}$ and $S\ge {10}^{5}$, respectively. Figures are plotted on a log–log scale for 2021.

**Figure 2.**Splitting of customers by the mean trade distance $\overline{D}$ for larger and smaller suppliers and normalised Mantel tests ${Z}_{n}$. (

**A**) provides a breakdown of the percentages of companies where suppliers that are smaller in relative terms (${S}_{s}<{S}_{c}$) have lower average trade distances than larger ones (${S}_{s}>{S}_{c}$), i.e., $\overline{D}({S}_{s}<{S}_{c})<\overline{D}({S}_{s}>{S}_{c})$ (dark cyan). When the opposite occurs, $\overline{D}({S}_{s}<{S}_{c})<\overline{D}({S}_{s}>{S}_{c})$, dark magenta is used for representation. (

**B**) shows the values for the normalised Mantel test (${Z}_{n}$) for the whole population (black at bottom left) as well as for each subset of the four areas resulting from the combination of areas above and below the symmetry break from Figure 1.

**Figure 3.**Trading proximity decreases with the years in all sectors. Industry sector analysis for geometric proximity between interacting firms, clustered by prefecture for 2000, 2010 and 2021. Each plot shows a histogram, where each prefecture is an item, for different bands $\mu \left(D\right)/\mu \left({D}_{random}\right)$ as calculated by the mean trade distance within the real network $\mu \left(D\right)$ and the equivalent values in the randomised network $\mu \left({D}_{random}\right)$. This is done for (

**A**) all industries and the following economic sectors: (

**B**) construction industry, (

**C**) manufacturing industry, (

**D**) wholesales and retailing industries, (

**E**) transport and communications industries and (

**F**) services industry. Magenta, cyan and navy blue bars in each plot refer to the years 2000, 2020 and 2021, respectively. Moreover, the overall mean ratio for each year is shown within the insets for each plot.

**Figure 4.**Trade distances generally follow a truncated power law at the prefecture level (generated by QGIS 3.6 software [18]). Normalised probability distributions of trade distance for each selected prefecture in 2000 and 2021, on a log–log-scale plot. Each curve of filled magenta squares, empty magenta squares, filled dark cyan circles and empty dark cyan circles shows the distribution below and above the radius for each prefecture in 2021 and 2000, respectively. The dotted lines indicate power law distributions. Each panel corresponds to the distribution generated by firms located in (

**A**) Japan and (

**B**) Hokkaido, (

**C**) Miyagi, (

**D**) Tokyo, (

**E**) Aichi, (

**F**) Osaka, (

**G**) Kyoto, (

**H**) Fukuoka and (

**I**) Okinawa prefectures, respectively.

**Figure 5.**Firms of different sizes closely resemble each other at longer distances, but have very distinctive patterns at shorter distances. Normalised probability distributions of trade distance for selected prefectures with fixed annual sales of customers (million yen) in 2021, on a log–log-scale plot. Each curve of filled shapes shows the distribution below the radius $D={10}^{\u2329R\u232a}$ km for each prefecture in 2021, whereas non-filled shapes are datapoints above the radius. The dotted lines indicate power law distributions. Each panel contains the normalised probability distributions ${P}_{t}/{P}_{r}$ for (

**A**) Japan as a whole, (

**C**) Miyagi, (

**D**) Tokyo and (

**H**) Fukuoka. Each curve of black circles and red triangles shows the normalised probability of existence of a trade within distance D in 2021, categorised by the amount of annual sales of companies as $S<{10}^{5}$ and $S\ge {10}^{5}$, respectively.

**Figure 6.**Pointwise contributions to the mutual information and structural clustering of prefectures (generated by QGIS 3.6 software [18]). The map on the left is coloured in accordance with the eight regions that emerge from the clustering method based on the pointwise positive contribution to the mutual information. The regions are aggregated into three groupings: the communitarian clusters (

**E**), corresponding to Kyushu (lime green), Chugoku (gold), Shikoku (saddle brown), Tohoku (dodger blue) and East Chubu (dim gray); the core clusters (

**D**) of the Tokyo sphere (navy blue) and Osaka sphere (dark orange); and the midway clusters (

**F**) (dark magenta). The panels in the bottom row show the normalised probability distributions ${P}_{t}/{P}_{r}$ with distance (similarly to Figure 5), where panel (

**G**) is a schematic trend-line representation of the three groupings. Panels (

**A**), (

**B**) and (

**C**) present the heatmaps of the pointwise contributions to the mutual information (calculated in accordance with Equation (14)) for the real world inter-firm trade network, the randomised network and the clustered regions, respectively.

**Figure 7.**Comparative structural clustering of prefectures by distinct methods (generated by QGIS 3.6 software [18]). The three maps at the top are coloured in accordance with the three highest-order clusters that emerge from each clustering method: the economic complexity (

**A**), the method applied in this research (i.e., the mutual information) (

**B**) and the officially defined administrative regions (

**C**). The colour coding is related to the clusters identified by the legends within the panels below. The panels at the bottom show the normalised probability distributions ${P}_{t}/{P}_{r}$ with distance (similarly to Figure 5).

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## Share and Cite

**MDPI and ACS Style**

Viegas, E.; Levy, O.; Havlin, S.; Takayasu, H.; Takayasu, M.
Spatial Constraints on Economic Interactions: A Complexity Approach to the Japanese Inter-Firm Trade Network. *Mathematics* **2024**, *12*, 1244.
https://doi.org/10.3390/math12081244

**AMA Style**

Viegas E, Levy O, Havlin S, Takayasu H, Takayasu M.
Spatial Constraints on Economic Interactions: A Complexity Approach to the Japanese Inter-Firm Trade Network. *Mathematics*. 2024; 12(8):1244.
https://doi.org/10.3390/math12081244

**Chicago/Turabian Style**

Viegas, Eduardo, Orr Levy, Shlomo Havlin, Hideki Takayasu, and Misako Takayasu.
2024. "Spatial Constraints on Economic Interactions: A Complexity Approach to the Japanese Inter-Firm Trade Network" *Mathematics* 12, no. 8: 1244.
https://doi.org/10.3390/math12081244