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ISSN : 1598-7248 (Print)
ISSN : 2234-6473 (Online)
Industrial Engineering & Management Systems Vol.18 No.3 pp.474-481
DOI : https://doi.org/10.7232/iems.2019.18.3.474

# Application of Modularity-Maximizing Graph Communities in Intersectoral Linkages: Case Study in Emerging markets in the Agricultural Sector

Faculty of Agriculture, Department of Agricultural Social Economics , Universitas Padjadjaran, Bandung, Indonesia
Corresponding Author, E-mail: y.deliana@gmail.com
June 3, 2019 June 16, 2019 June 24, 2019

## ABSTRACT

The network literature has shown that the great interest is to identify communities. There are many approaches for finding such partitions and one of them is by using modularity-maximizing graph. In this paper, graph application is used to determine cross-sector communities in the economy, especially related with agricultural sector. From I/O analysis, we derive intersectoral linkages and determine the leading sectors in the economy. Then, we use a particular natural approach called Leuvain modularity maximization to compute the modularity clustering across sectors. This paper reveals conditions for or properties of the maximum modularity of an intersectoral network. The number of clusters vary when the intersectoral network is changed. This paper shows that this approach creates interesting community structures for agricultural sector. Finally, we highlight the performance and quality of our approach versus standard I/O analysis.

## 1. INTRODUCTION

System of interacting entities can be described using the notion of graphs. Sector in the economy can be defined as an entity in the network. Networks in graph theory consists of nodes or vertices and edges between nodes (Ali and Haseeb (2019). There are many studies use network analysis to describe and identify any interaction, relationship and patterns from entities (Newma et al., 2006;Scott, 2000;Wasserman and Faust, 1994;Okoronkwo et al., 2014;Guimera and Amaral, 2005;Sharif and Butt, 2017). Another application of network analysis is for economy (Udoudo et al., 2016;Khan et al., 2017;Kareem et al., 2017;Blondel et al., 2008;Newman and Girvan, 2004;Ahmadi et al., 2018). For example, intersectoral relationship in the economy can be described using sector network. It can explain dependency in economic activity, credit performance, growth across sectors. Use the latest farm equipment and technology, including global positioning systems (GPS), geographic information systems (GIS), and unmanned aerial vehicles in crop production and land management.

To understand how the network is working, it is use-ful to identify communities. Communities can be defined as unusually densely connected sets of nodes (Khan et al., 2017;Gumel, 2017). It can describe groups of nodes that are closely related one to another that have community structure. As an example, sectors in economy can be groups according to their close relationship. Agriculture sector can have close relationship with manufacture sector with respect to economic activity. While in other aspect, agriculture sector can also have close relationship with service sector with respect to credit performance. If we can identify communities, with respect to one aspect, we can study the communities individually. There exist different properties in different communities. Economic activity and credit performance are some examples of properties. If we use global analysis, without considering the present of communities, then the result will be inap-propriate. But if we use detailed analysis of individual communities, it will bring us to more accurate results.

There are many studies proposed different approach to find partitions of a network, like spectral properties, flows, edge agglomeration, etc. (Kareem et al., 2017). The approaches differ in whether the number of communities or their size is pre-specified by the researcher or by the algorithm. Suppose we want to find the communities from sectors in the economy, with respect to their economic activity. We can determine using classification by primary and non-primary sectors. But, we can also use the algorithm to find the communities. The results will be based on the structural features of the network itself. It will find interesting to see how the algorithm create different communities that might differ from our thought. Today, there is another approach to measure the quality of a network partitioning into communities, called Leuvain modularity (Blondel et al., 2008). The modularity of a given clustering is the number of edges inside clusters, minus the expected number of such edge if the graph were random conditioned on its degree distribution (Newman and Girvan, 2004;Farzadnia et al., 2017). Many studies have shown modularity-maximizing clus-tering often identify interesting community structure in real network (Udoudo et al., 2016;Jenaabadi and Issazadegan, 2014;Osabohien et al., 2018;Newman, 2004;Clauset et al., 2004;Clauset, 2005;Bokhtiar et al., 2018, Muhammad, 2018). Using this algorithm, we can identify communities that agriculture sector belongs to. It might be interesting to find out pattern using Leuvain modularity.

Intersectoral analysis used Input-Output Model (I/O). The I/O table describes economic linkages among production sectors. I/O models provide a framework to examine a sector’s linkages with other sectors as well as its economy wide impact. We can use I/O model to analyze the input-output activities across sectors in the economy. As an example, in agriculture sector we can identify sectors that related to its input process, such as chemical industry, and what related to its output process, such as food processing industry. There are many sectors that can be related with agriculture sector, both in input and output process. Decision-making in agriculture management often relies on knowledge about the economic linkages and impacts of agriculture sector on the overall economy. Thus, quantitative assessment of agriculture sector’s economic linkage is crucial for policymakers to access the sector’s importance. The impacts of agriculture development are not contained within the sector but are transmitted to other sectors in the economy. This study aims to understand the sector communities in economy using modularity-maximizing graph (Torquato et al., 2018;Mendoza and Mendoza, 2018;Saurykova et al., 2018).

## 2. DATA AND METHODS

In this study, we use I/O table of Indonesia 2010. The I/O table developed by Statistics Office (BPS), in every five years. In the table, there are three assumption that are considered, i.e. homogeneity, proportionality, and additivity. Homogeneity means that each sector only produces single product or service (uniform). Proportionality means relationship between input and output in every sector is linear function. Additivity means effect total from production activity are summation of all effect of each activity. According to the I/O analysis, the production of a particular sector in the economy has two kinds of economic effects (Miller and Blair, 1985). As an example, if agriculture sector increases its output, there will be increasing demand on the sectors that produce the inputs that agriculture sector requires. This kind of linkage called backward linkage. This linkage determines the relationship on input process. On the other hand, increasing the output of agriculture sector will imply increasing supplies to be used as inputs. This kind of linkage is termed forward linkage. This linkage determines the relationship on output process.

An I/O model of a particular national economy can be represented in the following way

$X = ( I − A ) − 1 f$
(1)

where X is an n element sector output vector, f is an n element sector final demand vector, A is an n x n input coefficient matrix and I is an n dimensional identity matrix. Hirschman (1958) developed the idea of intersectoral linkages in this I/O framework. In this study, we will use Leontief supply-driven multiplier as a backward-linkage measure and forward-linkage measure. To measure backward linkage of each sector, this index can be used

(2)

where a(j) is the backward linkage index of sector-j. b(i,j) is the element of the inverse of Leontief matrix, n is the number of sectors. To measure forward linkage of each sector, this index can be used

(3)

where a(i) is the forward linkage index of sector-i. b(i,j) is the element of the inverse of Leontief matrix, n is the number of sectors.

In this study, we aggregate I/O table into 15 sectors in the economy as described in Table 1. These are the sectors that will be analyzed using intersectoral linkage. Agriculture sector as sector 1 originally come from sector 1 until 36. Using these 15 sectors in I/O table, we calculate the backward and forward linkage sector for each sector.

Using these indexes, we measure the total linkage which define as the total of backward and forward linkage. It measures the magnitude of the economic impact that can be caused by a sector. High index in total linkage indicates that sector is the key sector in the economy. Any change in key sector their production process can have high influence on other sectors that related to input or output process. In this study, we will concentrate to agriculture sector. We analyze how this sector has influenced to the economy.

Using, inverse Leontief matrix that we have in I/O analysis, it will be taken as an information about intersectoral relationship across sectors, described as the network. It is given as a directed graph G = (V;E). The adjacency matrix of G is denoted by A = (au,v), where au,v = av,u=1 if u and v share an edge, and au,v = av,u= 0 otherwise. The degree of a node v is denoted by dv. A clustering C={C1, …, Ck} is a partition of V into disjoint sets Ci. We use f(v) to denote the unique index of the cluster that node v belongs to. The modularity (Osabohien et al., 2018) of a clustering C is the total number of edges inside clusters, minus the expected number of such edges if the graph were a uniformly random multigraph subject to its degree sequence. To be able to compare the modularity for graphs of different sizes, it is convenient to normalize this difference by a factor of 1/2m, so that the modularity is a number from the interval [-1, 1]. If nodes u and v have degree du and dv, then any one of the m edges has probability 2du/2m´dv/2m of connecting u and v. By linearity of expectation, the expected number of edges between u and v is then du.dv/2m. Thus, the modularity of a clustering C is

$Q ( C ) = 1 / 2 m × s u m { u , v , ( a u , v − d u d v / 2 m ) × d [ f ( u ) , f ( v ) ] }$
(4)

where d denotes the Kronecker Delta, which is 1 if its arguments are identical, and 0 otherwise. Using Louvain approach, this paper tries to compute the modularity clustering across sectors and analyze the communities with agriculture sector.

## 3. RESULTS

Using 15 sectors in I/O table, we can calculate the inverse Leontief matrix to know the contribution of each sector in the economy. It uses the I/O table from the first quadrant for a 15-sector economy in Indonesia. Table 2 shows the input-output transactions among the 15 different production sectors where are registered in the uses of output (sales) in rows 1-15 and production costs in columns 1-15. Column or row 1 represent the agriculture sector. Using this I/O table, we can know how one sector correlated to other sectors in input or output process. For agriculture sector, five sectors outside it that used this sector as their highest production input are (4) manufacture, (1) agriculture, (9) hotel & restaurant, (6) construction and (5) health service sector. These sectors are related to agriculture sector since they use it for their input process. In the other side, agriculture sector has use input from other sectors. These are the five sectors that are used by agriculture sector the most: (4) manu-facture, (1) agriculture, (6) construction, (10) finance and (11) real estate and firm service sector. Agriculture sector in this I/O table consists of sectors from sector 1 to sector 36, from crops to forestry. This information explains these sectors seemly unrelated sector to/from agriculture sector. Thus, using nominal transaction, we find sectors that related most in agriculture sector, using intersection from input and output process in ascending order: (4) manufacture, (1) agriculture itself, and (9) hotel & restaurant sector.

Using this 15x15 I/O intersectoral matrix, we calcu-late the inverse Leontief matrix and find the linkage index for each sector. The linkage can be explained from two perspectives, forward and backward linkage. Table 3 shows the classification of the backward and forward linkages’ results. Strong backward sector is one that has higher index in backward linkage. Sector with relative backward linkages greater than the economy-wide average of the corresponding backward linkages of all sectors. Strong forward sector is one that has higher index in forward linkage. Sector with relative forward linkages greater than the corresponding economy-wide average of the forward linkages of all sectors. Potentially key sectors are those that are strong-ly connected to other industries both along their input demand and output supply chains, and thus have both forward and backward linkages greater than one.

Using forward and backward linkage index, we identify the sector position in intersectoral linkage as shown in Figure 1 in blue dot. Agriculture sector has high forward linkage but has low in backward linkage. It has larger total forward dependence on all sectors than the total forward dependence of all industries on this sector. Economic activity in agriculture sector have impact to other sectors that depend their input from this sector. Compare to agriculture sector, there are more sectors that are in weak linkage sector quadrant. There are also sectors that have higher forward linkage index than agriculture sector, i.e. (4) manufacture, (8) hotel & restaurant, and (5) utility sector. In the other hand, since agriculture sector is primary sector, it has low linkage in backward linkage. The same reason fishery sector. Sectors such as (7) trade, (5) utility, (8) hotel & res-taurant, and (6) construction are ones that have strong backward linkage compare to other sectors. These sectors have higher total backward dependence on all sectors than the total backward dependence on all sectors.

Next, there are two sectors that have both forward and backward linkage index higher than one i.e. hotel & restaurant and utility sector. Yet they have small magni-tude compare to others that have higher position in the map. It will be hard to ignore the influence of other sec-tors to the economy, even when they only have one strong linkage index. Therefore, we impose another measurement to identify key sectors. Using total linkage index, we can identify sectors that have high impact to the economy. These sectors have strong intersectoral linkage to other sectors. From Figure 2 in orange dot, we can know sectors that have the highest total linkage index. The top five sectors that are indicated as key sectors are (4) manufacture, (7) trade, (5) utility, (8) hotel & restaurant, and (9) construction sector. These key sectors indicate to be the determinant factor of multiplier output in the Indonesia economy. Manufacture and trade lie above any other sectors in the economy. Any change in these sectors can affect to the whole economy. In addition to these sectors, agriculture sector shows its influence in the economy through two sectors i.e. (4) manufacture and (9) hotel & restaurant sector.

Next, we use modularity-maximization graph to identify sector communities in the economy. Using metrics available in network application, we can have inter-sectoral linkage as shown in Figure 2. There are different label size, ranking size, and partition color across sector. In this graph, label size indicates the node degree. Here for each node the following measures were calculated: degree, in-degree and out-degree. The node degree is the number of relations (edges) of the nodes. We do not distinguish between in-degree (number of incoming neighbors) and out-degree (number of outgoing neighbors) of a vertex. For the figure, sectors that have more relations to other sectors in the economy. Sectors such as (2) fishery, (3) mining sector have weak relations to other sectors in the economy. In quite relevant to the fact that these sectors are primary sectors in the economy. There are not much I/O transaction between these sectors with other sector that does not use their output for its input process. Agriculture sector has more relationship to other sectors in the economy compare to them. This show how im-portant this sector can contribute to the output multiplier (Ali and Haseeb, 2019;Haseeb et al., 2018;Haseeb et al., 2019;Shabbir et al., 2018;Suryanto et al., 2018).

The ranking size for this graph is based on between central. Betweenness centrality is even more important statistics property of a network. This can be applied in intersectoral linkage in the economy. The between centrality of a node is an indicator of its centrality or im-portance in the network. It is described as the number of shortest parts from all the vertices to all the other vertices in the network that pass through the node in consideration (Brandes, 2001). From Figure 2, we can identify that bubble size for agriculture sector is as big as manufacture sector. The bigger the bubble size the more important this sector as the central of the network. Manufacture sector is in the central of the network. Sectors that are closed to this sector are considered to be as important in the network. Sector such as (1) agriculture, (2) mining, (3) fishery, (6) construction, (8) hotel & restaurants, (10) finance are ones that are closely to the central. These sectors are important to the economy since they located near the manufacture sector.

The partition in this graph is based on modularity-maximization. This community detection algorithm created a modularity class value for each node. For Figure 2, we can identify that there are two communities were found. The first community with green color node consists of (1) agriculture, (2) fishery, (4) manufacture, and (5) utility. Another community is in orange color code that consists of (3) mining, (6) construction, (7) trade, (8) hotel & restaurant, (9) transportation, storage and communication, (10) finance, (11) real estate & firm service, (12) government service, (13) education service, (14) health service, and (15) others sector. The community are created according to the Leuvain method. This result shows that agriculture sector bound in one community with fishery, manufacture and utility sectors. These sectors are have more densely connected from one to another in the economy compare to other sectors outside their community. It also reveals that agriculture sector has more connection to manufacture in input-output transaction. Knowing that, we need to consider agriculture sector as an important sector that drive the economy to higher growth. For Indonesia, agriculture and fishery sectors are ones that can give multiplier effect for manufacture sector. Our manufacture sectors have strong backbone with primary sectors, and these sectors are ones that will give more impact to the economy compare to mining sector. As we know, mining sector is in primary sector, but they are not in the same cluster as agriculture nor fishery sector. In addition, we also find out that the number of clusters vary when the intersectoral network is changed. Figure 2 only shows the intersectoral network that is corresponding to their input-output transaction.

## 4. CONCLUSION

Intersectoral linkage in the economy can be described using network analysis. To understand how this network is working, this study tries to identify communities that present among sectors using information on how closely one node to another that have community structure. The communities that we want to identify as based on the structural of the network itself. This study used Leuvain modularity to identify the network partition. Using I/O table of Indonesia 2010, we calculate forward and backward linkage indexes from 15 sectors of economy. From here, we identify sectors that have potentially to be key sectors in the economy. Using total linkages indexes, this study found that agriculture sector is not included in the key sectors. However, we identify how this sector can affect to the economy.

This study reveals that agriculture sector is close to the central node. This sector is as important as other sectors in the economy that are located near the manufacture sector in the network map. Sector such as manufacture, mining, fishery, construction, hotel & restaurant and finance are ones that are closely to the central. Using modularity metric, we can reveal that agriculture sector bound in one community with fishery, manufacture and utility sector. These sectors have more densely connected from one to another in the economy compare to other sectors outside their community. From this study, we need to consider agriculture sector as an important sector that drive the economy to higher growth. For Indonesia, agriculture and fishery sectors are ones that can give multiplier effect for manufacture sector.

## ACKNOWLEDGEMENT

I would like thank to Dr. Pantea Foroudi, Department Marketing Branding & Tourism, Middlesex University, The Burroughs, London UK for review and offered helpful suggestions

## Figure

Sectoral mapping of forward and backward linkage.

## Table

Aggregation of 15 sector in I/O table

Input-Output transaction among 15 sector economy in Indonesia 2010 (Million Rp.)

Classification of backward and forward linkages

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